Tax Evasion on a Social Network

We relate tax evasion behavior to a substantial literature on self and social comparison in judgements. Tax payers engage in tax evasion as a means to boost their expected consumption relative to others in their “local” social network, and relative to past consumption. The unique Nash equilibrium of the model relates optimal evasion to a (Bonacich) measure of network centrality: more central taxpayers evade more. The indirect revenue effects from auditing are shown to be ordinally equivalent to a related Bonacich centrality. We generate networks corresponding closely to the observed structure of social networks observed empirically. In particular, our networks contain celebrity taxpayers, whose consumption is widely observed, and who are systematically of higher wealth. In this context we show that, if the tax authority can observe the social network, it is able to raise its audit revenue by around six percent.


Introduction
Tax evasion is a signi…cant economic phenomenon. Estimates provided by the UK tax authority put the value of the tax gap -the di¤erence between the theoretical tax liability and the amount of tax paid -at 6.5 percent (H.M. Revenue and Customs, 2016). Academic studies for the US and Europe put the gap substantially higher, at around 18-20 percent (Cebula and Feige, 2012;Buehn and Schneider, 2016).
In this paper we link evasion behavior to a mass of evidence that people engage continually in comparisons -with others (social comparison) and with themselves in the recent past (self comparison -or "habit"). Utility, evidence for developed economies suggests, is in large part derived from consumption relative to these comparators, rather than from its absolute level (e.g., Ferrer-i-Carbonell, 2005;Luttmer, 2005;Clark and Senik, 2010;Mujcic and Frijters, 2013). The evolutionary processes that might explain this phenomenon are explored in Postlewaite (1998), Rayo and Becker (2007) and Samuelson (2004), among others.
Despite the overwhelming evidence of a concern for self and social comparison, these features have yet to be explored simultaneously in the context of the tax evasion decision. In this paper we provide a network model in which taxpayers are assumed to have an intrinsic concern for income relative to a benchmark that re ‡ects both self and social comparison. 1 Taxpayers in our model observe the consumption of a subset of other taxpayers (the "reference group") with whom they are linked on a social network. In this context, taxpayers may seek to evade tax so as to improve their standing relative to those they compare against. Taxpayers also benchmark their current consumption against its lagged values. The model exhibits strategic complementaries in evasion choices, so that more evasion by one taxpayer reinforces other taxpayers'decisions to evade also. Following the lead of Ballester et al. (2006), we utilize linear-quadratic utility functions to provide a characterization of Nash equilibrium. We show that there is a unique Nash equilibrium in which evasion is a weighted network centrality measure of the form proposed by Bonacich (1987). Network centrality is a concept developed in sociology to quantify the in ‡uence or power of actors in a network. It counts the number of all paths (not just shortest paths) that emanate from a given node, weighted by a decay factor that decreases with the length of these paths. In this sense, our contribution combines sociological and economic insights in seeking to understand tax evasion behavior.
Although the model is simple enough to admit an analytic solution, it is also su¢ ciently rich that it may be used to address a range of questions of interest to academics and practitioners in tax authorities. Here we focus on three such questions: …rst, we investigate -for an arbitrary network structure -how changes in the model's exogenous parameters a¤ect optimal evasion; second, we explore how the marginal indirect e¤ects arising from performing one extra audit vary across taxpayers in the network. Last, we investigate the value to a tax authority -in terms of additional revenue raised through audits -of knowing the structure of the social network. The analysis is performed on a class of generative networks that possess many of the empirically observed features of social networks -in particular allowing for highly visible celebrity taxpayers. Our results suggest that, if a tax authority moves from not observing the social network to observing fully the social network, it may increase audit revenues by around six percent. A stronger concern for social comparison by taxpayers increases the value of information relating to the social network, while a greater concentration of links within a social network decreases the value of such network information.
An important feature of our model is that it addresses explicitly the role of local comparisons on a social network. By contrast, the existing analytical literature on tax evasion allows only global (aggregate) social information to enter preferences: the global statistic that taxpayers are assumed to both have a concern for, and to be able to observe, is either (i) the proportion of taxpayers who report honestly (Gordon, 1989;Myles and Naylor, 1996;Davis et al., 2003;Kim, 2003;Traxler, 2010;Ratto et al., 2013); (ii) the average post-tax consumption level (Goerke, 2013); (iii) the level of evasion as a share of GDP (Dell'Anno, 2009); or (iv) the average tax payment (Mittone and Patelli, 2000;Panadés, 2004).
While reducing social information to a single statistic known to all taxpayers promotes analytical tractability, it is problematic in other respects. First, from the perspective of modelling with explicit social networks, assuming that taxpayer's observe aggregate-level information is implicitly the assumption that every taxpayer observes the behavior of every other taxpayer. As we adopt the convention that a link from i to j in the social network signi…es that i can observe j's consumption, full observability is equivalent to the assumption that the social network is the complete network (in which every taxpayer is directly linked to all other taxpayers). Yet there are reasons to think that relative consumption externalities are, in fact, heterogeneous across individuals. In particular, we know that people's reference group is typically composed of "local"comparators such as neighbors, colleagues, and friends (Luttmer, 2005;Clark and Senik, 2010). 2 Moreover, implicitly assuming a complete network implies that all taxpayers are equally connected socially, thereby ruling out, in particular, the existence of "stars" or "celebrities" whose consumption is very widely observed in the network. Yet, this feature of social networks may matter for the targeting of tax audits (Andrei et al., 2014).
