Supplement to ‘The Tax Evasion Social Multiplier: Evidence from Italy’ Roberto Galbiati∗
Giulio Zanella†
January 30, 2012
1
Derivation of linear-in-means equation
The derivation of the linear-in-means equation that links the theory and the empirical analysis, eig = δ0 + δ 1 Xi + δ 2 X g + δ 3
ag + Jeg ng
(1)
uses the following additional equations from Section 2 of the paper: the ad-hoc expression for the probability of an audit,
pig =
ng ag α1 + α0 Pr(yiR < yi | xi ) − Pr(yjR < yj | xj ), ng ng − 1 j=1,j=i
(2)
the linearized expression of the probabilities therein,
Pr(yiR < yi | xi ) = 1 − Pr(yi ≤ yiR | xi ) = 1 − β 0 yiR − β1 xi ,
(3)
the first-order condition from the taxpayer’s problem, ti (1 − pig ) = ti pig f − ∗
∂pig ti (1 + f) yi − yiR . R ∂yi
(4)
CNRS EconomiX and Sciences Po, Paris, France. E-mail:
[email protected] Corresponding author. University of Bologna, Italy, Department of Economics, Piazza Scaravilli 2, 40126 Bologna, Italy. E-mail:
[email protected] †
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and the the large group approximation, (ng − 1)−1
ng
yjR ≃ n−1 g
j=1,j=i
ng
i=1
yiR .
(5)
Replacing (3) into (2) and using (5), we obtain: ag α1 ng + α0 (1 − β 0 yiR − β1 xi ) − (1 − β 0 yiR − β1 xi ) ng ng − 1 j=1,j=i α1 ng ag = + α0 − α0 β 0 yiR − α0 β1 xi − α1 + (β y R + β1 xi ) ng ng − 1 j=1,j=i 0 i ag − α0 β 0 yiR − α0 β1 xi + α1 β 0 y R (6) ≃ (α0 − α1 ) + g + α1 β1 xg , ng
pig =
R where yR g and xg are the group-level averages of yi and xi , respectively. Replacing ∂pig /∂yiR = −α0 β 0 from this equation into (4) and rearranging, equation (4) becomes:
pig + α0 β 0 eig = (1 + f)−1 ,
(7)
where eig ≡ yi − yiR is concealed taxable income. Next, replace (6) into this expression. From the definition of concealed taxable income, we can also replace yiR ≡ yi − eig and y R g ≡ y g − eg , where eg denotes average concealed income in jurisdiction g. Rearranging, we obtain equation (1):
eig =
(1 + f)−1 + α1 − α0 ) 1 β + yi + 1 xi 2α0 β 0 2 2β 0 δ 1 Xi
δ0
α1 α1 β1 1 ag α1 − yg − xg + − + eg . 2α0 2α0 β 0 2α0 β 0 ng 2α0 δ3
δ2Xg
(8)
J
Notice that in this model the taxpayer conceals, ceteris paribus, 50% of any increase of taxable income, ∆y. Linearity plays a crucial role here. When taxable income increases, optimality requires reported income to increase. If pig was constant in equation (7) then the increase would be 100% the increase of taxable income. However, an increase of reported income leads to a decrease of the probability of an audit. Such a decrease is linear by assumption. Therefore, by reporting only 50% of the increase in taxable
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income and concealing the rest (i.e., ∆yiR = ∆ei = 0.5∆yi ), the left-hand side of (7) varies by −α0 β 0 (0.5∆y) − α0 β 0 (0.5∆y) + α0 β 0 (∆y) = 0 and the optimality condition still holds.
