Optimal Taxation and Social Networks Marcelo Arbex∗ and Dennis O’Dea† November 1, 2011

Abstract We study optimal taxation when jobs are found through a social network. This network determines employment, which workers may influence by engaging in social activities. We find that the optimal capital tax rate is zero, independent of labor market frictions. The optimal labor tax varies negatively with effective labor and network parameters play an important role in determining the optimal tax. When agents cannot engage in social activities, taxes are lower with more connected networks. When agents engage in social activities, and can affect both the intensive and extensive margin, labor taxes vary positively with employment and welfare is higher.

Keywords: Optimal Taxation, Social Networks, Labor Markets. JEL Classification: D85, E62, H21, J64.



Department of Economics, University of Windsor, 401 Sunset Ave., Windsor, ON, N9B 3P4, Canada. Email: [email protected]; † Department of Economics, University of Illinois at Urbana-Champaign. 484 Wohlers Hall, 1206 South Sixth St., Champaign, IL, 61820, US. Email: [email protected].

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1

Introduction

The importance of social networks in labor markets has long been understood. Networking plays a critical role in job searching and in improving the quality of the match between firms and workers. Access to information about job opportunities is influenced by social structure and individuals use connections with others (e.g., relatives, friends, acquaintances) to build and maintain information networks.1 Empirical research indicates that about half of jobs are obtained through networking and the other half are obtained through more formal methods (see Holzer, 1988; Montgomery, 1991; Topa, 2001, Gregg and Wadsworth, 1996; Addison and Portugal, 2001).2 The job network literature has shown that social networks have important implications for the dynamics of employment, as well as, the duration and persistence of unemployment (Calv´o-Armengol and Jackson, 2004). To the extent that networks can affect economic outcomes, the relevance of social networks for the design of government policies must be recognized and explored. The literature on optimal income taxation, however, has neglected the role of social networks in the labor market and has mainly focused on competitive or search labor markets. Well-known results in the theory of optimal labor taxation are that tax rates on labor should be roughly constant (Barro, 1979; Kyndland and Prescott, 1980; Chari and Kehoe, 1999), and labor taxes vary positively with employment (Zhu, 1992; Scott, 2007). Jones, Manuelli and Rossi (1997) show that optimal capital and labor income taxes are zero in the long-run, in an economy where labor services are a combination of human capital and worker’s time. Regarding the optimal capital tax rate, the Chamley (1986) - Judd (1985) result of optimal zero capital tax is, in general, verified. In this paper we examine if these results survive when the labor market is governed by job networks and how different network structures affect the structure of optimal taxation. In this paper, we study optimal tax policy in a model economy where the informational structure of the job market follows the classic epidemic diffusion model, surveyed in Vega1

See Granovetter (1995) and Ioannides and Loury (2004) for a recent survey. For instance, according to the National Longitudinal Survey of Youth (NLSY), more than 85% of workers use informal contacts when searching for a job (Holzer, 1988) and more than 50% of all workers found their job through their social network according to data from the Panel Study of Income Dynamics (Corcoran, Datcher and Duncan, 1980). On the firm side, between 37% and 53% of employers use the social networks of their current employees to advertise jobs according to data from the National Organizations Survey (Mardsen, 2001) and the Employment Opportunity Pilot Project (EOPP), respectively. According to the EOPP, 36% of firms filled their last opening through a referral (Holzer, 1987). 2

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Redondo (2007). Information about job opportunities arrives randomly; if an agent is unemployed, she will take the job. On the other hand, if the agent is already employed then she may pass job information along to a friend, relative or acquaintance who is unemployed. Each agent is connected to others through a social network. The strength of social ties among workers determines the probability their peers pass job information along. Unemployment results when individuals are unsuccessful in hearing about job opportunities either directly or through their peers in a network, or when jobs randomly break-up. Our model of the labor market is analytically simple and allows us to calculate well the longrun, average behavior of arbitrary networks, including power-law distributions and networks with the “small-worlds” properties of low diameter and high clustering. We consider several different classes of networks, and characterize their effect on optimal government policies. We do not consider endogenous network formation in this paper; workers invest in the strength of their ties to their peers, but they are endowed with these peers exogenously. The large scale structure of the network is fixed, and workers must choose to strengthen their social ties as well as to allocate time to labor. We consider the effect differences in the structure of the network have on optimal policy by comparing different complex networks, but do not directly endogenize network structure with a “network-formation” model. What is endogenous is the strength of social connections, rather than the pattern of social connections. In effect, while workers cannot directly choose who they do and do not know, they can choose how to exploit what connections they do have. This paper embeds the job network model into the general equilibrium framework and the design of optimal tax policy follows the Ramsey approach (Lucas and Stokey, 1983; Chari, Christiano and Kehoe, 1991). We consider an economy with a representative infinitely lived household. Each household consists of a continuum of family members, who either work or are unemployed. Employed workers receive a wage that is determined competitively, while agents without a job receive an unemployment benefit. Unemployed workers do not search for a job but rather learn about job opportunities through peers in their social network. Our model includes both intensive and extensive margins: employed family members decide how much they work (intensive), while family members without a job can engage in social activities that develop their social connections, increasing the strength of their ties to their peers and,

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ultimately, influencing their chances of finding a job (extensive). We allow time devoted to social networking to affect the job transmission rate between agents. That is, the rate at which job information is passed from employed workers to his unemployed peers in any period depends on how much effort agents spent on social networking in the previous period. This “network effort” intensity represents an additional trade-off for the agents: it improves their chances of becoming employed but at a cost of leisure. Once a worker finds a job, she is beyond the social network dynamics and no effort is devoted to improving her social contacts. We consider two distinct environments regarding the extensive margin: (i) agents cannot affect their participation rate (exogenous) and (ii) agents can influence it by devoting time to social networking (endogenous). The introduction of labor market frictions through job networks implies that the optimal tax policy should feature some response to unemployment and, consequently, to the dynamics of labor networks. The economy’s equilibrium unemployment rate is determined by the structure and properties of the job network; the steady state unemployment rate is decreasing in the job arrival probability, the job information transmission probability, and is increasing in the job break up probability. We show that the optimal capital tax is zero and the government relies on labor taxation to finance its expenditures and to pay unemployment benefits. Regarding the labor income tax, the distinction between intensity versus participation in the labor market is key for the distinction between the exogenous and the endogenous network process. When the labor market is completely exogenous, participation in the labor market is determined entirely by the network, where links and structure are exogenously given to the workers. Employed workers can, however, decide the intensity of their participation to maximize the family’s welfare. This is reached by having those with jobs working more, since it is uncertain that unemployed family members will find a job. The fact that the employment rate is higher in the exogenous case relaxes the government budget constraint in the sense that it can tax those working less, because the economy has higher, fixed, participation rate. In the endogenous case, finding a job is not a completely random process as agents can influence the rate at which they learn about job opportunities. In this case family members can affect both margins. That is, agents can decide how much to work (intensive margin) as well as the extent to which they participate in the labor market (extensive margin). By allocating

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time to improve their social contacts, an unemployed worker increases her chances of becoming employed. Employed family members tend to work less and the participation rate is lower, which requires the government to tax those with jobs more. Taxes are higher when the network process is endogenous, but welfare is higher as well. In this case, the higher welfare comes from the fact that family members are enjoying more leisure. This exercise allows us to compute a measure of the value of the networks, in terms of welfare. Although our results are obtained in the context of one particular model, we believe that the economic forces captured by our model are likely to apply much more broadly. The reason for this is that our model captures a very simple economic reality: in the presence of social networks, the intensive and extensive margins are substitutes. That is, effective labor - the combination of both intensive and extensive effects - is the key for the determination of optimal labor income tax. In other words, labor is a combination of employment, a stock, and worker’s time, a flow. The government cannot impose separate tax rates on these two inputs. The effective labor can be interpreted as the tax base and the optimal labor tax is lower the higher effective labor is. In the exogenous case, participation in the labor market is completely determined by the dynamics of the network. The extensive-margin effect dominates the intensive-margin effect and taxes are the lowest in the regular network and the highest in the empty network. The optimal income tax is higher in more connected job network economies because labor is more inelastically supplied when employment is high. Since the Ramsey planner is required to tax inelastic variables more heavily to minimize tax distortions, labor income tax rates vary positively with hours worked (Zhu, 1992; Scott, 2007). Contrary to the exogenous case, when agents are allowed to invest part of their time endowment in social networking, labor taxes decreases when labor supply increases (intensive margin) and increases along the extensive margin, i.e., labor tax is higher the higher the economy’s employment rate. The important element here is the ability agents have to influence their participation in the labor market and manipulate the extensive margin as well as the intensive margin. Our approach differs from traditional search models of labor markets, and previous attempts to introduce networks into labor markets, in important ways. In traditional models of search, the key friction is in the rate at which workers and firms find each other, and this can be

