JEL code: D92, D72, E20, E60, H26, H41,G11, O16, O41

Transcription

1 DEPARTMENT OF ECONOMICS ISSN DISCUSSION PAPER 5/07 CORRUPTION, UNCERTAINTY AND GROWTH Ratbek Djumashev # ABSTRACT Corruption in the publi setor erodes tax ompliane and leads to higher tax evasion. Moreover, orrupt publi offiials abuse their publi power to extort bribes from the private agents. In both types of interation with the publi setor, the private agents are bound to fae unertainty with respet to their disposable inomes. To analyse effets of this unertainty, a stohasti dynami growth model with the publi setor is examined. It is shown that deterministi exessive red tape and orruption deteriorate the growth potential through inome redistribution and publi setor ineffiienies. Most importantly, it is demonstrated that the inrease in orruption via higher unertainty exerts adverse effets on apital aumulation, thus leading to lower growth rates. Key words: Corruption, growth, publi goods, tax evasion, unertainty JEL ode: D9, D7, E0, E60, H6, H4,G, O6, O4 Building, Department of Eonomis, Monash University, Clayton, Vi Ratbek.Djumashev@Buseo.monash.edu.au # I thank Prof. Yew-Kwang Ng for insightful disussions that helped signifiantly in writing this paper. All remaining mistakes are mine. 007 Ratbek Djumashev All rights reserved. No part of this paper may be reprodued in any form, or stored in a retrieval system, without the prior written permission of the author.

2 CORRUPTION, UNCERTAINTY AND GROWTH INTRODUCTION Corruption usually means a deviation from what is onsidered to be normal or required by regulations or law. In other words, orrupt publi offiials distort rules and regulations. They do it to reate and apture rents for themselves. I argue that sine orruption is illegal the outomes of the orrupt deals are assoiated with risk. The orrupt bureaurats may try to reate a system of bribery and graft that is well-defined and aepted (or at least believed to be unavoidable) by the soiety. If it was possible then the burden reated by suh orruption would not be different from the publi setor burden imposed by means of taxation as all agents would be aware about the bribes and whom and when to pay. However, orruption is not something that bureaurat involved in it would display ostentatiously. Corrupt transations are landestine and therefore, risky. In the environment with orrupt bureauray the alloation of government permits and lienses is unpreditable. Therefore, the private firms output depending on suh permits and lienses are also subjet to unertainty. The another important aspet is that in the environment with highly orrupt and predatory bureauray there is always a risk that a private agent an be framed and extorted bribes by the publi offiials. For example, Polinsky and Shavell (00) studies orruption in the imposition of santions for violations of law and shows that the outomes for the private agents may vary quite signifiantly depending on the regulation and enforement struture and attitudes to risk. Put another way, orruption leads to inreased unertainty for the private agents, as their disposable inome may vary depending on the interations with the publi offiials. In this paper, I would like to investigate how this type of unertainty affets eonomi growth.

3 There is a quite extensive body of literature on the effets of inome volatility on saving and investment. The model introdued by Bewley (977) pioneered studies on growth impat of the labour inome risk. The Bewley-type models fous on the preautionary saving and wealth distribution aused by the labour inome unertainty (Aiyagari, 994; Calvet, 00; Huggett, 997). These studies infer that the labour-inome risk leads to lower interest rates and over-saving in the steady state. The models developed by Angeletos (005), Angeletos and Calvet (005) investigate the impat of idiosynrati risk in prodution and investment. These studies find that in the presene of the prodution unertainty, risk aversion dominates over the preautionary saving behaviour of the agents. The aforementioned Bewley type models are mainly built around the assumption that there is no aggregate unertainty. The analysis of the relationship between risk and growth at the aggregate level is also interesting. A group of related papers by Eaton (98), Gertler and Grinols (98), Grinols and Turnovsky (993) (998) among others address this question. They analyse how risk related to different aggregate variables affets growth. These models predit that volatility and the related risk adversely affet growth. Interestingly, not all empirial studies onfirm the negative assoiation between volatility and growth. At least the earlier findings of Kormendi and Meguire (985) and Grier and Tullok (989) based on ross-ountry data show that higher output volatility is orrelated with higher average growth rate. Analysing the US data, Tsai et al. (997) find no evidene on the relationship between output volatility and growth after aounting for the volatility assoiated with terms of trade, government expenditure, and monetary environment. Contrary to the foregoing results, Ramey and Ramey (995) onlude based on their empirial analysis that a higher maroeonomi volatility is assoiated with lower longterm growth. Aizenman and Marion (993) (999) also argue that there is signifiant orrelation between volatility and private investment. Although there are extensions of Bewley model in this diretion too; for example Krusell and Smith (998) 3