The only literature that has enriched the analysis of social information to allow for local social information is that which uses agent-based simulation techniques as an alternative to analytical methods. Models in this tradition nonetheless employ representations of social networks that appear to di¤er markedly from real world examples. A common property of the network structures employed (e.g., Korobow et al., 2007;Hokamp and Pickhardt, 2010;Bloomquist, 2011;Hokamp, 2014) is that the number of taxpayers who observe the behavior of each taxpayer is …xed, thereby ruling out the existence of highly-observed celebrity taxpayers. Other authors (e.g., Davis et al. 2003;Hashimzade et al., 2014Hashimzade et al., , 2016 utilize an undirected network, meaning that, if i is linked to j, then necessarily j is linked to i. Yet social networks display marked asymmetry in the direction of links (Foster et al., 2010;Szell and Thurner, 2010). 3 We o¤er a model that is both analytically tractable and that allows for local comparisons on an arbitrary social network. In this sense, our approach lies in the cleavage between existing analytical and agent-based approaches, and is complementary to each. 4 We perform simulation analysis on a class of generative networks that are not subject 2 More generally, relative consumption externalities may be viewed as a form of peer e¤ ect. In other contexts, generative models of peer e¤ects predict heterogeneous exposure. For instance, when job information ‡ows through friendship links, employment outcomes vary across otherwise identical agents with their location in the network of such links (Calvó-Armengol and Jackson, 2004). 3 Zaklan et al. (2008) and Andrei et al. (2014) are among exceptions that do explore more general network structures. 4 By extending analytical understanding of network e¤ects upon tax evasion -in particular being able to to the restrictions discussed above, and which are widely utilized to model network structures in the natural sciences. Our methodology in this regard, therefore, has applicability beyond the current context of tax evasion.
To our knowledge, no previous contribution allows simultaneously for both self and social comparison in the tax evasion decision. Goerke (2013), however, assumes an intrinsic concern for relative consumption by taxpayers. The primary focus of his contribution is, however, the derived impact on tax evasion from endogenous changes in labor supply, whereas we treat income as an exogenous parameter. In the remaining literature that considers a social dimension to the tax evasion decision, taxpayers are assumed to derive utility solely from absolute consumption, but react nonetheless to social information because they experience social stigma -the extent of which depends on the evasion of other taxpayers -if caught evading. The focus of much of this literature is on the potential for multiple equilibria, whereas our model yields a unique equilibrium. While a concern for relative consumption is compatible with the simultaneous existence of social stigma towards evaders, the two approaches di¤er in emphasis. Underlying the idea of social stigma is the concept of social conformity, in which agents seek to belong to the crowd, whereas the presumption of relative consumption theories is that individuals seek to stand out from the crowd. A literature relating to this point in the context of tax evasion supports the notion that social information impacts compliance behavior (Webley et al., 1988;De Juan et al., 1994;Alm and Yunus, 2009;Alm et al., 2017), but rejects social conformity as the underlying mechanism (Fortin et al., 2007).
A recent contribution that allows explicitly for self comparison in the tax evasion decision is Bernasconi et al. (2016). There are, however, important di¤erences in approach and results. In our model, a higher level of habit consumption is associated with higher evasion, for it generates a negative internality on myopic taxpayers: higher past consumption outcomes reduce present utility. To overcome this internality, taxpayers must gamble (evade) more.
Conversely, in Bernasconi et al., higher habit consumption reduces tax evasion. In their model taxpayers are far-sighted, so consumption internalities do not arise. Instead, higher habit consumption levels stimulate increased risk aversion, for taxpayers are assumed in…nitely loss averse over consumption levels below the habit level. By contrast, in our model, consumption may fall below its habit level in periods in which a taxpayer is audited and prove formal comparative statics properties of the model -we assist the interpretation of simulation output from related agent-based models. …ned.
The paper proceeds as follows: section 2 develops a formal model of tax evasion on a social network. Section 3 analyzes the comparative statics predictions of the model, and section 4 characterizes the indirect e¤ects of an audit, both on the future behavior of the audited taxpayer, and to the behavior of other, unaudited, taxpayers through the channel of social comparison. Section 5 considers the value of network information to a tax authority, and section 6 concludes. Proofs are collected in the Appendix.

Model
Let N be a set of taxpayers of size N . A taxpayer i 2 N has an (exogenously earned) income W i > 0. By law taxpayers should declare W i to the tax authority and pay tax (W i ), where : R 0 7 ! (0; W i ) is a non-decreasing function. If a taxpayer declares their true gross income, Taxpayers can, however, choose to declare less than their true income, thereby evading an amount of Taxpayer i is audited with probability p i 2 (0; 1) in each period.
Heterogeneity in the p i can arise, for example, if the tax authority conditions audit selection upon observable features of taxpayers. Audited taxpayers face a …ne at rate f > 1 on all undeclared tax, à la Yitzhaki (1974).