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directly influenced by workers and firms making costly investments in search. In our approach, workers can only indirectly influence the rate at which they find each other; much of the labor market is exogenous. Workers are endowed with peers in a complex social network and agents can make an effort to strengthen their social ties and increase their chances of learning of job opportunities. Calv´o-Armengol and Zenou (2005) extend the Calv´o-Armengol and Jackson (2004) model to provide a micro-foundation of the matching function. They allow job search by workers and firms, and explicitly model job information transmission between workers. Their approach differs from the present paper in that they do not consider complex network or workers who are heterogeneous in the number of peers they have, nor do they allow workers to invest in the strength of their social ties. The key network parameter for them is the number of links a worker has; essentially, they restrict their attention to regular networks. We show that with endogenous link strength, complex networks are very different than regular networks. This highlights the role of social activities, rather than search, in the labor market. Recent papers have shown that the introduction of search frictions changes the ChamleyJudd result of non-zero capital taxation, e.g. Shi and Wen (1999). Domeij (2005) finds that, if the government is constrained to use capital and labor taxation only, the capital tax will be non-zero as long as the worker’s bargaining power is different from the elasticity of search in the matching function. A non-zero capital tax is used to correct for distortions to labor market tightness, reducing frictions between workers and firms. In our model, there are no frictions between workers and firms; frictions arise solely from information transmission among workers, and firms have no active role in labor market search and cannot affect the employment rate.3 We show that regardless of the structure of the social network and the dynamics of the labor market, the optimal limiting capital tax rate is zero (Chamley, 1986; Judd, 1985). Capital taxation plays no role in reducing labor distortions and a zero tax on capital stimulates investment, raising output and consumption for all households in the long run. Regarding the optimal labor income tax, our results are not directly comparable to results obtained in the search environment. The structure of the labor market and its frictions are very different, as describe above, and we have to be cautious not to draw inappropriate comparisons. 3

Firms could have a more active role in the labor market, for instance, receiving information on the productivity of job applicants through current employees. Krauth (2004) analyzes a model economy in which networking arises because firms have limited information on the skill of job applicants. A firm’s current employees provide information on the job-specific skill of their friends, thus improving the likelihood of a productive match. See also Bramoull´e and Saint-Paul (2010).

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The paper proceeds as follows. In section 2, we characterize labor market dynamics governed by social networks and exogenous job separation. We discuss how the unemployment rate is affected by social activities and study the features of long run employment in networks. Section 3 presents our benchmark economy where the network process is exogenous, meaning that agents take as given both the structure of the network and the strength of their social ties and so cannot influence their chances of hearing about jobs. In Section 4 we study an environment where unemployed family members spend time to increase the strength of their ties to their peers and, ultimately, affect the rate at which job information is transmitted to them. We derive the optimal labor tax and show how job networking can affect the optimal labor tax through the employment rate. Section 5 presents a numerical exercise and Section 6 offers concluding comments.

2

Network Structure and Employment Rate

2.1

Demography and Network Structure

There is a continuum of infinitely lived agents whose total measure is normalized to one. The economy is populated by households who consume, save and work. An agent can be either employed or unemployed. Time evolves in discrete periods indexed by t and information about job opportunities arrives randomly and the job arrival process is independent across agents. All jobs are identical and employed workers receive a wage that is determined competitively, while agents without a job receive an unemployment benefit. The labor market is mediated by social networks and exogenous social ties between workers facilitate the transmission of job opportunities information. That is, unemployed workers learn about job opportunities through peers in their social network. The informational structure of the job networking follows the classic epidemic diffusion model, surveyed in Vega-Redondo (2007). The flow of agents between employment and unemployment status depends on a worker’s social network contacts and on an exogenous job separation rate. Each agent hears about a job opening with probability γ ∈ [0, 1]. If the agent is unemployed, she will take the job. On the other hand, if the agent is already employed then she may pass the information along to a friend, relative or acquaintance who is unemployed. The rate at

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which an employed worker passes information to each of her unemployed peers is given by v ∈ [0, 1]. The rate v is distinct from γ, and need not be directly derived from it. Let ρ be the exogenous job break up probability, which is independent across agents. Each agent may have peers to whom she passes information when employed, and from whom she may receive information when unemployed. These peers are connected to one another in a social network. A network is described by a symmetric matrix M , where mij ∈ {0, 1} denotes whether a link exists between agents i and j. That is, mij = 1 indicates that i and j know each other and mij = 0 otherwise. We assume that mij = mji , meaning that the relationship between i and j is reciprocal. The structure of this network m will determine how information flows throughout the network, and will have a large impact on each agent’s employment status. We are concerned with large networks, that is, a network among the continuum of agents, so that there are infinitely many nodes in this network. A key property of a network is its degree distribution {Dz }∞ z=1 , where Dz is the proportion of agents who have z peers. A network’s degree distribution summarizes much of its structure: whether there a some workers with many links, or not, and the relative prevalence of highly connected workers.4 In general, there may be many networks m consistent with a particular degree distribution {Dz }∞ z=1 . As networks grow large, much local information ceases to matter, so focussing on degree distributions is appropriate (Vega-Redondo 2007). In other words, we are not concerned about particular network structures but rather focus on large classes of networks sharing the same degree distribution. We may think of the actual network m as being a random draw from the set of networks having degree distribution {Dz }∞ z=1 . This is called the random network approach. In particular, we study the empty, regular, power-law and geometric degree distributions. The employment rate may be different for agents with different number of links (peers) z. The average employment rate can then be expressed as follows: ∫



nt =

(nzt Dz ) dz, z=1

where nzt is the employment rate at time t among agents with z links. Agents who have more links may expect to hear about jobs from their peers more often, and their employment status 4

Some important network properties, however, are not captured by the degree distribution, such as detailed local structures and clustering. For example, if workers who have a common peer are also likely to be connected themselves, this fact will not be captured by the degree distribution.

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will evolved differently than that of an unemployed agent with fewer links. To analyze the dynamics of employment, we apply the mean field approach, which assumes there are no correlations or neighborhood effects in information transmission. Our approach amounts to assuming the average state of the network is replicated locally, for every agent, so that the proportion of an agent’s peers who are unemployed is a common function of the economy-wide unemployment rate (Vega-Redondo 2007).5 The mean field approach relies on an assumption of homogenous mixing, i.e., there are no systemic differences between each worker’s local neighborhoods. This could be justified by imagining that a worker with z links does not have the same peers period after period, but continually draws new peers, randomly from the network. In that case, because the network is large, he could not infer anything about their employment status beyond the average in the network, and the mean-field approach is correct. Even without that formal assumption, the mean field approach has been shown in simulations to give good answers for the long-run dynamics in the networks we will consider (Vega-Redondo 2007, Jackson 2008). Following this approach, and suppressing the subscript t when there is no confusion, we can determine the law of motion for employed workers as follows:

n˙z = −ρnz + (1 − nz )[γ + (1 − γ)(1 − (1 − vθ)z )]

(1)

where employment is given by total labor force, normalized to one, minus the number of unemployed workers, i.e., nt = 1 − ut . The change in the level of employment has three main components. First, ρ percent of agents who are employed will lose their jobs. Second, a fraction γ of the unemployed agents will hear of a job themselves. Third, of those unemployed workers who do not hear of a job opportunity themselves, each of their z peers is employed with probability θ, and passes job information at rate v; the probability that at least one of their z peers passes them information is (1 − (1 − vθ)z ). We will consider v exogenous for the moment, but will later require workers to invest time and effort into maintaining their social relationships, so that v is a function of effort, et .6 5

This may not necessarily true, in general. Calvo-Armengol and Jackson (2004) showed that each worker’s employment status is correlated with that of his peers, so an agent who remembers his past status could infer the expected employment rates of his peers, and this need not be equal to the average state of the network. 6 Unlike the model of Calvo-Armengol and Jackson (2004), the rate at which job information arrives from a peer is independent of the rate at which job information arrives to the others connections that peer may have;

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The proportion of an unemployed worker’s peers who are employed (θ) will also depend on the employment status of the network as a whole. According to the mean field approach, we can define this probability in the following way: ∫



θ=

(nz ψz ) dz,

(2)

z=1

where ψz is the probability an agent’s peer has z links, which is given by ∫



ψz = z=1

where ⟨z⟩ =

∫∞ z=1

(

zDz ∫∞ (zDz ) dz z=1

)





dz = z=1

(

zDz ⟨z⟩

) dz,

(zDz ) dz is the average degree in the network. Note that ψz ̸= Dz , i.e., the

probability your peers have z links is not equal to the proportion of the population that has z links. This is because agents with many peers, and a large z, are disproportionately likely to be your peers. Plugging ψz into the definition of θ, equation (2), we have 1 θ= ⟨z⟩





(znz Dz ) dz. z=1

This implies that the probability an agent’s peers are employed (θ) depends on the average degree in the network ⟨z⟩, the number of links each of these peers have, z, the proportion of agents who have z peers (Dz ) and the employment rate among agents with z links (nz ). Hence, in this economy the employment rate nt follows a stochastic process and it is a function of the state of the network S, represented by the break-up probability (ρ), the job arrival probability (γ), the job transmission rate (v) and the degree distribution (Dz ).

2.2

Long Run Employment in Networks

Assume that the economy converges to a steady state. This implies that the change in the level of employment for each type of worker is equal to zero, i.e., n˙z = 0 for all z. The number of newly employed agents of each type z is exactly equal to the number of newly unemployed agents, and the economy will remain at this level of employment indefinitely. We consider this long run prevalence of employment to be the economy’s employment rate (Vega-Redondo in a large complex network, workers are not in competition with one another for job information.