4 Denizer, Iyugun, and Owen (000) state that orruption an be an important fator ontributing to volatility. Campos (00) argue that preditability of orruption is also important and plays signifiant role in determining its growth impat. The idea that the serey stemming out of the illegal nature of orruption imposes an additional burden on the eonomy has been indiated by Shleifer and Vishny (993). It is lear that serey of orrupt transations adds to the unertainty related to dealing with bureauray. This additional risk an ontribute to the overall volatility in the eonomy, sine due to orruption the volatility at the individual level inreases. The eonomi impat of orruption has been studied extensively (Aonia and d Amato, 006; Barelli and Pessôa, Barelli and Pessôa; Barreto, 000; Huntington, 968; Leff, 964; Lui, 985, 996; Mauro, 995, 998, 004; Rivera-Baitiz, 00; Rose- Akerman, 998, 999, 00, 004). However, the literature on orruption has negleted the growth impat of the unertainty reated by orruption. The aim of the paper is to takle this problem. The idea that the institutional struture may ause unertainty has been aptured in the model by Lin and Yang (00) and Eihhorn (006). They investigate a stohasti growth model with the unertainty aused by tax evasion. In this paper I essentially would like to extend their model by broadening the unertainty related to institutions. To analyse the effet of orruption-aused unertainty on the apital aumulation, I employ a stohasti dynami growth model with a publi setor. In the model, I suppose that the orrupt interations between the private and publi setors reate random shoks to private disposable inome. In partiular, the model of Lin and Yang (00) is extended by assuming a prodution funtion with a publi input and inome shoks engendered by the publi setor. Lin and Yang (00) find that the growth rate is always higher for the environment with tax evasion. Their finding hinges on the assumption that publi goods do not affet both onsumption and prodution, and the tax rates are not optimized. Eihhorn (006) shows that the optimizing government hooses higher tax rates and ahieves the same growth rates as if there were no tax evasion, or in other words, tax evasion is neutral to the rate of eonomi growth. 4

5 The unertainty in my model stems out not only from tax evasion but also from the predatory behaviour of the orrupt publi offiials. In my model, orruption takes plae both in tax olletion and publi good prodution. The rationale for this assumption stems from the fat that orruption always involves the publi setor, therefore, the effets of orruption should be onsidered as an integral part of the publi setor- private setor interations. It is a standard approah to aggregate the interations between the publi and private setors by means of two variables. Namely, the net taxes, T and the government spending G. Then it seems reasonable to analyse orruption as distortions reated with regards to these two variables. Based on this rationale we have the following distortions reated by orruption: First, the orrupt tax inspetors oneal tax evasion for the bribes paid by deteted tax evaders. Seond, the orrupt publi offiials abuse the authority given to them by attahing exessive red tape to the publi servies they are supposed to provide. It is supposed that exessive red tape is a set of unneessary proedures that has no produtive value for firms. The firms have to inur the burden of exessive red tape in order to obtain the essential publi servies. The orrupt offiials an derease the exessive red tape for the bribes paid by the firms. As a result, the orrupt offiials apture a part of firms profits. This inome redistribution from the firms to the orrupt offiials effetively imposes an illegal tax on the firms. The degree or extent of orruption depends on the quality of the institutions embodying the publi setor. Contrary to Lin and Yang (00), it is demonstrated that when the produtive inputs provided by the publi setor aounted for, tax evasion leads to suboptimal growth in the deentralised setting. The soial planner an inrease the tax rate to aount for the loss in tax revenue from evasion and thus voids the impat of tax evasion on growth. This result supports Eihhorn (006), though he derived his result in a setting different from mine. Then it is shown that an inrease in orruption via the assoiated 5