2
An additional social effect
We have assumed that enforcement congestion is the only externality generating social effects in tax evasion. In this case the endogenous interactions parameter is given by J = α1 /2α0 , which gives a precise structural interpretation to our estimate. How does this change if one allows for additional social effects? We consider here a simple extension to an endogenous local social norm. That is, we consider the following objective function: 1 2 (1 − pig )ti yiR + pig ti (yi + f yi − yiR ) + η(yiR − yR ν) , 2
(9)
to be minimized with respect to yiR , where y R ν is the average level of income reported in neighborhood ν, i ∈ ν. By neighborhood we mean any geographic aggregation. Objective function (9) is an extension of the one in (??) to the presence of an additive social component of preferences, 2 (1/2)η(yiR −y R ν ) . Any deviation from the average level reported in neighborhood ν–the social norm–implies a utility loss. Parameter η > 0 converts this social utility into tax dollars. There are four possible cases to be considered, in increasing degree of complexity. With some abuse of notation we will use the location indexes g and ν also to denote the respective spatial sets. The simplest case to analyze is ν = g, i.e. the neighborhood that defines the social norm and the R tax jurisdiction coincide. In this case yR ν = y g , and by repeating the steps leading to equation (8) we obtain: J≡
α1 β 0 + η(1 + f)−1 . 2α0 β 0 + η(1 + f)−1
(10)
This expression is more complicated that the corresponding expression when there is no social norm, i.e. η = 0, but the interpretation is not. Now the response to other’s behavior reflects social utility as well. In particular, if we keep assuming α1 < 2α0 , then J increases in η: the stronger the social norm, the more a taxpayer wants to report an amount close to the group average. Also notice that α1 β 0 + η(1 + f )−1 α1 > . 2α0 β 0 + η(1 + f)−1 2α0 3
Therefore, if η > 0 (so that what we identify is the left-hand side of this inequality) while we interpret our estimate as representing the case η = 0 then we are overestimating the contribution of the congestion externality to the social multiplier. In other words, our estimate should be regarded as an upper bound of the short-run effect of enforcement congestion in tax evasion. The second case is ν ⊂ g, i.e. the neighborhood that defines the social norm is a relatively small portion of the tax jurisdiction. In this case we can R R express the neighborhood-level average as yR ν = φy g − (φ − 1)y −ν , where φ = ng /nν is the jurisdiction to neighborhood population ratio, and y R −ν is average income reported in the remaining portion of the tax jurisdiction. R This case is more complicated because y R −ν is a function of yg . If we could R write yR −ν = θyg (i.e., the average in the portion of jurisdiction g outside ν behaves proportionally to the jurisdiction-level average) then we could still obtain a closed-form solution for J: J≡
α1 β 0 + η(φ − (φ − 1)θ)(1 + f)−1 . 2α0 β 0 + η(1 + f)−1
(11)
In this case, whether the parameter we identify underestimates or overestimates the true contribution of the congestion externality depends on whether θ < 1 or θ > 1, respectively. If θ = 1 then the behaviors in portion ν and in jurisdiction g at large are equal and we are back to equation R R (10). In the general case yR −ν = y −ν (y g ) a closed-form solution for the social multiplier in terms of underlying preference and (auditing) technology parameters may not exist. The third case is ν ⊃ g, i.e. the neighborhood that defines the social norm comprises the whole tax jurisdiction. This case is similar to the previous one, with the roles of g and ν inverted. That is, we can write R R yR ν = φy g + (1 − φ)y −g where −g now denotes the set-difference between ν and g. The possibility of deriving the interactions parameter in closedR form depends on the functional form, if any, that relates yR −g and y g . The additional complication is that if no such function exists then we get an extra term in the behavioral equation (1) that expresses the dependence of a taxpayer’s behavior on the behavior of a second reference group, −g, that exerts social pressure but in which the congestion channel is inactive. The last case is ν ∩ g = ∅ but ν and g non-overlapping. In this case we would surely get the dependence of individual behavior on the behavior of a second reference group exerting social pressure only–this group would again be ν g. This leaves us with jurisdiction g containing the remaining portion
4
of ν (i.e. ν ∩g) in full, which is equivalent to the second case analyzed above.
3
Short- vs. long-run multiplier
We emphasize in the paper that the social multiplier that we identify is the upper bound of the long-run effect, and that the latter may be substantially smaller than the short-run one. To further illustrate this point, consider Figure 1, an impulse response diagram for an exogenous shock to individual characteristics, Xi . Figure 1. Range of the dynamic multiplier effect in discrete time
In this figure, the shock is defined such that taxpayers, on average, conceal one extra dollar of taxable income. The figure illustrates the two extreme cases that could arise when time is discrete and resources cannot be adjusted immediately. Case (a) is when resources are never adjusted; hence, the long-run effect of the shock is given by the impact social multiplier γ itself. In this case, the initial effect of the shock is amplified permanently. Case (b) is when resources adjust fully after one period to restore the initial audit probabilities; in other words, the ag increase so that the probability of detection for each taxpayer is exactly the same as before the shock. In this case, the impact social multiplier tells us by how much the equilibrium concealed income increases during the period following the shock; yet after one period there is no multiplier effect. All possible dynamic paths of the multiplier effect are in between these two polar cases–that is, within the shaded region of Figure 1. 5
4
Is ours a random sample?