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2007). Setting n˙z = 0 in equation (1), we find that the steady state level of employment (n∗z ) satisfies

( n∗z

=1−

ρ 1 + ρ − (1 − γ)(1 − θ∗ v)z

) (3)

where θ∗ is given by equation (2). Together, the solution to these two equations for each z gives n∗z , which is averaged over z to get the employment rate n∗ : ∗





n =

(n∗z Dz ) dz.

(4)

z=1

For different degree distributions Dz , the long run steady state employment rate, equation (4), may have different solutions, with different characteristics and implications for optimal taxation. We focus on several well known classes of large, complex networks. As a baseline, we consider regular networks where every agent has the same number of peers, k. For these networks, Dz = 1 for z = k, and Dz = 0 for all other z, and every worker is exactly the same. For k = 0, this is the empty network, and may be taken as a worst case scenario, where each worker must hear of a job themselves, at the exogenous arrival rate γ. Regular networks are not realistic because they exhibit no heterogeneity among workers and no large scale structure. We consider two alternative models of large networks with heterogeneous workers: power-law and geometric degree distributions. Many models of social networks are described as deriving from linear growth in the number of agents in a society and preferential attachment in link formation as these agents arrive. In these models, we imagine the network growing over time. Workers arrive and choose to form some number of links to the workers already present in the network, with a preference for having links to workers with many links already. This preference is easy to justify, as well connected peers are more likely to be employed themselves, and thus prove to be a valuable source of job information. The limit of this process, as the number of workers goes to ∞, results in a power law degree distribution of the following form: Dz = (a − 1)z −a . In this kind of network, a few workers end up with many, many links, while most have relatively few. These networks have a number of attractive features, that match well many empirical social networks (Vega-Redondo 2007, Jackson 2008). Geometric networks are derived from a similar growth process, but where agents do not

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have a preference for links to agents with many links already. In this model, links are simply formed randomly among those agents already present, and in the limit, the degree distribution has the following form: Dz = ν 1−z log ν. These networks have a thinner tail than power-law networks, which exhibits a thick tail, with a relatively high proportion of agents who have many links. Let nqz , θq and nq denote the long run employment rate among agents with z links, the probability an unemployed worker’s peers are employed and the economy’s employment rate for a network q, respectively. Consider three classes of networks: Regular (R), Power-Law (P L) and Geometric (G). The employment rate of agents with z links in a network q = R, P L and G is given by

( n∗q z

=1−

ρ 1 + ρ − (1 − γ)(1 − θq v)z

) (5)

Table 1 presents the (steady state) expressions for θq and nq for each network considered here. Table 1 - Probability an unemployed worker’s peers are employed and the economy’s employment rate for a network

Regular P ower − Law Geometric

θq

nq

1 knR k k

nR k (e) ∫∞ (

= nR k ( ) ∫ ∞ 1 PL −a zn (a − 1)z ) dz z ⟨z⟩ z=1 ( ) ∫ ∞ 1 1−z znG log ν dz zν ⟨z⟩ z=1

) PL −a n (e)(a − 1)z dz z z=1 ) ∫∞ ( G nz (e)ν 1−z log ν) dz z=1

The economy’s employment rate for a regular network nR is equal to the employment rate R of the agents with k links, i.e., nR = nR k , where nk is the solution of the following expression:

( n∗R k

=1−

ρ k 1 + ρ − (1 − γ)(1 − nR k v)

) (6)

For the empty network, k = 0 and this expression simplifies to nE k=0 = γ/(ρ+γ). Unfortunately, for the power-law and geometric networks, no closed form solution to this system of equations (nqk , θq ) exists, and it must be characterized numerically. For each of these possible networks, the behavior of unemployment with respect to the job information process is straightforward. The unemployment rate is decreasing in the job opportunities arrival probability (γ) and there is a positive relationship between the exogenous job break up probability (ρ) and the equilibrium unemployment rate. The equilibrium employment 12

rate also depends on the average number of links ⟨z⟩ of agents in the network. In networks with a high ⟨z⟩, there are more agents with a higher number of peers, who have a higher (individual) employment rate. Employment is therefore higher for more connected networks.

2.3

Network structure and the employment rate

Before turning to the structure of the economy, it is instructive to consider how properties of the network labor market change with respect to the relationship between network parameters and the determination of the economy’s employment rate. For a baseline set of parameters (γ = 0.4, ρ = 0.15, λ = 0.05), the employment rate is increasing in the job opportunities arrival probability and decreasing in the the exogenous job break up probability (Figure 1 and 2). We observe a much lower employment rate in a empty network than in the power-law and geometric. Employment rate is the highest in the regular networks. In the case of power law and geometric networks, where there is heterogeneity in the number of links workers have, the equilibrium unemployment rate is decreasing in the number of links z. For these networks, because of the presence of workers with many many links, job information is disseminated more easily, which reduces unemployment. However, in these network the distribution of links is very heterogenous - we still can find people with few links. In a regular network everyone has the same number of links and this homogeneity leads to higher steady state employment rates. [ABOUT HERE] Figure 1 - Job Arrival Probability and Employment Rate [ABOUT HERE] Figure 2 - Break-up Probability and Employment Rate

2.4

Endogenous Network Process

We allow agents to invest time and effort in improving their social ties. Family members without a job can spend time in social activities, which develops their social connections, increasing the strength of their ties to their peers. A stronger connection to their peers results in more job information from their employed peers, and will improve their chances of finding a job while unemployed. The structure of the network is still exogenous and we assume that 13

the time devoted to social networking affects only the job transmission rate v. That is, the rate at which job information is passed from employed workers to his unemployed peers in any period depends on how much effort agents spent on social networking in the previous period. In allowing for social activities, we introduce an additional trade-off for the agents. Agents can improve their chances to become employed in the future, but at a cost in terms of current leisure. We denote this environment an endogenous network process.7 That is, the rate at which job information is passed from employed workers to his unemployed peers in time t depends on how much effort (et−1 ) agents spent on social activities in the previous period, i.e. v = v(et−1 ). The job transmission rate is determined according to the following decreasing returns to scale relationship technology:

v(et−1 ) = e1−λ t−1 , where λ measures the efficacy of this technology. When λ is close to 1, workers are able to build strong relationships with relatively little cost, in terms of time and foregone leisure. When λ is close to 0, maintaining social relationships is more difficult, and requires a greater investment of time. Once a worker finds a job, she is beyond the social network dynamics and no effort is devoted to improve social contacts. Viewing v as a function of the investment in social ties implies that the entire long run level of employment in the economy is also a function of it, i.e., nt = n(et−1 ). Figure 3 shows that, for each of these networks, the steady state employment rate is increasing in agent’s networking effort. Assuming that agents in these networks have the same average number links ⟨z⟩ = 5, employment is highest in the regular network. [ABOUT HERE] Figure 3 - Effort and Employment Rate The key differences between the networks manifest in the model as differences in n(e) and n′ (e). While the steady state employment rate is higher in the regular than in the geometric 7

While other means by which workers may affect their social network can be imagined, we believe this captures the role networks play in job matching; network formation models capture more the role networks play as an investment in social capital, which is beyond the scope of this paper.

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and power-law networks, for very low levels of effort, n′ (e) is larger in the geometric and powerlaw networks than in the regular network, so that effort is more effective in these networks for low levels of effort (see Figure A.1, Appendix). The equilibrium employment rate also depends on the average number of links ⟨z⟩ of agents in the network. In networks with a high ⟨z⟩, there are more agents with a large number of peers, who have a higher (individual) employment rate. For any given level of networking effort, economy’s employment rate will be higher in these denser networks. Figure 4 illustrate this result for the regular endogenous network case. Similar results are also observed for other networks. [ABOUT HERE] Figure 4 - Average Number of Peers and Employment Rate

3

A Benchmark Model: Exogenous Network Process

3.1

The Economy

In a typical household there is a measure nt of employed family members and a measure 1 − nt of unemployed individuals. Employed members supply labor hours lt and unemployed workers do not search for a job but rather learn about job opportunities through peers in their social network. We denote an exogenous network process as an environment where agents take as given both the structure of the network and the strength of their social ties. That is, an agent cannot influence his/her chances of hearing about jobs. This assumption will be changed in the next section. Preferences are represented by the following utility function

U=

∞ ∑

β t u(ct , ht )

(7)

t=0

where the instantaneous utility function u is increasing, concave and differentiable and β is the discount rate which lies in (0, 1). The variable ct is family consumption and the time endowment is normalized to 1 so that leisure is ht = 1 − nt lt . The timing of the model is as follows. At the beginning of each period, employed family 15

members - those that started the period with a job and those that just heard about and got a job - choose l. Then goods consumption c is determined. Households have two options with output they do not consume: they can invest in capital (k) or purchase government bonds (B). Next, employed family members are paid a wage (w) for the labor services, the unemployed receives unemployment benefits (b) and the family receives (tax free) interest (R) earnings on bonds and rental rate (r) of capital. The household takes as given government determined tax rates on labor (τ l ) and capital (τ k ) income. As in Calv´o-Armengol and Jackson (2004), we interpret the timing as one where job break-up occurs, essentially, at the beginning of the period. The sequence of real budget constraints reads as follows [ ] ct + kt+1 + Bt+1 = nt (1 − τtl )wt lt + (1 − nt )bt + 1 + (1 − τtk ) (rt − δ)) kt + Bt Rt

