Tax evasion, technology shocks, and the cyclicality of government revenues

Tax Evasion, Technology Shocks, and the Cyclicality of Government Revenues Jordi Caballé Universitat Autònoma de Barcelona and MOVE Judith Panadés Universitat Autònoma de Barcelona and MOVE March 7, 2011 Abstract This paper analyzes the behavior of the tax revenue to output ratio over the business cycle. In order to replicate the empirical evidence, we develop a simple model combining the standard Ak growth model with the tax evasion phenomenon. When individuals conceal part of their true income from the tax authority, they face the risk of being audited and hence of paying the corresponding …ne. Under the empirically plausible assumptions that the intertemporal elasticity of substitution exhibits a su¢ ciently small value and that productivity shocks are serially correlated, we show that the elasticity of government revenue with respect to output is larger than one, which agrees with the empirical evidence. This result holds even if the tax system displays ‡at tax rates. We extend the previous setup to generate larger …scal de…cits when the economy experiences a recession. JEL Classi…cation Number: H23, H26, O41 Keywords: Tax evasion, Technology shocks, Growth Financial support to both authors from the Spanish Ministry of Education through grant ECO200909847 and the Generalitat of Catalonia through grant SGR2009-00350 and, to the …rst author from the ICREA Academia program, is also gratefully acknowledged. Correspondence address: Jordi Caballe. Universitat Autònoma de Barcelona. Departament d’Economia i d’Història Econòmica. Edi…ci B. 08193 Bellaterra (Barcelona). Spain. E-mail: [email protected] 1. Introduction In this paper, we follow the approach introduced by Kydland and Prescott (1982) to study the role played by real technology shocks in driving business ‡uctuation. We will focus our analysis on the response of government revenue to technology shocks. The behavior of government revenue over the business cycle has received some attention in the empirical literature of recent years. It is well known that economic recessions tend to reduce the tax revenue and this makes di¢ cult for governments to fund their existing spending programs. Moreover, during expansion periods tax revenue increases and this creates a new additional political pressure on the government to increase public spending. Therefore, the empirical analysis of this question focuses on obtaining estimates of the income elasticity of tax revenue in order to …nd out whether tax revenues exhibit a more than proportional response to output ‡uctuations.1It is important to distinguish between the long-run income elasticity of tax revenue, which shows how revenues will grow over time as permanent income grows, and the short-run income elasticity of tax revenue, which shows how much revenues will ‡uctuate over the business cycle. For instance, Holcombe and Sobel (1997) estimate both the short-run personal income elasticity of tax revenue and the short-run personal income elasticity of the tax base for U.S. states and …nd that on average they are equal to 1.392 and 1.192 respectively.2 Hence, the average elasticity estimate suggests that a one percent increase in personal income should result in a 1.4 percent increase in the tax revenue. Recent studies by Dye and Merriman (2004) and Bruce et al. (2006) provide more accurate estimates that also support the idea that the short-run personal income elasticity of the tax base tends to be larger than one. The main objective of this paper is to provide a theoretical setup that can be consistent with these empirical …ndings. The standard Ak growth model with ‡at tax rates predicts that the government revenue to output ratio remains constant when a technology (or total factor productivity) shock takes place. Under ‡at tax rates, a technology 1See Dye (2004) for a review of this literature. 2Researchers distinguish between two tax measures when estimating elasticities: the tax base or the tax revenue. Tax revenue data by the type of tax is easily available for several developed countries but they can embed tax rate changes and this leads to a bias in the short-run elasticity estimator. 