Taxpayers are assumed to derive utility from their level of consumption relative to a reference level of consumption R it (the determination of which we shall discuss later). As is standard in agent-based modelling, although taxpayers live for multiple periods, each makes a succession of single-period decisions and so is "myopic". In this context, myopic behavior could be the result of cognitive limitations of the part of taxpayers. Consistent with this notion, and our emphasis on social networks, Manski (1991) and McFadden (2006) argue that individuals faced with dynamic stochastic decision problems that pose immense computational challenges may instead look to other individuals to infer satisfactory policies. 5 In each period, taxpayers behave as if they maximize expected utility, where utility is denoted by U (:). The expected utility of taxpayer i at time t is therefore given by where consumption in the audited state (C a it ) and not-audited states (C n it ) is given by: An obvious objection to this formulation is that it neglects entirely the possibility of absolute utility. Although an absolute component to utility surely exists, we note that measures of subjective wellbeing typically become uncorrelated with absolute income above a threshold of average national income estimated at $5,000 (in 1995, PPP) by Frey and Stutzer (2002). As most citizens of developed countries lie above this threshold, our model may be a reasonable approximation in such cases. Optimal evasion in period t is the solution to the problem subject to the Cournot constraint that reference consumption, R it , is taken as given. The …rst order condition for optimal evasion is therefore given by

Reference Consumption
Reference consumption, R it , is a function of self and social comparison. To formalize the notion of social comparison, we assume that each taxpayer observes the consumption of a non-empty set of taxpayers R i N , a set we term the reference group. A taxpayer, i, is said to be observed if their consumption is visible to at least one other taxpayer in the network, We represent the observability of consumption in the form of a directed network (graph), where a link (edge) from taxpayer (node) i to taxpayer j indicates that i observes j's consumption. Links are permitted to be subjectively weighted, for some members of the reference group may be more focal comparators than are others. The network is represented as an N N (adjacency) matrix, G, of subjective comparison intensity weights g ij 2 [0; 1], where g ii = 0. We normalize the g ij for each taxpayer to sum to unity: P j2R i g ij = 1. Taxpayers i and j with g ij > 0 are said to be linked. Accordingly, the reference group of taxpayer i is the set of all taxpayers to whom i is linked: R i = fj 2 N : g ij > 0g. For later reference, a network, G, in which there is a path (though not necessarily a direct link) between every pair of taxpayers is said to be connected. The set of connected networks we denote by C. A necessary condition for G 2 C is that all taxpayers belonging to N are observed.
To de…ne reference consumption we …rst introduce a latent variable, Z it , which re ‡ects self and social comparison. Speci…cally, Z it is the sum of a level of habit consumption h it re ‡ecting self comparison (weighted by h > 0), and a level of consumption s it re ‡ecting social comparison (weighted by s > 0): Habit consumption we specify as h it h it (C t 1 ). 6 To specify s it we …rst write expected consumption as We then set s it as the (weighted) mean of q jt over i's reference group (j 2 R i ). This weighted average is conveniently written as s it s it (q t ) = g i q t , where g i is the i th row of G, and q t is a N 1 column vector of the expected consumptions.
To form reference consumption, we embed Z it in a simple linear di¤erence equation, given by Under the speci…cation in (4) reference consumption adjusts towards Z it in each period, with the strength of this adjustment regulated by the parameter & R . In this sense & R may be interpreted as determining the persistence of shocks to reference consumption. In the special case & R = 1 there is full adjustment in every period, so R it = Z it , whereas, when & R = 0, R it is …xed at its initial value R i0 for all t.

Nash Equilibrium
Using (4) in the …rst order condition (3), we now solve for the unique Nash equilibrium of the model. To do this, we …rst de…ne a notion of network centrality due to Bonacich (1987), which computes the (weighted) discounted sum of paths originating from a taxpayer in the network: De…nition 1 Consider a network with (weighted) adjacency matrix G. For a diagonal matrix and weight vector , the weighted Bonacich centrality vector is given by b(G; ; ) = [I G ] 1 provided that [I G ] 1 is well-de…ned and non-negative. 6 In writing h it = h (C i;t 1 ) we adopt the convention, observed widely in the empirical literature, that habit depends only on consumption in the previous period. For examples of this approach see, e.g., Muellbauer (1988), Carroll and Weil (1994) and Guariglia and Rossi (2002). As, however, we shall embed habit consumption within a long-memory reference consumption level, past consumption levels nonetheless have persistent e¤ects.
In De…nition 1, the matrix speci…es discount factors that scale down (geometrically) the relative weight of longer paths, while the vector is a set of weights. In the present context the matrix [I G ] 1 is a form of social comparison multiplier. It measures the way in which actions by one taxpayer feed through into other taxpayers'actions. Ballester et al. (2006) show that [I G ] 1 will be well-de…ned, as required in De…nition 1, when I > (G) , where (G) is the largest absolute value of the eigenvalues of G. Intuitively, this condition is that the local externality that a taxpayer's evasion imparts upon other taxpayers cannot be too strong. If local externality e¤ects are too strong then the set of equations that de…ne an interior Nash equilibrium of the model have no solution. In this case, multiple corner equilibria can instead arise (see, e.g., Bramoullé and Kranton, 2007). Focusing on the case when local externality e¤ects are not too strong, we have the following Proposition: then there is a unique interior Nash equilibrium, at which the optimal amount of tax evaded is given by According to Proposition 1, in the case of linear-quadratic utility a taxpayer's optimal evasion corresponds to a Bonacich centrality on the social network M, weighted to re ‡ect a taxpayer's marginal utility of consumption. 7 By this measure, taxpayers that are more central in the social network evade more. The uniqueness of equilibrium evasion follows intuitively from the observation that, under linear-quadratic utility, each taxpayer's best response function is linear in the evasion of every other taxpayer. The social network M transforms the underlying comparison intensity weights, g ijt , by a factor [1 i > 0 that re ‡ects potential heterogeneity in audit probabilities across taxpayers. It follows that, in the special case that all taxpayers face a common audit probability, as might occur if a tax authority has committed to a policy of random auditing, no adjustment to the underlying comparison intensity weights is warranted. In this case, therefore, optimal evasion is a weighted Bonacich centrality measure on the untransformed network G: Corollary 1 Under the conditions of Proposition 1 and setting p i = p for all i 2 N , the unique interior Nash equilibrium for evasion is given by What if utility is not linear-quadratic? For an arbitrary twice-di¤erentiable utility function we may generalize the model by considering the …rst order linear approximation around a Nash equilibrium to a set of (potentially non-linear) …rst order conditions of the form in (3).