(8)

where δ is the rate at which capital depreciates each period, rt is the real rate of return on capital and





nt =

(nzt Dz ) dz, z=1

where nzt is the employment rate among agents with z links in period t. The total household income is divided evenly among all individuals, so that family members perfectly insure each other against variation in labor income. Or, alternatively, we can assume that agents can insure themselves against earnings and unemployment uncertainty and, for this reason, wage earnings are interpreted as net of insurance costs (Merz, 1995; Andolfatto, 1996; Faia, 2008). Employed and unemployed family members consume the same amount and capital allocation and bonds purchase is a family decision. Firms produce a single good and maximize profit taking factor prices as given. Production technology is a constant returns Cobb-Douglas specification so that output (y) is

yt = F (kt , lt ) = (kt )α (nt lt )1−α

(9)

where α ∈ (0, 1) is the capital income share. Firms operate under perfect competition and earn zero profits in equilibrium. Factors of production are paid their marginal products, i.e. Fk (t) = rt = α (kt )α−1 (nt lt )1−α and Fl (t) = wt = (1 − α) (kt )α (nt lt )−α nt , where Fk (t) and 16

Fl (t) denote the marginal product of capital and labor, respectively, and wt the wage rate for labor. Differently than models of search that allow firms to influence the number of workers to be hired through vacancies posted (Domeij, 2005), here the number of workers employed nt is entirely defined by the dynamics of the network. The government faces the budget constraint

gt + (1 − nt )bt = nt τtl wt lt + τtk (rt − δ)kt + Bt+1 − Bt Rt .

(10)

where gt denotes government consumption, which is assumed to be exogenously specified. The government finances its expenditures by levying taxes on labor and capital and issuing government bonds. The economy as a whole faces the following aggregate resource constraint

ct + kt+1 + gt = F (kt , nt lt ) + (1 − δ)kt .

3.2

(11)

A Network Competitive Equilibrium

A representative household, taking prices, taxes and the social network structure as given, chooses {ct , kt+1 , lt , Bt+1 } to solve

max

∞ ∑

{ct ,kt+1 ,lt ,Bt+1 }

β t u(ct , ht )

(P.1)

t=0

subject to (8), (1) and k0 , B0 , and n0 given. Let u(t) = u(ct , ht ) and likewise for ui (t), where i = 1 for consumption and i = 2 for leisure. The necessary conditions for an interior solution of the family’s maximization problem are given by

u2 (t) = u1 (t)(1 − τtl )wt Rt = [1 + (1 − τtk )(rt − δ)]

(12) (13)

Equation (12) is the standard equation showing how the income labor tax affects the laborleisure choice and equation (13) is the no-arbitrage condition for capital and bonds. Definition 1 A network competitive equilibrium is a policy Υ = {τtl , τtk }∞ t=0 , government 17

¯ = {gt , bt }∞ , household’s allocations x = {ct , kt+1 , lt , Bt+1 }∞ , a price system spending G t=0 t=0 P¯ = {wt , rt , Rt }∞ t=0 and the state of the network variables {ρ, γ, v, Dz } such that given the policy, government spending, the price system and the state of the network, the resulting household’s allocation choice maximizes the consumer’s utility and satisfies the government’s budget constraint, the economy’s resource constraint and market clearing conditions.

3.3

Ramsey Equilibrium

At the beginning of each period, the government announces its program of tax rates and individuals behave competitively.8 The objective of the social planner is to choose values of its fiscal instruments such that the agent’s utility is maximized. The Ramsey problem is a programming problem of finding the optimum within a set of allocations that can be implemented as a competitive equilibrium with distortionary taxes. In other words, the Ramsey problem is to choose a process for tax rates {τtl , τtk }, which maximizes social welfare and satisfies (8) and an implementability constraint (see Chari and Kehoe, 1999). In this paper the unemployment benefit is exogenously given and the planner does not choose it optimally. To derive the implementability constraint, we use family’s first order conditions and the intertemporal budget constraint, which yields the following expression: ∞ ∑

β t (u1 (t) [ct − (1 − nt )bt ] − u2 (t)nt lt ) = W0

(14)

t=0

where W0 = u1 (0) (T0 k0 + R0 B0 ). Definition 2 A Ramsey equilibrium in this economy is a policy Υ, an allocation rule x and a price rule P¯ that satisfy the following two conditions: (i) the policy Υ maximizes (7) subject to the government budget constraint (10) and the state of the network {ρ, γ, v, Dz } with allocations and prices given by x and P¯ and (ii) for every Υ′ , the allocation x(Υ′ ), the price rule P¯ (Υ′ ) and the policy Υ′ constitute a network competitive equilibrium. Proposition 1 In a network competitive equilibrium, the household’s allocations and the date 0 policy Υ0 satisfy the economy’s resource constraint (11), the law of motion for employed 8

We follow the majority of the literature in assuming that the government can commit to follow a longterm program for taxing labor income. We assume that there are institutions that effectively solve the time inconsistency problem so that the government can commit to the tax plan it announces in the initial period.

18

workers (1) and the implementability constraint (14). Furthermore, given household’s choices and Υ0 , prices and policies can be constructed for all dates, which together with the choices and date 0 policies constitute a network competitive equilibrium. The solution to the Ramsey problem is an allocation that maximizes social welfare, subject to the restriction that it can be decentralized as a competitive equilibrium with taxes. The proof of Proposition 1 follows directly from Chari and Kehoe (1999) and is omitted. It has two parts: (i) allocations in a competitive equilibrium must satisfy the implementability constraint (14) and the resource constraint (11) and, conversely, (ii) any allocation satisfying (14) and (11) can be decentralized as a competitive equilibrium. In this environment, these two constraints fully describe the set of competitive allocations that can be attained through feasible government policies and an allocation from the centralized problem can be implemented as a competitive equilibrium. The planner’s maximization problem can thus be written as follows:

max

∞ ∑

{ct ,kt+1 ,lt }

β t u(ct , 1 − nt lt )

(P.2)

t=0

subject to (14), (11), equation (1) and g¯, τ0l , τ0k , k0 , n0 given. Notice that the implementability constraint, equation (14), is similar to the objective function in the sense that they are both discounted infinite sums of terms. Thus, given the Lagrange multiplier, η, - which, of course, is endogenous - it is possible to rewrite (P.2) as

max

{ct ,kt+1 ,lt }

∞ ∑

β t W (ct , nt , lt ; η) − W0

(P.3)

t=0

subject to the ”flow” constraints from (P.2) where

W (ct , nt , lt ; η) ≡ u(ct , 1 − nt lt ) + η (u1 (t) [ct − (1 − nt )bt ] − u2 (t)nt lt ) .

3.4

Optimal Tax Rates

We first show that when the government only has access to capital and labor income taxes and the network structure and information transmission is taken as given by the workers, the

19

optimal limiting capital tax rate is zero and the optimal labor income tax is positive. We assume constant government spending over time, i.e. gt = g, ∀t ≥ 0. The first order conditions with respect to c, k and l for problem (P.3) evaluated at the steady state are, respectively Wc∗ = ϕ

(15)

1 = β [1 + (Fk∗ − δ)] Wl∗ = ϕFl∗

(16) (17)

where ϕ is the Lagrange multiplier on the resource constraint (11). The optimal limiting capital tax rate is zero as in Chamley (1986) and Judd (1985), a result that can be verified by comparing the first order condition of the family’s problem with respect [ ] to capital, evaluated at the steady state, i.e. 1 = β 1 + (1 − τ ∗k )(Fk∗ − δ) and equation (16). These two conditions directly imply that τ ∗k = 0. A comparison of the first order conditions of the family problem and the Ramsey problem with respect to consumption and labor reveals that the optimal labor income tax is given by

τ ∗l = 1 −

Wc∗ /u∗1 Wl∗ /u∗2

(18)

where Wc∗ ≡ u∗1 + η ∗ [u∗1 + u∗11 C ∗ − u∗21 n∗ l∗ ], Wl∗ ≡ u∗2 + η ∗ [u∗2 + u∗12 C ∗ − u∗22 n∗ l∗ ] and C ∗ = c∗ − (1 − n∗ )b and where n∗ is the measure of employed family members in a steady state, defined as in (4). Notice that (18) is not an explicit expression for the optimal tax rate, since the Wc∗ , Wl∗ depend on endogenous variables. Since n∗ is determined by the dynamics of the social network, the optimal labor tax will also display a response to changes in the network process. The following proposition summarizes the above results. Proposition 2 If the solution to the Ramsey problem converges to a steady state, the labor market is governed by social networks and the network dynamics are exogenous, then in a steady state, the tax rate on capital is zero, i.e., τ ∗k = 0, and the optimal labor income tax is given by (18). In order to derive expressions for optimal labor taxes we need to make an assumption regarding preferences. The case of additively separable preferences is of particular interest 20

because it allows us to solve for the optimal labor tax analytically. We assume the instantaneous utility function is of the form U (ct , ht ) = ln ct + κ ln ht . When the utility function is additively separable (u12 (t) = u21 (t) = 0) in leisure, it implies that leisure is a normal good and labor is taxed. Evaluating (18) for this functional form we obtain:

τ ∗l = 1 −

1 + η ∗ (1 − n∗ ) (b/c∗ ) 1 + η ∗ (1 − n∗ l∗ )−1

(19)

The effect of employment, which is completely exogenous at this point, on the optimal labor tax will also depend on how labor supply l∗ and consumption c∗ react to changes in the employment rate n∗ . Note that the family’s problem first order conditions with respect to consumption and leisure form a system of equations such that c and h can be solved in terms of φ (the Lagrangian multiplier on the household’s budget constraint) and ω, where ω = (1−τ l )w. For our purpose we are interested in the compensated labor supply response with respect to a change in ω holding φ constant. From this procedure we get ∂ht φt u11 (t) = <0 ∂ωt u22 (t)u11 (t) − u212 (t)

(20)

This expression represents the compensated labor-supply response when the tax rate changes (the substitution effect). This substitution effect captures the distortionary effect of the laborincome tax. That is, a higher labor tax increases leisure and lowers labor supply (1 − ht ) and thus lowers the tax base. For our additively separable utility function, the compensated elasticity of labor supply is given by ϵ = (∂h/∂ω) /(ω/h) = −(1 − τ l )w(1 − nl)ϕ−1 φ. There are two effects of the employment rate on the elasticity of labor supply. The employment rate impacts this elasticity directly and implies that when employment is high, labor is more inelastically supplied. On the other hand, a high employment indirectly decreases labor supplied, indicating that labor is more elastic. These two effects are associated with the two - intensive and extensive - margins in the model. If the net effect of a high employment rate on the elasticity of labor supply is positive, labor is more inelastically supplied when unemployment is low. In this case, the Ramsey planner taxes inelastic variables more heavily to minimize tax distortions and labor income tax rates vary positively with employment (Zhu, 1992; Scott, 2007). On the other

21

hand, if the net effect is negative, labor income taxes vary negatively with employment.

4

Optimal Taxes and Endogenous Network Process

In this section, we extend the benchmark economy described in Section 3: agents are allow to invest time to improve their social ties. Family members without a job can spend time in social activities, which develops their social connections, increasing the strength of their ties to their peers. We denote this environment an endogenous network process.9 Incorporating this change, we write the family’s problem as follows

max

∞ ∑

{ct ,kt+1 ,lt ,et ,Bt+1 }

β t u(ct , ht )

(P.4)

t=0

subject to (1) ct + kt+1 + Bt+1 = n(et−1 )(1 − τtl )wt lt + (1 − n(et−1 ))bt + Tt kt + Bt Rt for all t ≥ 0, (2) nz (et ) = (1 − ρ)nz (et−1 ) + (1 − nz (et−1 ))[γ + (1 − γ)(1 − (1 − v(et−1 )θ)z )] for all t ≥ 0, (3) ht = 1 − n(et−1 )lt − (1 − n(et−1 ))et for all t ≥ 0, (4) k0 , B0 , and n0 given.

The equilibrium conditions for this family’s problem are represented by [ ( ) ] l (1 − n(et−1 ))u2 (t) = βn′ (et−1 ) u1 (t + 1) (1 − τt+1 )wt+1 lt+1 − bt+1 − u2 (t + 1)(lt+1 − et+1 ) (21) in addition to equations (12) and (13). Equation (21) states that the utility cost of network effort (LHS) equals the discounted (expected) gain from successfully finding a job, where the gain of one additional worker equals the additional consumption gain in period t + 1 less the leisure cost of working and not spending time in social networking. Definition 3 A network competitive equilibrium, in the presence of an endogenous network ∞ ¯ process, is a policy Υ = {τtl , τtk }∞ t=0 , government spending G = {gt , bt }t=0 , household’s allo9

While other means by which workers may affect their social network can be imagined, we believe this captures the role networks play in job matching; network formation models capture more the role networks play as an investment in social capital, which is beyond the scope of this paper.

22

∞ ˆ cations xˆ = {ct , kt+1 , lt , et , Bt+1 }∞ t=0 , a price system P = {wt , rt , Rt }t=0 and the state of the

network variables {ρ, γ, v, Dz } such that given the policy, government spending, the price system and the state of the network, the resulting household’s allocation choice maximizes the consumer’s utility and satisfies the government’s budget constraint, the economy’s resource constraint and market clearing conditions. For completeness, we present the government budget constraint and resource constraint for this economy, respectively

gt + (1 − n(et−1 ))bt = n(et−1 )τtl wt lt + τtk (rt − δ)kt + Bt+1 − Bt Rt , ct + kt+1 + gt = (kt )α (n(et−1 )lt )1−α + (1 − δ)kt .

4.1

(22) (23)

Ramsey Equilibrium

Definition 4 A Ramsey equilibrium for an economy where the network process is endogenous is a policy z, an allocation rule xˆ and a price rule Pˆ that satisfy the following two conditions: (i) the policy z maximizes (7) subject to the government budget constraint (22) and the state of the network {ρ, γ, v, Dz } with allocations and prices given by xˆ and Pˆ and (ii) for every z′ , the allocation xˆ(z′ ), the price rule Pˆ (z′ ) and the policy z′ constitute a network competitive equilibrium. The fact that agents can invest time in social networking leads to two important changes for the formulation of the Ramsey problem with respect to (i) the implementability constraint and (ii) the intertemporal choice of labor supply. The key element here is the relationship between effort in the current period and payment of taxes in the future, in the case an agent successfully finds a job. Again, we use family’s first order conditions and the intertemporal budget constraint problem (P.4) - to derive the following the implementability constraint (see Appendix A.1 for derivation details): ∞ ∑ t=0

( [ ]) n(et ) β u1 (t) (ct − bt ) − u2 (t) n(et−1 )(lt − et ) + (1 − n(et−1 )) ′ = Z0 n (et ) t

] [ where Z0 = u1 (0) n(e−1 )(1 − τ0l )w0 l0 − n(e−1 )b0 + T0 k0 + R0 B0 − u2 (0)n(e−1 )(l0 − e0 ). 23

(24)

The Ramsey problem for this economy can be described as maximizing the welfare of the representative family given feasibility, the government’s budget constraint and the first order conditions from both the household’s and the firm’s maximization problems as well as the household’s budget constraint. The basic idea here is that the first order conditions should be used as defining prices and tax rates given allocations. Hence, these conditions, along with prices and taxes as choice variables, need not be explicitly included in the planner’s problem. We first assume that taxes at time zero (τ0l , τ0k ) are given. Next, the equivalent of (12) and (13) for problem (P.4) can be used to compute τtl and τtk , respectively. This process leaves (21) as a condition that must be imposed. Notice that we restrict the tax code to impose a tax on the output of the effective labor activity (τtl ) and this tax affects both the static choice of labor supply (lt ) and the dynamic choice of networking effort (et ). It is necessary to guarantee that, given an allocation, the tax τtl from (12) and (21) coincide (Reinhorn, 2009; Domeij, 2005; Jones, Manuelli and Rossi, 1997). Imposing this equality is equivalent to requiring Φ(t) = 0, where Φ(t) = Φ(c, l, e, b) = (1 − n(et−1 )) u2 (t) − βn′ (et ) (u2 (t + 1)et+1 − u1 (t + 1)bt+1 )

(25)

Proposition 3 The household’s allocations and the date 0 policy Υ0 , in a network competitive equilibrium satisfy the economy’s resource constraint (23), the law of motion for employed workers (1), the implementability constraint (24) and a constraint on labor income taxes, equation (25). Furthermore, given household’s choices and Υ0 , prices and policies can be constructed for all dates, which together with the choices and date 0 policies constitute a network competitive equilibrium. Proof. See Appendix A.2. The planner’s maximization problem can thus be written as follows:

max

{ct ,kt+1 ,lt ,et }

∞ ∑

β t u(ct , 1 − n(et−1 )lt − (1 − n(et−1 ))et )

(P.5)

t=0

subject to (24), (25), (23), (1) and g¯, τ0l , τ0k , k0 , n0 given. The first order conditions of the planner’s problem are not the same in the first period and subsequent periods which implies that this Ramsey problem is non-stationary. Since our goal is to study this economy in the 24

steady state, we will focus our attention on the first-order conditions for period 1 and onwards. To save space, the first-order conditions for period zero are not presented. Introduce the auxiliary function for this problem

Z(ct , lt , et , nt ; ηˆ) ≡ u(ct , 1 − n(et−1 )lt − (1 − n(et−1 ))et ) ]) ( [ n(et ) . +ˆ η u1 (t) (ct − bt ) − u2 (t) n(et−1 )(lt − et ) + (1 − n(et−1 )) ′ n (et ) where ηˆ is the Lagrange multiplier on (24). Evaluating the first order conditions of problem (P.5) at the steady state, and after some manipulation, we obtain 0 = Zc∗ + µ∗ (1 − n(e∗ ))u∗21 − κ ∗

(26)

0 = −1 + β [Fk (k ∗ , n∗ l∗ ) + (1 − δ)]

(27)