1 shock a¤ects symmetrically output and government revenue since government revenue is a constant proportion of output. Therefore, the standard Ak growth model with ‡at tax rates does not o¤er a plausible explanation for the empirical evidence as the value of the short-run income elasticity of tax revenue predicted by the model is equal to one. There are several candidate explanations for the high empirical income elasticity of tax revenue. One obvious explanation consists of dispensing with the assumption of constant marginal tax rates and considering instead a progressive tax schedule on income. Clearly, as the average tax rates increase with income the government revenue will increase more than the aggregate income. Another alternative explanation for the high income elasticity of government revenue relies on the behavioral responses to income shocks. When the economy deteriorates, individuals might increase their savings and reduce consumption, especially of items like durable goods. Then, after the economy starts to recover, they might make some of the purchases that had previously been put o¤ during the recession. If the government collect taxes on consumption, then the previous behavior of consumption along the business results in a high elasticity of revenue. In this paper we provide an alternative mechanism generating the desired pattern of cyclicality of government revenue. This mechanism complements the previous ones since relies exclusively on a di¤erent assumption, namely, the existence of tax evasion under a ‡at tax rate on income. We will show that even this simple tax structure is able to generate an income elasticity of tax revenue larger than one under serially correlated productivity shocks when the value of the inverse of the intertemporal elasticity of substitution (IES, henceforth) is larger than one. Of course, under a progressive tax system our mechanism based on tax evasion will reinforce the previous result and, thus, the government revenue will overshoot even more as a response to a productivity shock. The same can be said if taxes were imposed on other procyclical endogenous variables like consumption. Note that our model displays an income elasticity of government revenue larger than one even for economies having tax systems characterized by ‡at tax rates.3 3In this respect, it should be mentioned that during the last decade some countries made an important reform of their system of income taxation. They replaced their previous progressive tax structure by a pure ‡at tax rate. For instance, Russia, Serbia, Iraq, Slovakia and Ukraine set a ‡at tax rate of 13%, 14%, 15%, 19% and 13%, respectively. 2 In order to endow the standard Ak growth model with tax evasion, we assume that individuals have to choose in each period the amount of income they want to consume and the amount of income they want to evade. When individuals conceal part of their true income from the tax authority, they face the risk of being audited and hence of paying the corresponding …ne. Both taxes and …nes determine individual saving and the rate of capital accumulation. Thus, two types of shocks coexist in this model: the aggregate shock, which is given by changes in the total factor productivity of the economy and the idiosyncratic shock, which is introduced by means of the tax inspection policy. The main result of our analysis says that, when technology shocks are serially correlated, the value of the IES fully determines the behavior of the government revenue to GDP ratio. In particular, when the inverse of IES is larger than one, the government revenue increases more than output in the presence of a positive technology shock. In this case, the elasticity of tax revenue with respect to GDP is larger than one, which is consistent with the aforementioned empirical regularity. Moreover, when either the IES is equal to one or technology shocks are not serially correlated, the unitary elasticity of tax revenue is recovered. The intuition of this result lies in the fact that, when shocks are serially correlated, an increase in current total factor productivity means that the expected total productivity and, thus, the expected return of investment in the next period will be higher. Therefore, saving will increase or decrease depending on the value of the IES. Moreover, under tax evasion, underreporting the true income is also a mechanism that allows individuals to transfer present income to the future. This means that, if individuals decide to save more (less) as a response to a real business shock they will also decide to evade more (less) taxes and this will result in less (more) revenues raised by the government. In the next section we develop the basic dynamic model of tax evasion. In Section 3, we will discuss the implications of a technology shock on the government revenue to GDP ratio. In section 4, we extend our model to cope with the implications for the budget de…cits run by the government. Some …nal remarks conclude the paper. 