The resulting set of equations are given by where^ t is again a vector of weights for the di¤erent taxpayers, and J is a matrix of coe¢ cients measuring how actions interact. By appropriate decomposition of J, therefore, a solution to the equation system in (5) is a Bonacich centrality measure of the form contained in De…nition 1.
The model we have outlined is su¢ ciently rich that it may be used to address a range of questions of interest to academics and practitioners in tax authorities. Subsequent sections will consider how changes in the exogenous parameters a¤ect evasion; how the marginal revenue e¤ects that arise from performing one extra audit vary across taxpayers in the social network; and the value of knowing of network information to a tax authority. To study these questions, however, requires a controlled environment that, in particular, abstracts from the stochastic perturbations generated by tax authority audits. Accordingly, in a second corollary of Proposition 1, we …nish this section by de…ning the notion of steady state, in which C it = C i;t 1 = C n it for all i 2 N . 8 This corresponds to the state the model converges toward if the auditing process is temporarily "switched o¤": Corollary 2 In a steady state of the model consumption satis…es C SS = C n;SS = X + E SS , where steady-state evasion, E SS , is given by the vector of Bonacich centralities, b(M; ; SS ),

Comparative Statics
Under linear-quadratic utility the model exhibits strategic complementaries in evasion choices: expected utility is supermodular in cross evasion choices. An advantage of this feature of the model is that we may employ the theory of monotone comparative statics (Edlin and Shannon, 1998;Quah, 2007) to analyze, in a straightforward way, the qualitative (sign) implications of changes in the underlying exogenous parameters for an otherwise arbitrary social network satisfying condition (ii) in Proposition 1. 9 We consider a steady state of the model and analyze a permanent marginal increase in a parameter z. The model is then allowed to adjust to a new steady state. This marginal response of the steady-state level of evasion to a change in z we denote by dE SS i =dz. Because the e¤ects of habit are not contemporaneous, the full adjustment to a new steady state (dE SS i =dz) comprises a contemporaneous adjustment (@E SS i =@z) and a delayed adjustment. To analyze the sign of the full adjustment, we …rst prove a Lemma that links the signs of the contemporaneous and full adjustments.
According to Lemma 1, the sign of the full and contemporaneous adjustments of steadystate evasion are related. The condition in the Lemma encompasses a number of cases; in 8 In this sense, analysis at a steady-state of the model removes unhelpful time-dependencies from the analysis. It also permits the underlying behavioral trends predicted by the model to be observed visually and calculated algebraically. These trends are otherwise obscured, both algebraically and in visual output, by uninformative noise. 9 For an excellent introduction to monotone comparative statics methods see Tremblay and Tremblay (2010). particular, it holds if (i) X i is independent of z (@X i =@z = 0); and (ii) if @X i =@z and @E SS i =@z both take the same sign. The intuition for the condition in the Lemma is that it is su¢ cient to ensure that the contemporaneous adjustment of evasion, @E SS i =@z, and of steady-state consumption, @C SS i =@z, are of the same sign. In the event, however, that z induces opposing contemporaneous responses from X i and E SS i the overall e¤ect on steady-state consumption may take either sign, so Lemma 1 does not apply.
With Lemma 1 in hand we prove the following Proposition: Proposition 2 Under the conditions of Proposition 1 it holds at an interior Nash equilibrium that: We begin with the results for the pair of parameters fa; bg belonging to the linear-quadratic utility function. Noting that the coe¢ cient of absolute risk aversion is given by A (z) = a [b az] 1 > 0, increases in a associate with decreased risk aversion, while increases in b associate with increased risk aversion. Consistent with this observation, increases in a cause optimal evasion to increase, while increases in b decrease optimal evasion.
An increase in one's own audit probability lowers optimal evasion, as does an increase in the audit probability of another taxpayer in the social network. The latter result is a weak inequality, but can be strengthened to a strict inequality if G 2 C. When a taxpayer's audit probability increases they decrease their evasion, thereby decreasing the evasion required of other taxpayers to maintain a given level of expected relative consumption. Albeit with di¤erences in economic interpretation, these results are in line with those of models of tax evasion that introduce social concerns through a social norm for compliance. As is standard, an increase in the …ne on undeclared tax reduces optimal evasion.