As mentioned in the paper, our sample is potentially selected: one expects tax cheaters to be oversampled in a collection of tax audits. To illustrate, for e ≥ 0, if we denote by F (e) the distribution of concealed income in the population of taxpayers and by G(e) the distribution of concealed income in the sample of audited taxpayers, then our own model predicts that G(e) first-order stochastically dominates F (e). Thus, audited taxpayers tend to be drawn from tax cheaters, and those who conceal more income are more likely to be drawn. In a sample of tax audits, therefore one should have (relative to the underlying population) relatively few observations with no concealed income and relatively many observations with above-average concealed income. What are the consequences of such a selection process for inference? Unfortunately, the selection criteria applicable to our data set are not public information so we can proceed only by speculating. As illustrated in Section 4 of the paper, the identification method we use requires tax jurisdictions to be units of observation and sets the between- and within-jurisdiction variance of concealed income as dependent and independent variables, respectively. Hence our question is: What is the consequence of this selection process on such variances? To fix ideas, consider the following example. Suppose the taxpayers audited in jurisdiction g were actually drawn from the truncated distribution Fg (e | e > Tg ), where Tg > 0 is some threshold. Although the truncated distribution usually has lower dispersion than the original one, this is not true in general. Therefore, we cannot be sure of what happens to the within-jurisdiction variance in the selected sample. Similarly, although the truncated distributions Fg (e | e > Tg ), g = 1, ..., G, have higher mean than the original ones, the variance of these means across jurisdictions could be larger or smaller than the corresponding variance before truncation. Hence we cannot be sure of what happens to the between-jurisdiction variance, either. Matters become even more complicated when the selection rule is not as clear-cut as in this example. This situation could be interpreted as nonclassical measurement error in both the dependent and the independent variables. Therefore, if our sample were selected then we would incur a bias that is difficult to quantify, a fortiori to correct. We claim that this is not an issue because our sample is not much different from a random sample of the relevant population. We first offer three pieces of evidence in support of this claim and then an interpretation. The first two pieces of evidence are micro in nature: they compare available observables in our data with available data in the underlying population of self-employed individuals. The third piece of evidence will consist of a macro 6
argument. First, if we allow for a 1% error margin in reporting income (i.e., define tax cheaters as those individuals who report less than 99% of their taxable income), then in our data about 63% of taxpayers are cheaters and 37% are not. An honest taxpayer in a collection of tax audits is, ex post, a “mistake” made by part of the tax authority. The high number of such mistakes in our data suggest that, when the data were collected, the tax auditors in Italy were not performing well in terms of selecting likely tax cheaters. The Italian government estimates that 39% of self-employed individuals and businesses reported their taxable income correctly in 2005. This estimate is not based on a sample of audits but rather on the universe of self-employed workers and businesses.1 Although the behaviors reflected by this estimate and our data are almost two decades apart, the 2005 estimate is nearly equal to the corresponding figure in our data. Along this dimension, our data seems to be generated by random audit. Second, we have information on the industry to which each individual in our sample belong. These industries are: agriculture, food, mining, manufacturing, wholesale trade, retail trade, transport and communication, credit and insurance, services, and professional services. We were also able to recover, at the regional level, the corresponding distribution of self-employed workers during the period in which the data were generated. Comparing the two distributions in each region, we conclude that they are statistically indistinguishable. Formally, we perform Kolmogorov—Smirnov tests by region, and we can never reject the hypothesis that the distribution of selfemployed across industries is equal to the distribution of audits by industry. The smallest p-value is 0.83, and most exceed 0.99. One may object that what matters is not the distribution of cheaters or the distribution of audits across types of economic activity but, instead, who is targeted conditionally on being a cheater in a certain sector and to what extent these targeted individuals cheat. Our third and final piece of evidence addresses this objection. Schneider (2005) estimates that, at the end of the 1980s (i.e., when our data were collected), the size of the shadow economy in Italy was 22.8% of GDP. Denote by E the size of the shadow economy and by N the size of the nonshadow economy; Y will denote GDP. Since official GDP estimates in Italy (and elsewhere) include the shadow economy, Y = E + N. Schneider’s estimate is E/Y = 0.228. Now consider the National Income and Product Accounts definition of GDP from the income side, according to which GDP is the sum of employees’ labor income 1
Nicoletta Cottone, “I controlli del 2005.” Il Sole 24 Ore, 19 June 2007.