6 inome unertainty leads to less apital aumulation. This results in further deterioration of growth. The struture of the paper is as follows: first, the setup of the basi model is desribed, then the impliations based on the optimal solution obtained for the model is analysed. Based on the results, the basi model is extended to the environment with orruption to analyse its growth effets. THE BASIC MODEL Let us start with the simple ase. First, we want to examine the growth effet of tax evasion under the assumption that the government provides an essential produtive input to private prodution as in Barro (990). In this assumption my model is different from the model used in Lin and Yang (00) and Eihhorn (006), for they assume unprodutive publi servies. In addition, we an assume that the probability of detetion of a tax evader is endogenous and depends on the rate of tax evasion. Let us onsider a losed eonomy with ex-ante idential infinitely-lived agents with zero population growth. Eah agent has a measure of utility defined by a funtion on private onsumption. The utility funtion is given by u () = ln() () Assume that the agents maximize their expeted utility 0 { ρ 0 } U(0) = E ln( ( t)) exp( t) dt () where ρ is the onstant rate of time preferene. Further, when it does not distort the underlying idea we omit the time argument. Assume the prodution funtion has the following form: y a a = Ak g (3) where y is output per apita, A is TFP oeffiient, k is per worker apital, g is per worker publi input. It is assumed that the prodution funtion is stationary within the planning horizon. The government imposes an inome tax at a flat rate τ. To inrease their disposable inome the agents evade taxes by underreporting their true inome. We assume that the 6

7 agent reports only (- ey ) of his total inome y in per apita terms. To ombat tax evasion the government audits taxpayers randomly and depending on the evasion rate detets the evasion. The joint probability of being audited and deteted is given by p. This probability is expressed as p = φe (4) where e < is the tax evasion rate, φ (0,) is the institutional parameter that aptures the effetiveness of the bureauray inluding tax administration. The deteted evader pays bak the due tax liability and some additional fine. This penalty is determined by a penalty rate θ =+s, whih inludes the tax evaded and a surharge for s. The tax paid by the agent is then either T = τ (- e) y+ θτ eywith probability p, or T = (- e) ywith probability q=- p. Consequently, the expeted tax payment for the agent is expressed as T = τ (-e)y+pθτ ey (5) The random part of the agent s inome an be desribed by the return on one unit of tax evaded: with probability p the return equals to r = - s, and with probability q= - p, equals r =. Then the expeted return on a unit of tax evaded is found as r = p (- s) + (- p) = - p( + s) (6) B y denoting θ =+s we re-write (6) as r = - pθ (7) Then the agent s inome is given by y d = ( τ ) y+ ( pθτ ) ey+ w τ (8) where the first two terms stand for the deterministi part of the inome, w = σ eyw is the stohasti part, σ is a onstant, W is defined as a Wiener proess. Later, we will disuss the rationale for this assumption. By introduing a notation τ =τ ( re), the agent s inome an be put as e y d = ( te) y+ wt (9) The variane of the return on tax evasion is given by the following : σ = pqθτ (0) t The derivation is given in Appendix 7