The parameter, s , which measures the extent to which taxpayers care about social comparison, is positively associated with evasion. Taxpayers impose a negative externality upon other taxpayers when their expected consumption increases, and the size of this externality is directly regulated by s . The greater the externality, the more evasion is pushed upwards in the struggle among taxpayers to maintain relative consumption. The parameters h it and h , which both re ‡ect the role of self comparison, are also positively associated with evasion, but the economic intuition (relative to social comparison) di¤ers. Whereas social comparison generates negative externalities, self comparison generates negative internalities: past consumption outcomes a¤ect negatively the evaluation of current consumption. To overcome this internality, taxpayers must seek a present consumption level that exceeds h it , which entails attempting greater evasion. 10 The e¤ects of self and social comparison therefore interact positively: the desire to out-consume one's reference group induces evasion, which then pushes up past consumption (in expectation), causing a further increase in evasion on account of the concern for self comparison.
The …nding for X j in Proposition 2 governs how a taxpayer's evasion responds to changes in the income of other taxpayers. This cross e¤ect is always non-negative (and strictly positive if G 2 C), for W j enters optimal evasion only through X j (W j ) and @X j (W j ) =@W j > 0, so sign (@E it =@W j ) = sign (@E it =@X j ). This e¤ect arises as one taxpayer becoming richer implies that, to preserve their level of expected relative consumption, other taxpayers must evade more. The role of own-income is the only case where Lemma 1 does not apply, for @X i =@z and @E SS i =@z are of opposite signs. Empirically, evasion and income are positively related (Clotfelter, 1983;Baldry, 1987). Accordingly, in the simulation analysis, we calibrate the model to be consistent with this evidence.

Indirect E¤ects
Consider a single audit to a taxpayer i that perturbs the steady state of the model. The revenue e¤ects this generates are commonly broken down three ways: the direct e¤ect, D i , is the tax recovered contemporaneously with the audit that would otherwise have been evaded; the own indirect e¤ect (I ii ) refers to the additional revenue that arises from future changes in evasion behavior by the audited taxpayer i, while the cross indirect e¤ect (I ij ) refers to the additional future revenue that arises from spillover (or "ripple") to the evasion behavior of the (unaudited) taxpayer j (j 6 = i). Cross indirect e¤ects occur in our model through the channel of social comparison: perturbations in the consumption of other taxpayers as a result of audit activity a¤ect the reference consumption of other "local" taxpayers. The aggregate indirect e¤ect, i , is the sum of the indirect e¤ects across N from an audit of i, i = P k2N I ik . As own and cross e¤ects are hard to distinguish empirically, studies in this tradition focus solely on the aggregate e¤ect. Our model, however, allows us to characterize indirect e¤ects at the individual level, and to relate heterogeneity in these e¤ects to di¤erences in individual characteristics.
The pattern of indirect e¤ects across taxpayers may di¤er markedly from the pattern of direct e¤ects. To see this, suppose there is a taxpayer who has high evasion but is not observed by any other taxpayer, and another who has lower evasion, but is widely observed. Choosing to audit the former taxpayer will yield a higher direct e¤ect, but the cross indirect e¤ects will all be zero, potentially making the latter taxpayer a more gainful choice. Understanding the pattern of indirect e¤ects is important as empirical evidence points to their dominance over direct e¤ects: Dubin et al. (1990), for instance, estimate the aggregate indirect e¤ect of an audit to be, on average, six times larger than the direct e¤ect in the US. 11 Let I i denote the vector of indirect e¤ects, whose j th entry corresponds to the indirect e¤ect I ij ; and denote the vector of aggregate cross indirect e¤ects, whose j th entry corresponds to j . In many contexts what is important to tax authorities is measures that rank audit revenue e¤ects across taxpayers, for then audit resources may be targeted towards the highest ranked taxpayers (see, e.g., Hashimzade et al., 2016). We therefore formally introduce the notion of ordinal equivalence: two column vectors A and B are ordinally equivalent (written

Audit Targeting and Network Structure
Can tax authorities observe links in social networks? Although surely the full gamut of links cannot be observed, importantly, there exist some individuals -celebrities -for whom it is common knowledge that many people observe them. Also, even for non-celebrities, the idea that tax authorities know at least something about people's associations is becoming more credible with the advent of "big data". The UK tax authority, for instance, uses a system known as "Connect", operational details of which are in the public domain (see, e.g., Baldwin and McKenna, 2014;Rigney, 2016;Suter, 2017). Connect cross-checks public sector and third-party information, seeking to detect relationships among actors. According to Baldwin and McKenna (2014), the system produces "spider diagrams"linking individuals to other individuals and to other legal entities such as "property addresses, companies, partnerships and trusts." The IRS is known to have also invested in big data heavily, but has so far been much more reticent in revealing its capabilities.
Given the discussion above, in this …nal section we consider the business case for investing in the means to acquire information about social networks. Can such knowledge be used to raise 12 As an immediate corollary of Proposition 3 the vector of aggregate indirect e¤ects satis…es , where the i th entry of is given by additional revenue, and, if so, how much extra might realistically be raised? We also address the related questions of how the value of network information varies with the topological properties of the network, and with the assumed level of concern for social comparison. We begin by developing a theoretical framework for analyzing rigorously these questions, and then perform simulations of this framework to obtain numerical estimates.