0 = Zl∗ + µ∗ (1 − n(e∗ ))u∗22 − κ ∗ Fl (k ∗ , n∗ l∗ ) { [ ]} 0 = Ze∗ + µ∗ u∗22 (1 − n(e∗ ))2 − βn′ (e∗ )(l∗ − e∗ ) (1 − n′ (e∗ )e∗ − n(e∗ ))

(28) (29)

+ βµ∗ {u∗2 [n′′ (e∗ )e∗ + n′ (e∗ )] − u∗1 n′′ (e∗ )b + u∗12 n′ (e∗ )(l∗ − e∗ )b} − βκ ∗ Fl (k ∗ , n∗ l∗ )n′ (e∗ )l∗ where µ, κ are the Lagrange multipliers on the condition (25) and resource constraint (23), respectively and Zc∗ , Zl∗ and Ze∗ are defined as follows: ]) ( [ n(e∗ ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ = (1 + η + ηˆ u11 (c − b) − u21 n(e )(l − e ) + (1 − n(e )) ′ ∗ n (e ) ( [ ]) n(e∗ ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ Zl = (1 + η ) u2 + ηˆ u12 (c − b) − u22 n(e )(l − e ) + (1 − n(e )) ′ ∗ n (e ) Zc∗



) u∗1



Ze∗ = (1 + η ∗ ) u∗2 [(1 − n(e∗ )) + βn′ (e∗ )(l∗ − e∗ )] ( ) n(e∗ )n′′ (e∗ ) ∗ ∗ ∗ ∗ − ηˆ u2 (1 + β)n(e ) + (1 − n(e )) [n′ (e∗ )]2 ( ( [ )]) n(e∗ ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ′ ∗ ∗ ∗ ∗ + ηˆ [(1 − n(e )) + βn (e )(l − e )] u12 (c − b) − u22 n(e )(l − e ) + (1 − n(e )) ′ ∗ n (e ) Proposition 4 If the solution to the Ramsey problem converges to a steady state, the labor market is governed by social network and the network dynamics are endogenous, then in steady state, the tax rates ares as follows: τ ∗k = 0,

25

τ ∗l = 1 − [(Zc∗ /u∗1 ) / (Zl∗ /u∗2 )]. Proof. See Appendix A.3.

5

Numerical Results

5.1

Parameterization and Solution Method

We now use numerical methods to simulate calibrated versions of the model we have been examining. We use these results to illustrate our main points and to explore further the relationship between network structures and labor taxation. The model is calibrated such that the steady state is consistent with United States data. We assume the instantaneous utility function U (ct , ht ) = ln ct + κ ln ht and set the weight on leisure, κ, to match initial steady-state labor force participation n∗ l∗ − (1 − n∗ )e∗ = 0.68. We set the discount factor β = 0.96 which implies a rate of time preference of 4 percent on an annual basis. Production technology is a constant returns Cobb-Douglas specification of the form F (kt , lt ) = (kt )α (nt lt )1−α and we set α = 0.30 and a depreciation rate of 0.04. Regarding parameters related to the network structure, there are no available estimates in the literature. Our strategy is to choose the job arrival and the job break up probability such that the steady state employment rate in the empty network (our baseline), i.e. nR k=0 = γ/(ρ + γ), is consistent with the annual employment rate in the United States. The exogenous job separation probability ρ is set to 0.15 (Andolfatto, 1996) and the probability for a worker of finding a job γ is set equal to 0.4. The rate at which an employed worker passes information to each of her unemployed peers v plays an important role in our analysis. For the endogenous network process, we assume v(e) = e1−λ , where λ is initially set to 0.05. We set v = 0.0858 for the exogenous case to match effort e = 0.0754 in the endogenous regular network to allow comparisons between these two network processes. For what follows the average number of peers in each network is the same. We consider two models of large complex networks with heterogeneous workers and set a = 2.25 and ν = 1.284 for the power-law and geometric degree distributions, respectively. These numbers are calibrated such that all three networks regular, power-law and geometric - have the same average number of peers, i.e., ⟨z⟩ = 5. We investigate the sensitivity of the results by considering a range of alternative values for the 26

network parameters γ, ρ and λ. We have shown that in the presence of social networks where unemployment arises due to frictions in information transmission among workers, the optimal capital tax rate is zero and labor tax revenue only finances government expenditures. We calculate the solution of the optimal tax problem for the calibrated version of the model described above and follow Domeij (2005)’s calibration strategy and solution method. The initial capital and labor tax rates are set to 30 percent, i.e., τ0k = 0.30 and τ0l = 0.30 (Carey and Tchilingurian, 2000). Initial government debt, B0 , is assumed to be zero and government purchases, gt , are such that the steady-state ratio of government purchases to GDP generated by the model with initial policy is 20 percent of GDP. Unemployment benefits b are also constant and set equal to 0.04. The method to solve for the Ramsey equilibrium is standard in the literature and we briefly describe it. We assume that the economy converges to a steady-state and the Ramsey equilibrium is characterized by a system of non-linear equations. We solve for the Ramsey equilibrium by adjusting the multipliers η and ηˆ on the implementability constraints, equations (14) and (24), for the exogenous and endogenous cases, respectively, until these constraints are satisfied.

5.2

Optimal Labor Tax Rates and Social Networks

The results for our benchmark parameterization when the social network process is exogenous are presented in Table 2. As discussed in Sections 3 and 4, the optimal capital tax is zero and the government relies on labor taxation only to finance its expenditures. The empty network is the case where agents hear about job opportunities themselves and have no access to information from their peers. As expected, the employment rate is lower in the empty network and employed family members work more. Leisure h∗ = 1 − n∗ l∗ is relatively constant across all networks. Employed agents tend to work less in the regular network while the family enjoys higher consumption and welfare. Regarding the labor income tax, we observe two opposite effects. First, the higher the labor supply the higher the optimal labor tax. Second, the labor income tax is decreasing in the economy’s employment rate. This result illustrate the relevance of the intensive and extensive margins in the determination of our results. Notice that for the purpose of the government what is relevant is the effective labor n∗ l∗ . In our model it can be interpreted as the tax base,

27

and the optimal labor tax is lower, the higher effective labor is. In this case, the extensivemargin effect dominates the intensive-margin effect and taxes are the lowest in the regular network and the highest in the empty network. Table 2 - Social Network: Exogenous Optimal Tax and Allocations: Average Number of Peers ⟨z⟩ = 5 Empty

Regular

Geometric

Power-Law

0.2231

0.2195

0.2217

0.2212

−17.2463

−17.2305

−17.2402

−17.2376

Consumption c∗

0.7784

0.7798

0.7790

0.7792

Labor l∗

0.9229

0.8465

0.8919

0.8795

Leisure h∗

0.3288

0.3278

0.3284

0.3283

Employment rate n∗

0.7273

0.7941

0.7530

0.7637

Effective Labor n∗ l∗

0.6712

0.6722

0.6716

0.6717

Labor Income tax τ l∗ Welfare U ∗

Next, we investigate how our results change when agents are allowed to invest part of their time endowment in social activities to strengthen their social ties. Table 3 shows the optimal allocations and labor income tax for the endogenous network process. Compared to the exogenous case, for the same average numbers of peers ⟨z⟩ = 5, labor taxes are higher in the endogenous case. The economy’s employment rate is lower but consumption and welfare are higher. Even though unemployed agents make a positive effort to improve their chances to hear about jobs from their peers, employed workers work less and the family’s leisure is higher. Once again we observe that the optimal labor income tax responds differently depending on the two margins. Contrary to the exogenous case, labor taxes decrease when labor supply increases (intensive margin) and increases along the extensive margin, i.e., labor tax is higher the higher the economy’s employment rate. However, as in the exogenous case, labor taxes respond negatively to effective labor n∗ l∗ . In the endogenous case, the intensive-effect dominates the extensive-effect. The distinction here between intensity and participation is key for the distinction between the exogenous and the endogenous network process. When the labor market is completely exogenous and agents cannot influence their chances of hearing about job opportunities, participation in the labor market is determined entirely by the network, where links and structure 28

are exogenously given to the workers. Employed workers can, however, decide the intensity of their participation to maximize the family’s welfare. This is reached by having those with jobs work more. The fact that the employment rate is higher in the exogenous case relaxes the government budget constraint in the sense that it can tax those working less, because the economy has higher participation rate. In the endogenous case, finding a job is not a completely random process as agents can influence the rate at which they learn about job opportunities, and the tax rate will influence this decision. In this case family members can affect both margins. That is, agents can decide how much to work (intensive margin) as well as the extent to which they participate in the labor market (extensive margin). By allocating time to improve their social contacts, an unemployed worker increases his chances of becoming employed. Employed family members tend to work less and the participation rate is lower, which requires the government to tax more those who have jobs. Nevertheless, family’s welfare is higher and agents are better off. In this case, the higher welfare comes from the fact that family members are enjoying more leisure and we can compute a measure of the value of the networks, in terms of welfare. Table 3 - Social Network - Endogenous Optimal Tax and Allocations: Average Number of Peers ⟨z⟩ = 5

Labor Income tax τ l∗ Welfare U ∗

Empty

Regular

Geometric

Power-Law

0.2231

0.2980

0.2235

0.2244

−14.7085

−14.5902

−17.246 −16.4481

Consumption c∗

0.7784

0.7401

0.8031

0.8062

Labor l∗

0.9229

0.7375

0.8177

0.8208

Effort e∗

0.0000

0.0653

0.0097

0.0092

Leisure h∗

0.3288

0.4051

0.3928

0.3937

Employment rate n∗

0.7273

0.7879

0.7395

0.7357

Effective Labor n∗ l∗

0.6712

0.5810

0.6046

0.6038

We see that welfare is higher in the geometric and power-law networks than in the regular network, when social ties are endogenous, despite the higher employment rate in the regular network. This is due to the fact that, at the low levels of effort that are optimal, the employment rate is more responsive to effort in the geometric and power-law networks. That is, n′ (e) is larger in these networks, than in the regular network. The household is able to devote less time 29

to networking, and still enjoy a relatively high level of employment. This allows the household to obtain most of its leisure from unemployed family members, and have employed family members supply more labor. The optimal tax is lower, and consumption is higher, leading to higher welfare than in the regular network. Comparing the two kinds of network dynamics and across network structures, it is clear that social networking is welfare enhancing.