8 It is assumed that r >0 and thus risk-averse taxpayers evade taxes, as the expeted return is positive. Sine in our model all agents are idential ex-ante, all agents evade taxes as soon as the return to tax evasion is positive. Thus, the value of after-tax inome is random and depends on being aught and penalized for tax evasion or being suessful in the at of tax evasion. Assuming that the probability of audit and detetion is random for the taxpayer, we an onlude that this gamble of tax evasion has a binomial distribution. Then we an assume that the deision proess of the taxpayers follows the Markov proess. It is said that a stohasti proess x t has the Markov property if k and t, the probability of outome in period t + is onditional on the state in period t : P ( x x x x Pr x ). r t+ t, t,..., t k) = ( t+ x t That is the future distribution of the states depends only on the urrent state. This assumption is based on the following logi. Let us ask ourselves this question: If the tax evader is aught then should his future behaviour be affeted and is the probability of the detetion in the future for this partiular taxpayer different? Let us try to elaborate on this. Assume that the tax administration reates a blak-list or prior tax-evasion reords of the offenders and audits them with a higher probability in the next year, whih means for those who has not been deteted in the urrent year the probability of detetion dereases even further, so that, they should inrease their evasion. Thus, rational taxpayers should take aount of this strategy employed by the tax administration and should not evade in the following year if aught in the urrent year. Therefore, the ostly audit arried out by the tax administration would turn out in vain, and would be ineffiient, as the other taxpayers not aught in the previous period would inrease their evasion rate in the next period. It is evident, for the tax administration it is an effiient strategy to keep its audit random and with some probability (whih depends on the ost of onduting an audit) equally faed by all taxpayers. That effetively means that the probability of audit and detetion of this year should be independent from the similar events in other years. 8

9 Based on this logi, the administration path annot be non-stationary. If we assume non-stationarity of the tax administration and taxpayer s behaviour that essentially means that we are assuming that if one aught for tax evasion the taxpayer s propensity to evasion dereases signifiantly forever. Then the government an in fat arry out tax audit of all taxpayers (as we assume a finite number of agents) over some finite number of years and thus signifiantly derease tax evasion or eliminate it for good. Yet, the real life experiene tells us that tax evasion is persistent even in the ountries with very strong administrative mahinery. Based on the Markovian nature of the ontrol variables it an be shown that the umulative hange of the a gent s inome ( yt y0) follows a binomial distribution as we have only two outomes. The random proess, when the number of steps goes to infinity, onverges to a Wiener proess (also known as the Brownian motion). By asertaining that the tax evasion proess is the Wiener proess we open a new avenue for analysis, as it allows us to employ a well-established stohasti dynami optimization approah. Lin and Yang (00) first apply this methodology to apital aumulation problem with tax evasion and ome up with a stohasti model of eah individual s apital evolution. In my ase I follow their approah; however, the first and seond moments of the linear stohasti Itô differential equation are different as I am assuming a different funtional form for the prodution proess. The households in their pursuit of utility maximization fae a resoure onstraint in deiding what part of their inome to onsume and what part to save. As the households are faing stohasti disposable inomes depending on the suess of tax evasion, the amount of apital aumulation also follows a stohasti proess. Based on our assumption that the random part of the disposable inome of the agents follows the Wiener proess, the amount of apital aumulation is given by the following linear stohasti Itô differential equation: dk = [(- τ + rτe) y - ] dt + ( σey) dw () where σ is the standard deviation of the normalized proess of random return on tax evasion, W is the Wiener proess. As in Chang (004); Dixit and Pindyk (994) 9