Theoretical Framework
To understand the value of network information to a tax authority we seek to understand the additional revenue that a tax authority could raise when moving from a position of not observing the social network to a position of fully observing the social network. To focus attention solely on the role of network information, we shall assume the tax authority observes all other information. In particular, the tax authority observes (i) the functional forms h (:) and X (:), which govern the formation of, respectively, habit consumption and legal disposable income; (ii) the set of audit probabilities fp i g i2N ; and (iii) the parameters Let R i be a measure of the additional revenues accruing from an audit of taxpayer i, potentially allowing for both direct and indirect e¤ects. In analogous fashion, letR i be the best prediction of R i of a tax authority that cannot observe the social network. We consider a tax authority that, in each period, utilizes the vector of taxpayer income declarations, d i , to choose a set N A N of taxpayers to be audited. The set N A is taken to be chosen to maximize additional predicted revenue,R A P j2N AR j , subject to a binding budget constraint. The IRS, for instance, ranks tax returns for the purpose of audit selection by assigning each return a "DIF score"that re ‡ects the potential revenue gain from auditing it (see, e.g., Alm and McKee, 2004;Plumley and Steuerle, 2004;Hashimzade et al., 2016).
How does the tax authority computeR i , conditional on observing an income declaration d i ?
To address this question, let the taxpayer's income declaration by denoted as d it . Using this notation, we may write evasion as E it = (W i ) (d i ), thereby giving the income declaration as The functiond it (W i ) in equation (6) gives the optimal disclosure d it for a taxpayer with income W i . Of relevance to our purpose, however, is the inverse function , which gives the true income W i of a taxpayer who optimally declares an income d it . Accordingly, if the tax authority observes fully the network, then on receipt of the taxpayer's declaration d it it can infer true income, W i , exactly. If, however, the tax authority does not observe the social network, thend 1 it (d it ) must be computed with respect to the tax authority's prior beliefs about the network, rather than with respect to the true network, causing taxpayer income to be inferred imprecisely. To capture this point formally, let G denote the set of all feasible networks G, for a given N . We then writeR i (G) as the expectation of R i over the set of all feasible networks: As the set G is unmanageably large, we approximate equation (7) bŷ where E H2G (H) is a network whose adjacency matrix has the (common) value [N 1] 1 for all o¤-diagonal elements. Thus, the tax authority's uncertainty is represented by setting all comparison intensity weights (both within and across taxpayers) to a uniform value.
Computing estimates forR i (G) in this way, our analysis centers on the the statistic which measures, for a given social network G, the percentage increase in audit revenue that is achieved when a tax authority moves from not observing the social network to full observing it. To obtain numerical estimates of R (G) we now simulate the model.

The Social Network
To generate the social network, we follow the approach of network scientists, who utilize a class of network models, known as generative models, to investigate complex network formation (see, e.g., Pham et al., 2016). In this modelling paradigm, complex networks are generated by means of the incremental addition of nodes and edges to a seed network over a long sequence of time-steps. Two processes governing the node/edge dynamics in generative models have been shown to generate features consistent with a multitude of social, biological, and technological networks (see, e.g., Redner, 1998;Adamic and Huberman, 2000;Jeong et al., 2000;Ormerod and Roach, 2004;Capocci et al., 2006). The …rst -the node-degree (or preferential attachment) process -makes the probability that a new addition to the network observes an existing taxpayer, i, a positive function of i's degree (the number of taxpayers who already observe i). The second -the node-…tness process -makes the probability that a new addition to the network observes an existing taxpayer, i, a positive function of i's …tness (an exogenous and time-invariant characteristic of node i).
At step s of the generative process consider a taxpayer i with degree d is , and …tness i > 0.
The separate node-degree and node-…tness processes are entwined in a single process by allowing the probability that said taxpayer i is observed by the taxpayer added at step s to be proportional to the product i A (d is ), where A (:) is an increasing function. Important special cases of this approach include that of Barabási and Albert (1999), who assume i to be equal across taxpayers; and that of Bianconi and Barabási (2001), who assume A (d) = d. In allowing for a role for node-…tness in social network formation, we are able to account for the observation that, empirically, celebrity taxpayers are surely not drawn at random from the distribution of income, but rather belong systematically to the upper tail. TV and sports stars, whose consumption habits are widely reported, are also some of the richest members of society. To replicate this feature, we equate node-…tness with income: i = W i . We specify the distribution function of W i across taxpayers to satisfy a power law, consistent with a large body of empirical evidence (e.g., Coelho et al., 2008).