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(W ), gross operating surplus and gross mixed income (P , which includes self-employment income), and net taxes on production and imports (T ); thus, Y = W + P + T . Transactions involving the government cannot be part of the shadow economy, so T is not part of E. Note also that employees in Italy (and elsewhere) have extremely limited opportunities to cheat on labor income taxes because of third-party reporting. In effect, the only such opportunity is working “off the books”. The Italian statistical office estimates that about 15% of employees were “irregular in Italy” at the beginning of the 1990s (Istat, 2004), which means that all or part of their labor income was not taxed. Because illegal employees tend to work in low—value-added occupations (in Italy, almost half of them are in agriculture and construction), we assume that these 15% (irregular employees) accounted for 10% of the labor income of all employees. This is the part of W that we can attribute to E, so the remaining part of E (i.e., E − 0.1W ) must be included in P . In Italy at he end of the 1980s, W was about 44% of GDP and P was about 47% of GDP; hence, W/Y = 0.44 and P/Y = 0.47. We can now calculate the unreported business and self-employment income in Italy at the end of the 1980s: E − 0.1W E − 0.1W P 0.228 − 0.044 = ÷ = ≈ 0.39. P Y Y 0.47 Self-employed individuals and small businesses normally conceal more income than larger businesses and corporations taken together. For the United States, for example, Slemrod (2007) reports a tax gap of 57% for nonfarm proprietor income and 52% for the self-employment tax–versus 17% for the corporate income tax. We can therefore view 39% as a lower bound on concealed income by self-employed individuals in Italy at the end of the 1980s. If our sample substantially overrepresented tax cheaters, then the ratio of concealed to total income in our sample would be much higher than the lower bound we estimate.2 In our data, this ratio is 43.6%. Moreover, Bernasconi and Marenzi (1997) combine a representative expenditure survey and tax data for year 1991 and estimate that, in Italy, concealed income relative to taxable income is 53% for small businesses and 30% for professionals. Their sample contains about 8 times as many professionals as small businesses, and the income of the former is about 1.7 times that of the latter; hence a back-of-the-envelope calculation suggests that the average concealed income in the aggregate category (i.e., what we define as self2
The relevant denominator is now total income–not taxable income as in Table 1– because we are comparing our sample with GDP, which contains both taxable and nontaxable income.
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employed individuals) is about 48%. These three numbers–a lower bound of 39% from our previous calculation, 48% from Bernasconi and Marenzi, and 46% in our data–are in striking accordance with each other: they suggest that our sample is roughly representative even in terms of the amount of concealed income. In sum, when taken together these three pieces of evidence suggest that our sample is representative with respect to the distribution of cheaters and noncheaters, to the distribution of self-employed workers across industries, and to the overall amount of undeclared income. In other words, they suggest that our sample is not far from a random sample of taxpayers.