2. The Model Let us consider a competitive economy in discrete time with a continuum of ex-ante identical individuals who are uniformly distributed on the interval [0;1] :Each indi3 vidual ihas access to a common technology represented by the production function yi;t =Atki;t where At>0is the random total factor productivity (TFP), yi;t is the output per capita of individual iand ki;t is the capital per capita of individual iin period t.4We assume that capital fully depreciates after one period. We assume that the stochastic process of strictly positive TFP shocks fAtgfollow a logarithmic autoregressive process, ln At+1 =ln At+ut+1;(2.1) where 2[0;1] and ut+1 is i.i.d. and normally distributed with zero mean and variance 2:Note that the realization of TFP shocks are the same for all individuals. Therefore, production is exposed to macroeconomic (or non-idiosyncratic) TFP shocks. Output can be devoted to either consumption or investment. After production has taken place, each individual idecides both his consumption ci;t and the amount xi;t of declared income, and then pays the corresponding income tax at the rate 2(0;1) :If he is inspected by the tax enforcement agency, the total amount of unreported income is discovered and the taxpayer has to pay a penalty at the ‡at rate  > 1;which is imposed on the amount of evaded taxes (as in Yitzhaki, 1974).5Inspection of a particular individual is an event that occurs with probability p2(0;1) :We also assume that p < 1in order to ensure positive tax evasion. The amount of output remaining after consumption has taken place and taxes and (potential) penalties have been paid constitutes the capital stock ki;t+1 that is used for production in the next period. Therefore, the budget constraint of an audited individual is Atki;t xi;t  (Atki;t xi;t) = ci;t +ki;t+1; whereas the budget constraint of a non-audited individual is Atki;t xi;t =ci;t +ki;t+1: We assume that the amount of taxes collected by the tax agency is devoted to …nancing government spending that enters into the instantaneous utility of individuals 4See Rebelo (1991) for a model where the Ak production function arises endogenously when physical and human capital are perfect substitutes. In this case the capital stock kembodies both types of capital. 5If the penalty rate were smaller than one, tax evasion would be encouraged by the tax authority. 4 in an additive way. Therefore, the marginal rate of substitution of private consumption between two arbitrary periods is not a¤ected by the level of government spending. Since consumers take as given the path of government spending, the utility accruing from this spending can be suppressed from the consumers’objective function. Individuals are assumed to maximize the following expected discounted sum of instantaneous utilities: 1 X s=0 tEt[U(ci;t+s)] ;(2.2) where 2(0;1) is the discount factor and Et[]is the conditional expectation given the information available at period t. We assume that the instantaneous utility function is isoelastic, U(ci;t) = (ci;t)1 1; where the parameter value plays the usual double role as the value of the (constant) relative risk aversion index and as the value of the inverse of the IES. The amount of unreported income in period tfor each individual iis i;t =Atki;t xi;t : Hence, we can use the previous budget constraints to write the stochastic law of motion of capital per capita as ki;t+1 =8 > > < > > : (1 )Atki;t ci;t (1)i;t;with probability p; (1 )Atki;t ci;t +i;t;with probability (1 p); or, equivalently, ki;t+1 = (1 )Atki;t ci;t +i;thi;(2.3) where hiis a random variable with the following probability function: f(hi) = 8 > > < > > : pfor h= 1 ; 1pfor h= 1; (2.4) for all i2[0;1] :Moreover, the variables hiare independently distributed across individuals. Note that E(hi) = 1 p > 0as we have assumed that p < 1. We de…ne the net true income per capita as ni;t = (1 )Atki;t:(2.5) 5 Then, using (2:3) we can write ni;t+1 as ni;t+1 = (1 )At+1 (ni;t ci;t +i;thi):(2.6) Taking ni;t as the state variable for individual iin period t, and ci;t and i;t as the control variables, the Bellman equation for the stochastic dynamic problem faced by this individual in period tbefore knowing if he is going to be audited or not is V(ni;t) = Max fci;t; i;tg((ci;t)1 1+Et[V(ni;t+1)]);(2.7) where ni;t+1 satis…es (2:6) :It is well known that the value function for this problem is the isoelastic function, V(ni;t) = D 1(ni;t)1with D > 0(see Hakansson, 1970): Therefore, using (2:6) and computing the conditional expectation Et[V(ni;t+1)], the optimization problem faced by a taxpayer with initial after-tax true income ni;t becomes Max fci;t; i;tg((ci;t)1 1+EtD 1[(1 )At+1 (ni;t ci;t +i;thi)]1);(2.8) Di¤erentiating with respect to the control variables ci;t and i;t;we obtain the following …rst order conditions for the previous problem: (ci;t)=DEth((1 )At+1)1(ni;t ci;t +i;thi)i;(2.9) and Et[(1 )At+1 (ni;t ci;t +i;thi)]hi= 0:(2.10) Using the independency between At+1 and hiand the distribution of the random variable higiven in (2:4) ;condition (2:9) becomes (ci;t)=D(1 )1t(1 p) (ni;t ci;t +i;t)+p(ni;t ci;t +(1 )i;t); (2.11) with tEth(At+1)1i; while condition (2:10) becomes (1 p) (ni;t ci;t +i;t)=p(1) (ni;t ci;t +(1 )i;t):(2.12) Solving for ci;t and i;t in the system composed of equations (2:11) and (2:12), we obtain ci;t =tni;t;(2.13) 6 and i;t = (ni;t ci;t);(2.14) where t=1 1 + D(1 )1t(1 