5.3

Sensitivity Analysis

In this section we investigate how the optimal tax rates and allocations are affected by a variety of parameter changes. Although we could perform sensitivity analyses for all parameters, we restrict our attention to those pertinent to social networks. We present results for the endogenous regular network regarding the arrival and break-up probability, efficacy of effort and average number of peers. For other networks, results are similar, and are available upon request. Tables A.4, A.5 and A.6 in the Appendix A.4 present detailed results for each exercise. As the probability of a worker finding a job increases, welfare and the economy’s employment rate increases. A high job arrival probability means that unemployed agents have a higher chance of finding a job. The case of γ = 0.80 is quite illustrative. At this rate, agents will become employed almost certainly and their optimal behavior is to exert a very small effort to connect with their peers. This higher arrival probability leads to a higher participation rate and allows employed to work less and enjoy more leisure. The higher employment rate is however not enough to compensate for the drop in labor hours and the government has to increase the labor income tax to finance its given level of expenditures. Compared to the benchmark case, a higher break-up probability ρ leads to welfare losses for two reasons: (i) it reduces the fraction of family members with a job and (ii) it forces those without a job to exert more effort to find a job through their social networks and those (fewer) with a job to work harder. The increase in hours worked is not enough to compensate the drop in the participation rate and the optimal tax rate increases. For a small λ, it is harder to maintain social relationships and requires a greater investment of time. The fraction of family members working is smaller and they work less, which implies for the government the need to tax them at a higher rate. The optimal labor tax is decreasing in the efficacy of network effort. As λ gets closer to 1, workers are able to build strong

30

relationships with relatively little cost in terms of leisure. More unemployed workers find jobs which increases the economy’s employment rate, family’s consumption and welfare. Finally, we analyze how the average number of peers affect our results. From our baseline ⟨z⟩ = 5, i.e., each worker has on average five peers, we observe that consumption and welfare increase as the average number of peers increases. With more peers that can potentially pass information about job opportunities, unemployed workers dedicate less time to social networking and effort falls. For large numbers of peers, employed agents work less, as the fraction of the family employed increases. As ⟨z⟩ increases, effective labor n∗ l∗ also increases allowing the government to reduce the labor income tax. Notice that as the average number of peers increases, we observe a positive correlation between employment rate and labor supply. As ⟨z⟩ grows larger, the correlation between n∗ and l∗ becomes negative. The employment rate of the economy increases and employed workers tend to work less. With more peers on average, unemployed workers have a higher chance to hear about a job and more will become employed. More family members with jobs allow employed workers to work relatively less. For the government, effective labor, or the tax base, increases and taxes on labor income can be reduced

6

Conclusion

This paper studies optimal labor income taxation in the presence of social networks. The unemployment rate is determined by the dynamics of the labor market, which is governed by social networks. The optimal labor income tax is negatively related to effective labor, which is determined by labor supply response along the intensive margin (hours or intensity of work on the job) and along the extensive margin (employment rate). When agents cannot influence their participation in the labor market, which is instead exogenously determined by the social network, the optimal labor tax is lower in more connected networks (higher employment rate). On the other hand, if the network process is endogenous, labor taxes are positively related to the economy’s employment rate. Overall, people are better off in the endogenous case (welfare is higher), even though taxes are higher when compared to the exogenous case. The optimal limiting capital tax rate is zero, independent of the labor market frictions. Our approach provides a simple way to measure the value of social networks, in terms of social welfare. 31

The exercise that we carry out in this paper provides a new insight into the relationship between taxes and labor market dynamics and should be seen as an illustration of a much broader line of research. A natural extension of our model is to allow firms to have a more active role in the labor market, for instance, receiving information on the productivity of job applicants through current employees. We pursue this in future research.

Appendix [ABOUT HERE] Figure A.1 - Effort and Marginal Employment Rate

A.1. Derivation of the Implementability Constraint To derive the implementability constraint, equation (24), first premultiply the family’s budget constraint in period t with the associated Lagrangian multiplier β t φt and sum over all periods t≥0 ∞ ∑

t

β u1 (t) [ct + kt+1 + Bt+1 ] =

∞ ∑

t=0

[ ] β t u1 (t) n(et−1 )(1 − τtl )wt lt + (1 − n(et−1 ))bt + Tt kt + Bt Rt

t=0

(30) Using first-order conditions with respect to capital and bonds to eliminate the after-tax return on capital and bonds we obtain ∞ ∑

β t u1 (t) [ct ] =

t=0

∞ ∑

[ ] β t u1 (t) n(et−1 )(1 − τtl )wt lt + (1 − n(et−1 ))bt + A00

(31)

t=0

where A00 = u1 (0) [T0 k0 + B0 R0 ]. Multiplying equilibrium equation (21) by β t+1 u1 (t + 1) we get l β t+1 u1 (t+1)(1−τt+1 )wt+1 lt+1 = β t+1 u1 (t+1)bt+1 +β t+1 u2 (t+1)(lt+1 −et+1 )+β t u2 (t)

(1 − n(et−1 )) n′ (et )

and then multiplying it by n(et ) yields ∞ ∑

∞ ∑ ] [ l β t+1 n(et )u2 (t + 1)(lt+1 − et+1 )(32) β t+1 n(et )u1 (t + 1) (1 − τt+1 )wt+1 lt+1 − bt+1 = t=0 ∞ ∑

t=0

+

β t+1 n(et )u2 (t)

t=0

(1 − n(et−1 )) . n′ (et )

Notice that the right-hand-side of equation (31) can be written as ] [ u0 (0) n(−1)(1 − τ0l )w0 l0 + (1 − n(−1))b0 ∞ ∞ ∑ ] ∑ [ t+1 l β t u1 (t)bt . + β u1 (t + 1)n(et ) (1 − τt+1 )wt+1 lt+1 − bt+1 + t=0

t=0

32

Substituting (32) into (31) and after some manipulation, we obtain the implementability constraint for this problem equation (24): ∞ ∑ t=0

( [ ]) n(et ) β u1 (t) (ct − bt ) − u2 (t) n(et−1 )(lt − et ) + (1 − n(et−1 )) ′ = A0 n (et ) t

[ ] where A0 = u1 (0) n(e−1 )(1 − τ0l )w0 l0 − n(e−1 )b0 + T0 k0 + R0 B0 − u2 (0)n(e−1 )(l0 − e0 ).

A.2. Proposition 3 Proof. To show that any allocation that satisfy equations (23), (24) and (25) can be decentralized as a network competitive equilibrium we use these allocations together with the family’s and firm’s first-order conditions to construct the corresponding prices and taxes. The rental rate rt is given by the firm’s first-order condition with respect to capital. The capital tax τtk is determined using the family’s and firm’s first-order condition with respect to capital, and implicitly defined by ( ( ) ) u1 (t) = 1 + 1 − τtk [F1 (t + 1) − δ] βu1 (t + 1) The wage rate wt and the labor tax rate τtl are determined by substituting equation (21) into the firm’s first-order condition with respect to labor, obtaining ]−1 [ 1 (1 − n(et−1 )) = u1 (t+1)F2 (t+1)lt+1 u1 (t + 1)bt+1 + u2 (t + 1)(lt+1 − et+1 ) + u2 (t) l n′ (et ) (1 − τt+1 ) (33) The family’s first-order condition with respect to labor for period t + 1 is 1 u1 (t + 1) = F2 (t + 1) l u2 (t + 1) (1 − τt+1 )

(34)

Combining (33) and (34) and rearranging we obtain Φ(ct , lt , et , ct+1 , lt+1 , et+1 ) = (1 − n(et−1 )) u2 (t) − βn′ (et ) (u2 (t + 1)et+1 − u1 (t + 1)bt+1 ) which is equivalent to equation (25). The labor tax τtl is implicitly defined by both (33) and (34) and to ensure that the labor taxes implied by these two conditions coincide the constraint (25) is imposed in the Ramsey problem. To show that any network competitive equilibrium allocations satisfy equations (23), (1), (24) and (25), we proceed as follows: (a) The resource constraint, equation (11), is implied by the family’s and government’s period-by-period budget constraints, thus feasibility is satisfied; (b) Premultiply the family’s budget constraint in period t with the associated Lagrangian multiplier β t φt and sum over all periods t ≥ 0. We proceed by solving for taxes and prices as a function of allocations using the family’s and firm’s first order conditions. This results in the implementability constraint, equation (24); (c) Since, by definition, the labor tax rate τ l satisfies both (33) and (34), the allocations also satisfy the intertemporal constraint on labor taxes, equation (25).