In our implementation we generate networks of N = 200 taxpayers, starting from a seed network composed of two interlinked taxpayers. Each taxpayer incrementally added to the network is linked to members of the existing network according to the outcome of …ve random draws under the probability distribution d is discussed above. Note, however, that these draws are with replacement, so a taxpayer may be linked multiple times to another. As the model of section 2 allows for only a single, albeit weighted, link between taxpayers, we use the frequency of links realized by the generative process to construct the comparison intensity weights. Speci…cally, let # ij 2 N denote the number of times taxpayer i is linked with j by the generative process. If # ij = 0 then taxpayers i and j are not linked. If # ij 1 then taxpayers i and j are linked, and the intensity of the link is given by

Model Functions and Parameters
Having now described the social network, we specify the remaining model functions and parameters. Habit consumption is speci…ed as h it = C i;t 1 , and, to make concrete the vector of predicted income,Ŵ, we specify the tax system as a linear income tax, where 2 (0; 1). We may then write evasion as E it = [W i d it ] and the legal disposable income level as X (W i ) = [1 ] W . Next, we show that the vectorŴ takes the form of a weighted Bonacich centrality: Lemma 2 Under the conditions of Proposition 1, and with a linear income tax, the income of a taxpayer who declares income optimally according to (6) is given bŷ We now choose the audit probabilities. AlthoughŴ (G) in Lemma 2 controls for the systematic e¤ects of the p i , nonetheless the quantitative …ndings are not wholly immune to the values chosen. The restriction to interior equilibria, however, pins down the implied pattern of the p i across taxpayers. To see this, suppose a taxpayer will belong to N A (i.e., p i = 1) then they will optimally choose the corner solution E i = 0. Similarly, if a taxpayer will not belong to N A (i.e., p i = 0) they will optimally choose the other corner solution, To avoid a corner solution, therefore, no taxpayer must belong to N A for sure. The only instance in which random selection of the set N A is optimal is when theR i are equal across all taxpayers, such that every taxpayer is predicted to be an equally attractive audit target. We therefore choose the p i to equalize theR i , where we take R i to be the total additional revenue accruing from an audit of taxpayer i, allowing for both direct and indirect e¤ects (R i = D i + i ). 14 As we …nd multiple such solutions for the p i , we choose the one corresponding as closely as possible to an observed level of steady-state evasion of 20 percent, as is broadly consistent with the empirical evidence for developed countries cited in the Introduction. 15 As the memory parameter, & R , governs -independently of the other parameters -the rate of convergence of evasion back to its steady-state level, it may be calibrated using a growing body of empirical evidence (Gemmell and Ratto, 2012;DeBacker et al., 2015DeBacker et al., , 2017Advani et al., 2017;Mazzolini et al., 2017) that own indirect e¤ects persist for approximately four years after an audit. Figure 1 shows the temporal pro…le of the deviation in evasion from its steady-state level following an audit. It may be seen that, when interpreting periods as years, for & R = 0:9 the model predicts a speed of convergence that matches closely the empirical evidence. A level of …ne f 2 [1:5; 2] is broadly consistent with practice in the UK and US, and we set f = 1:75. As an approximation to income tax schedules in many developed countries, we set = 0:3.
Values for the parameters fa; b; s g cannot be chosen with reference to an existing empirical evidence base. As such, the values we use in the reported results -fa; b; s g = f2; 80; 1:5gare intended only to be interpreted as illustrative. We have, however, examined the model over a range of parameter values consistent with the conditions in Proposition 1. Although, for brevity, we do not report the results of these additional simulations, they suggest that our qualitative …ndings are not a¤ected by the precise settings of these parameters.
14 This is an objective setting of R i in the sense that $1 of direct revenue is valued equivalently to $1 of indirect revenue. We note as an aside, however, that tax authorities often place greater weight on direct revenue than on indirect revenue, for the former is tangible, whereas the latter is much less tangible: it must be inferred against a counterfactual that is never actually observed. 15 Solutions for the p vector are obtained numerically to an arbitrary degree of precision using the augmented Lagrange multiplier method with a sequential quadratic programming (SQP) interior algorithm (see, e.g., Bertsekas, 1982).
Using the parameter values speci…ed above, the empirical observation that tax evasion is increasing in income permits us to place a lower bound on h , which measures the concern for habit. As may be veri…ed by di¤erentiation in Proposition 1, optimal evasion is linear in X i -and thereby in W i -and the gradient of this linear relationship is itself linearly increasing in h . Figure 2 depicts the gradient @E SS i =@W i for di¤erent values of h ; @E SS i =@W i > 0 for h > 0:9995. Satisfying this lower bound, we choose to align h with s by setting h = 1:5 in the main results. 16

Results
To compute an estimate of R (G), as de…ned in (8), we simulate the model in the manner described above. As, however, R (G) is de…ned for a single G it has the potential to be misleading when the realized G is unrepresentative of the generative process. Moreover, computation of R (G) is su¢ ciently burdensome to make its calculation for many di¤erent G infeasible. To mitigate this source of variability, therefore, we instead form G as the average of multiple independent iterations of the generative process. 17 We then obtain R (G) = 6:01: Our result in (9) is that knowledge of the network increases audit revenues by six percent.
Thus, in a way we have made precise, knowledge of the structure of social networks is of value to tax authorities.
How does the estimate in (9) respond as a function of s , the weight attached to social comparison? Figure 3 depicts estimates for R (G) as s is varied on the interval [0; 3]. For s = 0 the evasion decision is based entirely on private information, and the network plays no role. In this case the tax authority can predict evasion perfectly even when not observing the network (implying R (G) = 0). As s is increased above zero taxpayers exhibit increasing concern for social comparison, with the implication that network information increases in value, for it becomes more critical to predicting patterns of evasion across taxpayers.
It is also of interest to understand the systematic e¤ects of network structure. Figure 4 depicts the estimates for R (G) as is varied on the interval [0; 1]. Note that regulates the importance of node-degree (preferential attachment) in the network formation process: high values of produce networks with a highly concentrated distribution of links, implying the existence of a small number of extremely visible taxpayers. It is seen that as the concentration of links is increased the value of network information decreases monotonically. This …nding might appear counterintuitive at …rst glance, for the level of heterogeneity between the most and least visible taxpayers in the network grows with , seemingly making network e¤ects more pronounced. The widening gap at the extremes of the distribution of links masks, however, the reality that the variability across taxpayers in reference consumption actually diminishes with . As increases a growing proportion of taxpayers have the same few celebrity taxpayers in their reference group. As everyone is, by degrees, comparing to the same set of taxpayers, the amount of variability in the social information obtained by each taxpayer decreases. This reduction in variance permits a tax authority to target audits e¤ectively, even when not observing the network, thereby reducing the value of obtaining network information.