5
Derivation of conditional moments
For each group g that comprises Ng individuals in the population in question, we have a random sample of size ng ≤ Ng . Denote with esg the sample mean of concealed income in group g, and manipulate the data in terms of within-group (w) and between-group (b) deviations from the respective means, with the cross-group mean conditioned on observable group-level information, that is W1g and W2g : ng 2 1 1 eig − esg , Ng ng − 1 i=1 s 2 1 1 b Gg ≡ eg − E (eig | W1g , W2g ) − − Ng Gw g. ng Ng
Gw g ≡
(12) (13)
This means that Gw g is simply the within-group sample variance of tax evasion, normalized by population size, while Gbg is the square deviation of group average evasion in the sample from the conditional population mean, minus a correction term to account for the discrepancy between sample and population means. The role of this correction term is made clear below. The purpose of these statistics is to derive an estimator based on the variance of concealed income at different levels of aggregation. Notice that after using b individual-level data to construct Gw g and Gg , the analysis uses only these transformations and (W1g , W2g ), i.e. aggregate group-level data: jurisdictions, not individual taxpayers, are the units of observation. The normalized conditional within-group variance of concealed income, within the jurisdictions defined by (W1g , W2g ), can be computed by taking the conditional expectation of (12). In the following we will use the following equation derived in Section 4.2 of the paper:
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eig = γαg + εi + (γ − 1) εg . Assume, without loss of generality, that E (εi | Ng , ng , W2g ) = 0.3 Then, the conditional mean of concealed income is: E (eig | W1g , W2g ) = γE (αg | W1g , W2g ) . Taking the conditional expectations of Gw g yields:
= = = = =
=
3
E Gw g | Ng , ng , W1g , W2g
ng 1 1 2 (eig − eg ) | Ng , ng , W1g , W2g E Ng ng − 1 i=1
ng 1 1 2 E (γαg + εi + (γ − 1) εg − γαg − εg − (γ − 1) εg ) | Ng , ng , W1g , W2g Ng ng − 1 i=1
ng 1 1 2 E (εi − εg ) | Ng , ng , W1g , W2g Ng ng − 1 i=1
ng 1 1 2 2 ε + εg − 2εi εg | Ng , ng , W1g , W2g E Ng ng − 1 i=1 i ng ng 1 1 E ε2i | Ng , ng , W1g , W2g + E ε2g | Ng , ng , W1g , W2g Ng ng − 1 Ng ng − 1 ng 1 −2 E (εi εg | Ng , ng , W1g , W2g ) Ng ng − 1 2 ng 2 ng ng 1 1 1 E εi | Ng , ng , W1g , W2g + E εi | Ng , ng , W1g , W2g Ng ng − 1 Ng ng − 1 ng i=1 ng ng 1 1 −2 E εi εj | Ng , ng , W1g , W2g Ng ng − 1 ng j=1
Notice that this is an assumption about the theoretical mean. The sample mean, as 1 assumed above, is εg = εi . ng i
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2 ng ng 1 1 1 2 σ + = E εi | Ng , ng , W1g , W2g Ng ng − 1 ε Ng (ng − 1) ng i=1 ng 1 −2 E εi εj | Ng , ng , W1g , W2g Ng (ng − 1) j=1 ng 1 1 1 ng σ 2ε + ng (ng − 1) σ εε = σ 2ε + Ng ng − 1 Ng (ng − 1) ng 2 1 −2 σ ε + (ng − 1) σ εε Ng (ng − 1) ng 1 1 1 σ 2 − σ εε = σ 2ε − σ 2ε − σ εε = ε , Ng ng − 1 Ng (ng − 1) Ng Ng σ 2ε and σ εε . By the law of where we abbreviate σ 2ε (W2g ) and σ εε (W
2g ) with W W iterated expectations E Gg | W2g = E E Gg | Ng , ng , W2g | W2g . That is: 2
w σ ε − σ εε E Gg | W1g , W2g = E | W1g , W2g ≡ Vgw , Ng as claimed in Section 4.2 of the paper.