A.3. Proposition 4 Proof. The result of Judd (1985) and Chamley (1986) that capital taxes are zero in the limit follows directly from an evaluation of the first order conditions of the family’s and the Ramsey’s

33

problems with respect to capital. Once again, τ ∗k = 0 regardless the labor network structure or dynamics. Next, we discuss the result for the labor income taxes. Notice that the multipliers η, µ and κ appear on equations (26), (28) and (29). First, we reduce these three expressions to one where only the Lagrangian multiplier on the implementability constraint, η, appears. Manipulating (26) and (28), we obtain: (Zl∗ − Zc∗ Fl (k ∗ , n∗ l∗ )) u∗21 κ = + Fl (k ∗ , n∗ l∗ )u∗21 − u∗22 Zl∗ − Zc∗ Fl (k ∗ , n∗ l∗ ) µ∗ = (1 − n(e∗ )) [Fl (k ∗ , n∗ l∗ )u∗21 − u∗22 ] ∗

Zc∗

(35) (36)

Substituting (35) and (36) into (29), we get 1 β(1 − n(e∗ ))n′ (e∗ )l∗ (Zl∗ u∗21 − Zc∗ u∗22 ) − (1 − n(e∗ ))Ze∗ u∗21 + Zc∗ Γ∗ = Fl (k ∗ , n∗ l∗ ) Zl∗ Γ∗ − (1 − n(e∗ ))Ze∗ u∗22

(37)

where [ ] Γ∗ = u∗22 (1 − n(e∗ ))2 − βn′ (e∗ )(l∗ − e∗ ) (1 − n′ (e∗ )e∗ − n(e∗ )) +βu∗2 [n′′ (e∗ )e∗ + n′ (e∗ )] − βu∗1 n′′ (e∗ )b + βu∗12 n′ (e∗ )(l∗ − e∗ )b Manipulating (37) further and using the fact that Ze∗ = βn′ (e∗ )l∗ Zl∗ , we obtain Zc∗ 1 = Fl (k ∗ , n∗ l∗ ) Zl∗

(38)

Expression (38) together with the family’s problem (P.4) first order conditions imply that τ ∗l = 1 −

u∗2 Zc∗ u∗1 Zl∗

(39)

For a broad class of preferences, we argue that the labor income tax is non-zero. Labor taxes would be zero in the limit under two possibilities: either ηˆ∗ = 0 in which case, τ ∗l = 0 and the solution is first best, or ηˆ∗ ̸= 0 in which case τ ∗l = 0 if and only if ( ) n(e∗ ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ [u11 u2 − u12 u1 ] (c − b) = [u21 u2 − u22 u1 ] n(e )(l − e ) + (1 − n(e )) ′ ∗ (40) n (e ) In general, (40) will not be satisfied and the optimal labor income tax is given by (39).

A.4. Sensitivity Analysis The first column of each table below shows the results for our benchmark parameterization for the arrival probability and the break-up probability to allow comparisons. We then vary one parameter at a time while all others are kept at their benchmark levels.

34

Table A.4 - Endogenous Regular Network Optimal Tax and Allocations: Average Number of Peers ⟨z⟩ = 5 γ # = 0.40 Arrival Probability Break-up Probability # ρ = 0.15 γ = 0.50 γ = 0.60 γ = 0.80 ρ = 0.10 ρ = 0.40 ρ = 0.80 Labor Income tax τ l∗ 0.2980 0.3024 0.3126 0.2933 0.2926 0.2824 0.3096 ∗ Welfare U −16.448 −16.415 −15.685 −14.869 −16.336 −17.060 −19.089 ∗ Consumption c 0.7401 0.7252 0.7279 0.7448 0.7436 0.7086 0.6423 Labor l∗ 0.7375 0.7055 0.6667 0.6320 0.6903 0.8514 0.9722 ∗ Effort e 0.0653 0.0436 0.0200 0.0000 0.0619 0.1380 0.2156 Leisure h∗ 0.4051 0.4279 0.4564 0.4678 0.4049 0.4250 0.4438 ∗ Employment rate n 0.7879 0.7983 0.8096 0.8421 0.8486 0.6125 0.4502 ∗ ∗ Effective Labor n l 0.5810 0.5632 0.5398 0.5322 0.5857 0.5214 0.4376

Table A.5 - Endogenous Regular Network Optimal Tax and Allocations: Average Number of Peers ⟨z⟩ = 5 Efficacy of Networking Effort # λ = 0.05 λ = 0.20 λ = 0.40 λ = 0.80 l∗ Labor Income tax τ 0.2980 0.2394 0.1995 0.1645 Welfare U ∗ −16.448 −15.982 −15.446 −15.122 Consumption c∗ 0.7401 0.7641 0.7923 0.8151 ∗ Labor l 0.7375 0.7527 0.7641 0.7433 Effort e∗ 0.0653 0.0478 0.0273 0.0046 ∗ Leisure h 0.4051 0.3917 0.3772 0.3628 ∗ Employment rate n 0.7879 0.7951 0.8082 0.8563 Effective Labor n∗ l∗ 0.5810 0.5984 0.6175 0.6364 Table A.6 - Endogenous Regular Network Optimal Tax and Allocations: Average Number of Peers ⟨z⟩ ⟨z⟩# = 5 ⟨z⟩ = 6 ⟨z⟩ = 10 ⟨z⟩ = 20 Labor Income tax τ l∗ 0.2980 0.2745 0.2273 0.1885 ∗ Welfare U −16.448 −16.543 −16.288 −16.216 Consumption c∗ 0.7401 0.7444 0.7655 0.7785 ∗ Labor l 0.7375 0.7450 0.7569 0.7602 ∗ Effort e 0.0653 0.0628 0.0516 0.0377 Leisure h∗ 0.4051 0.3954 0.3780 0.3649 ∗ Employment rate n 0.7879 0.7942 0.8088 0.8269 Effective Labor n∗ l∗ 0.5810 0.5912 0.6127 0.6284

⟨z⟩ = 50 0.1639 −15.394 0.8112 0.7544 0.0245 0.3573 0.8470 0.6382

⟨z⟩ = 100 ⟨z⟩ = 200 0.1532 0.1532 −15.242 −15.175 0.8162 0.8183 0.7468 0.7428 0.0171 0.0111 0.3572 0.3573 0.8575 0.8632 0.6404 0.6419

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[21] Jones, Larry E.; Rodolfo E. Manuelli; and Peter E. Rossi (1997). “On the optimal taxation of capital income,” Journal of Economic Theory 73(1), 93–117. [22] Judd, Kenneth L. (1985) “Redistributive taxation in a simple perfect foresight model,”Journal of Public Economics 28(1), 59–83. [23] Krauth, B. (2004). A dynamic model of job networking and social influences on employment. Journal of Economic Dynamic and Control 28, 1185-1204. [24] Kyndland, F. and E. Prescott (1980). A competitive theory of fluctuations and the feasibility of stabilization policy, in ”Rational Expectations and Economic Policy” (Stanley Fischer, Ed.), University of Chicago Press, Chicago, IL. [25] Lucas R., Stokey N., 1983. Optimal fiscal and monetary policy in an economy without capital. Journal of Monetary Economics 12, 55-93. [26] Mardsen, Peter V. (2001). Interpersonal ties, social capital, and employer staffing practices, in Social capital: theory and research, ed. Nan Lin, Karen Cook and Ronald S. Burt, Mew Brunswick, NJ, Transaction Publishers. [27] Merz, M. (1995). Search in the labor market and the real business cycle, Journal of Monetary Economics 36, pp. 269–300 [28] Montgomery, James D. (1991) ‘Social networks and Labor-Market outcomes: Toward an economic analysis.’ The American Economic Review 81(5), 1408–1418. [29] Reinhorn, Leslie J. (2009) “Dynamic Optimal Taxation with Human Capital,” The B.E. Journal of Macroeconomics: Vol. 9: Iss. 1 (Topics), Article 38. [30] Scott, A. (2007). Optimal taxation and OECD labor taxes. Journal of Monetary Economics 54, 925-944. [31] Topa, Giorgio (2001) ‘Social interactions, local spillovers and unemployment.’ The Review of Economic Studies 68(2), 261–295 [32] Vega-Redondo, Fernando (2007). Complex Social Networks. Econometric Society Monographs. ESM 44. Cambridge University Press. [33] Shi, S., Wen, Q., 1999. Labor market search and the dynamic effects of taxes and subsidies. Journal of Monetary Economics 43, 457–495. [34] Zhu, X. (1992). Optimal fiscal policy in a stochastic growth model. Journal of Economic Theory 58, 250-289.

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Optimal Taxation and Social Networks

Nov 1, 2011 - We study optimal taxation when jobs are found through a social network. This network determines employment, which workers may influence ...

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