Conclusion
Tax evasion is estimated to cost governments of developed countries up to 20 percent of income tax revenues. We link the tax evasion decision with a large literature on the role in individual decision-making of self and social comparison. Previous studies have focused on only one of these forms of comparison, and social comparison has been restricted to comparisons at the aggregate, rather than local, level. Moreover, the network structures that have been employed in these models possess few of the topological properties of observed social networks.
In this paper we have sought to provide an analysis that addresses these issues. Taxpayers compare their consumption with others in their social network, and also to their own consumption in the recent past. In making social comparisons, each taxpayer makes "local" comparisons on their part of the social network. Engaging in tax evasion is a tool by which taxpayers can seek to increase their consumption relative to others, and relative to their own prior consumption. In this setting, we show that a linear-quadratic speci…cation of utility yields a unique solution for optimal evasion corresponding to a weighted Bonacich centrality measure on a social network: by this measure, taxpayers that are more central in the social network evade more.
Our model provides a rich framework for understanding how a variety of variables, some under the control of tax authorities, will in ‡uence evasion behavior. Although optimal evasion depends in quite a complex way on the underlying parameters, we are able to sign unambiguously its comparative statics for all but one of the exogenous parameters. Allowing for self comparison makes ours a dynamic model in which audits have long-lasting impacts.
This permits a rich decomposition of the direct and indirect e¤ects of an audit. Although the precise value of indirect e¤ects is algebraically complex in our model, we show that a relatively simple Bonacich centrality measure is ordinally equivalent to the vector of individual indirect e¤ects. We believe the techniques developed in this regard may be used to understand more generally the predictions of a range of related social network models.
We show that network information allows a tax authority to better predict the likely revenue bene…ts from conducting an audit of a particular taxpayer. We obtain numerical estimates of this e¤ect using a social network that permits, in particular, the existence highly-observed "celebrity" taxpayers belonging systematically to the upper tail of the income distribution. Although our precise numerical estimates must be treated with care due to uncertainty over some of the parameter values of the model, our results point to an important role for network e¤ects: audit revenues are increased by around six percent when network information is known, relative to when it is not.
We …nish with some possible avenues for future research. First, the comparative statics exercises we have performed are by no means exhaustive: it would, for instance, also be of interest to investigate systematically the e¤ects of adding or removing links within the social network. Second, while we have focused on tax evasion, it seems possible to extend the model to consider tax avoidance behavior, or indeed criminal activity more generally. While these extensions must await a dedicated treatment, we hope our contribution at least clari…es the role of self and social comparison in driving tax evasion behavior on a social network.
Proof of Proposition 1. Under linear-quadratic utility equation (3) can be solved to give optimal evasion at an interior solution as , is positive by the assumed restrictions on a. Using (4), optimal evasion in (A.1) is written in full as where necessarily 1 p i f > 0 at an interior optimum. As the right side of (A.5) is negative, a su¢ cient condition for (A.5) to hold is that @E SS i @z @X i @z 0, which is the condition given in the Lemma.
Proof of Proposition 2. We begin by …rst computing the sign of @E it =@z, where z is a placeholder for each variable given in Proposition 2. Observe that E it and E jt (j 6 = i) are complementary actions. We have With this result we are able to utilize the theory of monotone comparative statics. In particular, we establish globally the sign of the derivative @ 2 E (U it ) = [@E it @z] for each exogenous variable z. It then follows that if @ 2 E (U it ) = [@E it @z] 0 for all i, then @E it =@z 0, and if @ 2 E (U it ) = [@E it @z] 0 for all i, then @E it =@z 0. Di¤erentiating in (1) we obtain The exception is the exogenous variable a, for which we show that @ 2 E (U it ) = [@E it @a] is signed locally to an interior equilibrium. Under a set of regularity conditions -that utility is C 2 and concave, U (:) > 0 for positive values of the argument, and that the problem has a unique solution that obeys the …rst order conditions and varies smoothly with the variable of interest (a here) - Quah (2007, p. 420) shows that signing @ 2 E (U it ) = [@E it @a] local to the (unique) interior maximum is su¢ cient to determine the equilibrium sign of @E it =@a. As these regularity conditions hold in the current context, we utilize this approach to establish the equilibrium sign of @E it =@a. We obtain We now utilize Lemma 1. The variables z 2{a; b; f; p i ; p j ; X j ; h ; s ; h it } satisfy @X i =@z = 0, giving the sign of dE SS i =dz as the sign of @ 2 E (U it ) = [@E it @z] above. For X i we have @X i =@z > 0 and @E SS i =@z < 0, hence Lemma 1 does not apply.

Proof of Proposition 3.
Suppose, in a steady state of the model, that taxpayer i is audited at time t = t a , and otherwise no audits take place. In period t a + 1 his/her evasion deviates below the steady-state level, re ‡ecting the consumption shock in period t a . Beyond period t a + 1 the taxpayer's evasion converges back to its steady-state level at a rate determined by & R . These deviations in the evasion behavior of taxpayer i generate a matching pattern of deviation from steady state, followed by uniform convergence, in the evasion of i . As Noting that the second order condition for (A.1) to de…ne a maximum is 2 a f1 + [f 2] p i f g < 0, it follows that i > 0. Using (4), (A.6) is written in full as