p)(1 + )+p(1 (1))1= ;(2.15) and =1p p(1) 1= 1 1+(1) 1p p(1) 1= >0:(2.16) Applying the envelope theorem, that is, U0(ci;t) = V0(ni;t);it must hold that c i;t =Dn i;t :(2.17) Substituting (2:13) in (2:17) and using (2:15) we obtain D=1 1 + D(1 )1t(1 p) (1 + )+p(1 (1))1= : Therefore, solving for Din the previous equation we get D="1 1((1 )1tH)1= # ;(2.18) where H= (1 p) (1 + )+p(1 (1)): Substituting (2:18) into (2:15) ;and using (2:13), and (2:14) ;we get the following consumption and evasion policies: ci;t =h1(1 )1Ht1=ini;t;(2.19) and i;t = (1 )1Ht1= ni;t:(2.20) Note that, when p = 1, we have that = 0 and, hence, H= 1. Therefore, when p = 1;individuals do not evade taxes, i;t = 0 for all i2[0;1] :Moreover, under this full enforcement policy conducted by the tax agency, the optimal consumption policy is the one appearing in absence of tax evasion, ci;t =h1(1 )1t1=i(1 )Atki;t: 7 In order to obtain the value of the aggregate after-tax true income nt+1 in equilibrium, which is given by (2:6) ;we compute nt+1 =Z[0;1] ni;t+1di = (1 )At+1 "Z[0;1] ni;tdi Z[0;1] ci;tdi +Z[0;1] i;thidi# = (1 )At+1 "Z[0;1] ni;tdi Z[0;1] ci;tdi + Z[0;1] i;tdi! Z[0;1] hidi!# = (1 )At+1 [ntct+(1 p)t]; where the third equality follows from the independence between the variables hiand i;t at the beginning of period t; whereas the last equality comes from the law of large numbers for a continuum of i.i.d. random variables, according to which R[0;1] hidi = E(hi)=1p; and from the de…nitions of aggregate consumption ctR[0;1] ci;tdi, aggregate evasion tR[0;1] i;tdi; and aggregate after-tax true income ntR[0;1] ni;tdi. In consequence, as follows from (2:19) and (2:20) ;the aggregate values of consumption and evaded income are ct=1(1 )1Ht1= | {z } t nt;(2.21) and t= (1 )1Ht1= nt= (1 t)nt:(2.22) In order to analyze the e¤ect of a TFP shock on evaded income and on consumption, we must compute the value of t:Given that the random variable ut+1 is normal and thus the technology shock At+1 is log-normal, the conditional expectation t Eth(At+1)1iis equal to t=A(1) texp (1 )22 2:(2.23) The next section discusses the e¤ect of a TFP shock on both the amount of evaded income and the government revenue to GDP ratio. 3. E¤ects of TFP shocks In order to analyze the e¤ect of a technology shock on government revenue to output ratio, we should …rst compute the e¤ect of an increase of the TFP value Aton the evasion to income ratio t=yt:Since aggregate output satis…es yt=Atktand the the 8 (4.3) that (1 + gt)1=A1 t [(1 )H]1= A(1)= t1(1 + (1 p)) exp (1)22 2 so that Et1[1+gt)]1=Et1A1 t [(1 )H]1= A(1)= t1(1 + (1 p)) exp (1)22 2 =A t1e2=2 [(1 )H]1= A(1)= t1(1 + (1 p)) exp (1)22 2 =1 [(1 )H]1= A= t1(1 + (1 p)) exp (13+2)2 2;(4.6) where the second equality comes from the fact that Et1A1 t=A t1e2=2; and the third comes from some straightforward simpli…cation. Therefore, using (4.5) and (4.6), the amount of government spending in date tis Gt=yt1 Et1[1+gt)]1=At1kt1 Et1[1+gt)]1 =At1kt1[(1 )H]1= A= t1(1 + (1 p)) exp (1 3+2)2 2 =A(+)= t1kt1[(1 )H]1= (1 + (1 p)) exp (1 3+2)2 2: Note that the government spending in tdepends on the values of two variables known at1;namely, the capital kt1and the the TFP shock At1: Concerning the e¤ective government spending to GDP ratio in period t, note that Gt yt =Gt (1 + gt)yt1 = (1 + gt)Et1[1+gt)]1= [(1 )H]1= A= t1(1 + (1 p)) exp (13+2)2 2 At[(1 )H]1= A(1)= t1(1 + (1 p)) exp (1)22 2=A t1e2=2 At ;(4.7) where the second equality comes from (4.5) and the third from (4.3) and (4.6). 15 As we have shown in the previous section, the government revenue to GDP ratio can ‡uctuate in each period twith the technological shock Atin the presence of tax evasion (i.e., when p < 1) even if the tax rate remains constant across periods (see (4.9)). Moreover, we have just seen in this section that the government spending to output ratio at talso ‡uctuates with the shock Atas the amount of government spending was decided in period t1: Concerning the …scal de…cit to GDP ratio, we can compute GtRt ytfrom (4:7) and (4:9) :Note from (4:7) that the government spending to GDP ratio strictly decreases with the innovation shock in At:However, the government revenue to output ratio increases (decreases) with the innovation shock in Atif  > 1(<1) when  > 0;while it does not vary with Atif either = 0 or = 1:Therefore, we get the following result: Proposition 4.1. For a given value of At1;the government de…cit to output ratio GtRt ytis decreasing in the value Atof TFP if 1and  > 0. Moreover, the same result holds for all  > 0when = 0. Proof: Note that, if 1and  > 0;then the government revenue to output ratio weakly increases with Atand, since the government spending to GDP ratio strictly decreases with Atfor a given value of At1;the result immediately follows. When = 0;the government de…cit is decreasing since the government revenue to output ratio is not a¤ected by changes