Similarly, the between-jurisdiction conditional variance can be computed by taking the conditional expectation of (13), as expressed in equation (??): E Gbg | Ng , ng , W2g
1 1 2 W = E (eg − E (eig | ν, ω)) | Ng , ng , W2g − E − Ng Gg | Ng , ng , W2g ng Ng
1 1 2 = V (eg | Ng , ng , W2g ) − E − σ ε − σ εε | Ng , ng , W2g , (14) ng Ng
where V denotes variance. Denote by εsg the sample mean of individual characteristics in group g, as opposed to population mean εg . Then we can write the first term in this equation, i.e. the between-group conditional variance of concealed income, as:
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= = =
=
V (eg | Ng , ng , W2g ) ng 1 V (γαg + εi + (γ − 1) εg ) | Ng , ng , W2g ng i=1 V γαg + εsg + (γ − 1) εg | Ng , ng , W2g γ 2 V (αg | Ng , ng , W2g ) + V εsg | Ng , ng , W2g + (γ − 1)2 V (εg | Ng , ng , W2g ) +2γC αg , εsg | Ng , ng , W2g + 2γ (γ − 1) C (αg , εg | Ng , ng , W2g ) +2 (γ − 1) C εsg , εg | Ng , ng , W2g ng Ng 1 1 2 2 2 γ σα + V εi | Ng , ng , W2g + (γ − 1) V εi | Ng , ng , W2g ng i=1 Ng i=1 ng Ng 1 1 +2γσ αε + 2γ (γ − 1) σ αε + 2 (γ − 1) C εi , εi | Ng , ng , W2g ng i=1 Ng i=1 1 2 ng − 1 1 2 Ng − 1 σε + σ εε + (γ − 1)2 σ ε + (γ − 1)2 σ εε + ng ng Ng Ng 1 2 Ng − 1 σ + σ εε +2γσ αε + 2γ (γ − 1) σ αε + 2 (γ − 1) Ng ε Ng
= γ 2 σ 2α +
= γ 2 σ 2α + 2γ 2 σ αε + (γ − 1)2 + 2 (γ − 1) + 1 σ εε σ2 − σ σ 2 − σ εε εε + ε + (γ − 1)2 + 2 (γ − 1) ε ng Ng σ 2 − σ εε σ 2 − σ εε 2 = γ 2 σ 2α + 2γ 2 σ αε + γ 2 σ εε + ε + γ −1 ε ng Ng 2 1 1 2 2 2 2 2 2 σ ε − σ εε , = γ σ α + 2γ σ αε + γ σ εε + γ + − σ ε − σ εε (15) Ng ng Ng
where we have again, for brevity, omitted the argument of the conditional variances and covariances. Now replace (15) into (14):
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E Gbg | Ng , ng , W2g σ 2ε − σ εε 1 1 2 2 2 = γ σ α + 2σ αε + σ εε + + − σ ε − σ εε Ng ng Ng
1 2 1 − −E σ ε − σ εε | Ng , ng , W2g . ng Ng
Finally, take the expectation of this expression, conditional on W2g : E E Gbg | Ng , ng , W2g W2g | 2
σ 2ε − σ εε σ ε − σ εε 2 2 2 = γ σ α + 2σ αε + σ εε + +γ E | W2g Ng Ng
1 1 2 +E − σ ε − σ εε | W2g ng Ng
1 1 2 −E E − σ ε − σ εε | Ng , ng , W2g | W2g . ng Ng
Applying the law of iterated expectations, the LHS reduces to E Gbg | W1g , W2g , and the last two terms on the RHS cancel out. Therefore this equation reduces to γ 2 σ 2α (W2g ) + 2σ αε (W2g ) + σ εε (W2g ) + Vgw ≡ Vgb ,
as claimed in the paper.
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Some useful properties of the covariance
Three basic properties of the covariance are used in the previous derivations as well as in Section 4.1 of the paper. We summarize them here for convenience. First, for any sequence of n random variables Xi with common variance σ 2X and covariance σ XX :
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n 1 V Xi n i=1
= = =
1 n2
nσ 2X
+2
n
C (Xi , Xj )
i
1 n (n − 1) 2 nσ X + 2 σ XX n2 2 1 2 (n − 1) σX + σ XX . n n
Second, for the same sequence and another random variable Y whose covariance with any random variable in the series is σ Y X , n n 1 1 C Y, Xi = C (Y, Xi ) = σ Y X . n i=1 n i=1 Third, if we extend the sequence to N > n, then:
C
n N 1 1 Xi , Xi n i=1 N i=1
=
n N 1 1 C (Xi , Xj ) C (Xi , Xi ) + n N i=1 i
n
= =
1 1 2 nσ X + (nN − n) σ XX nN 1 2 N −1 σ + σ XX . N X N
References [1] Bernasconi, M. and A. Marenzi (1997). “Gli effetti redistributivi dell’evasione fiscale in Italia”, Bank of Italy, Convegno sulle ricerche quantitative per la politica economica. [2] Istat (2004). La misura dell’economia sommersa secondo le statistiche ufficiali, Anno 2002. Istituto nazionale di statistica, Roma. [3] Schneider, F. (2005). “Shadow Economies Around the World: What do We Really Know?” European Journal of Political Economy, 21, 598-642. [4] Slemrod, J. (2007). “Cheating Ourselves: The Economics of Tax Evasion.” Journal of Economic Perspectives, 21(1), 25-48.
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