in At;while the government spending to output ratio strictly decreases with Atfor a given value of At1: The previous result agrees with the empirical evidence since tell us that, under the empirically relevant case with 1and  > 0;…scal de…cits increase when the current rate of growth is lower than the expected one. Note in this respect that, as we have shown at the beginning of this section, the deviation of the actual rate of growth in period tand its expectation at t1for a given value At1in period t1is fully explained by the realization Atof the TFP in period t: However, for the empirically most implausible case  < 1;if TFP shocks are positively correlated,  > 0;the overall e¤ect on the public de…cit to GDP ratio is ambiguous. In this case the revenue to GDP ratio decreases when there is a positive shock on TFP, which coupled with the decrease in the government spending to GDP ratio, gives raise to an ambiguous e¤ect on the government de…cit to output ratio. 16 Let us …nish this section with some comments about the selection of the tax rate when the amount of government spending is chosen a period in advance. Note that we have bee implicitly assuming in our previous analysis that the selection of tax rates is subject to less discretion than government spending, that is, that tax rates are set for longer periods than the amount of government spending. In fact, we were making the extreme assumption that the value of the tax rate was exogenously given. One way of rationalize this assumption and make it consistent with balanced budget in the long run consists of assuming that the government (or the legislative body) chooses at date 0, before observing any technological shock, the tax rate in order to minimization of the unconditional expected square of the government de…cit to output ratio. Therefore, the objective of the government is to choose the tax rate in order to minimize EGtRt yt2 This target is fully achieved achieved when ERt yt=EGt yt; which according to the government spending objective becomes ERt yt=: (4.8) as, from the law of iterated expectations, E(Gt/yt) = E(Et1(Gt/yt)) = . Combining (3:2) with (3:1) we obtain the government revenue to GDP ratio Rt yt =(1 p)[H(1 )]1= A(1)= texp (1 )22 2!:(4.9) The unconditional expectation (i.e., the expectation at the initial date 0 before observing any realization of the TFP shock) of the previous government revenue to GDP ratio can be easily computed by taking into account the following unconditional expectation: EA(1)= t= exp 2(1 )22 22(1 2)!: Plugging the previous expression in the unconditional expectation of the ratio (4:9) ; we get ERt yt=(1 p)[H(1 )]1= exp 2(1 )22 22(1 2)!exp (1 )22 2! 17 =(1 p)[H(1 )]1= exp (1 )22 21 + 2 (1 2)!: It is immediate to see that the previous expectation is strictly increasing in the tax rate and tends to 1 as converges to 1 and to a negative number when approaches 0. Thererefore, there exists a unique value of the tax rate solving the equation (4.8) for 2(0;1). This is the tax rate that balances the government budget in (unconditional) expected terms and that is kept constant for all periods in our analysis. 5. Final Remarks In this paper, we have shown that, by introducing tax evasion in the standard Ak model growth with ‡at tax rates, it is possible to obtain an elasticity of tax revenue with respect to output larger than one, which agrees with the empirical evidence. Therefore, tax evasion o¤ers by itself an explanation to the high income elasticity of government revenue that complements other explanations relying either on progressive income taxation or on taxes imposed on procyclical variables. we have extended the model to account for the cyclical behavior of …scal de…cits when government has a target concerning the value of its spending relative to GDP. we show that, under a plausible parameter restriction, …scal de…cits become larger in recessions. We have used for our analysis a very simple model of capital accumulation where the static portfolio choice model of tax evasion presented by Allingham and Sandmo (1972) has been extended to a dynamic setup.8In this framework, consumers’decisions about how much income they want to report not only a¤ect their present consumption but also their future consumption. Therefore, the response of consumers to positive TFP shocks a¤ects both the tax evasion decision and government revenue. In this setup, we have shown how the e¤ect of a positive technology shock on the government revenue to GDP ratio is fully characterized by the value of IES parameter when TFP shocks are serially correlated. In particular when the IES exhibits a su¢ ciently small value, a positive technology shock makes individuals to lower more than proportionally their amount of evaded income in order to maintain a smooth path of consumption over time. 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