10 We employ the dynami programming method in dealing with the optimal ontrol problems we fae here. We digress here to desribe the method following Dixit and Pindyk (994).. Dynami programming We start with the desription of theory of dynami programming in an environment with unertainty modelled as a disrete Markov proess and then later show that the theory an be extended to ontinuous-time Wiener proess. Sine we are interested in the deision of the households, we onsider the household hoosing its ontrol variables and affeting its state variables. The urrent value of the state variable k t is known, but all future values kt+, kt+... are random variables. It is assumed that the values of the state variable follow the Markov proess. At time t the household an hoose its ontrol variable value t. The state and ontrol variables at time t determine the household s utility ut( kt, t). The kt and t affet the probability distribution of the future states. The onditional umulative probability distribution of the state in the next period is given by Φ ( k k, ) t t+ t t. The main idea employed in the dynami programming is to split the deision sequene into two steps: the urrent period and all the future periods. Assume that the urrent time is t and the state is k t Let us denote by It( k t) the expeted present value of all of the households utility flows, when the household makes all deisions optimally from this point onwards. When the household hooses the ontrol variable t, it obtains urrent utility of ut( kt, t). At the next period ( t + ), the state is determined by k t +. From that point in time onwards the sum of optimal utility flows are given by It+ ( kt+ ). At the urrent period t the households treats it as an expeted value Et[ It+ ( kt+ )]. Dixit and Pindyk (994), pp

11 In the present-value terms, the overall utility is given by uk (, ) + E[ I ( k )]. t t ρ t t + t + + Then the households hooses We write it as to maximize the lifetime utility, whih is equal to I ( k ). t t t It( kt) = max u( kt, t) + Et[ It+ ( kt+ )]. () + ρ The expression () in fat is Bellman s Priniple of Optimality, whih laims that if the poliy is optimal then the remaining hoies represent an optimal poliy with respet to the problem starting at the state that resulted from the initial state due to whatever was the initial ation. In other words, at any given period the agent needs to hoose only the urrent ontrol value optimally. Equation () is also alled the Bellman equation. In a more general form the Bellman equation is written as I( kt ) = max u( k, ) + E[ I( k' k, )] (3) + ρ where instead of and we use k and k' as it ould be any of possible states, and also kt k t + the I( k t ) is written without time index as the value funtion has the same form for all periods, by the virtue of the stationarity property. The Bellman equation an be adjusted for the ase when time is ontinuous. Assume that time hanges in inrements of ukt (,, ). The utility over the time of Δ t is then ukt (,, ) rate per unit of time, the total disounting over the time of Bellman equation then an be written as Δ t. Then the instantaneous utility is given by Δt. Sine ρ is the disount Δ t is given by + ρδ t. The Ik (, t ) = max (,, ) [ ( ',, )] ukt Δ t + ρ t EIk t +Δ t k + Δ Multiplying by (+ ρδ t) and rearranging yields { ρ } { uk ( t t ρ t E I} ρδ tikt (, ) = max ukt (,, ) Δ t( + Δ t) + EIk [ ( ', t+δt) Ikt (, )] = max,, ) Δ ( + Δ ) + [ Δ ] Dividing this equation by Δt and taking the limit with regards to Δt 0, we get

12 ρi( kt, ) = lim ukt (,, ) + EdI [ ] dt However, we need the Bellman equation that reflets the optimal deision proess, when the state variable is defined by the Itô stohasti differential equation as in ( ). Assume that the state variable, per apita apital stok k, follows a Wiener s proess and given by (4) dk = a( k, t) dt + b( k, t) dw (5) Then the Itô s lemma states that for a funtion I( ktwhih, ) is twie differentiable in k and at least one in t the differential di is given by I I I di = dt + dk + ( d t k k Taking into aount (5) we write k ) (6) I I I I di = + a( k, t) + b ( k, t) dt b( k, t) dw + t k k x Applying the Itô s lemma to the value funtion we get (7) I I I EIk [ ( +Δ kt, +Δ t) k, ] = Ikt (,) + + akt (,,) + b(,,) kt Δ t+ o( Δ t) t k k where o( Δ t) stands for the terms that tend to zero faster than Δt. Using this result we an write (4) as ρikt (, ) = max ukt (,, ) + + akt (,, ) + b( kt,, ) t k k I I I The speifi form of akt (,,) and bkt (,,) is found by omparing (5) and () or akt (,,) = ( τ + rτey ) and bkt (,,) = σey. That yield the following state (8) dynamis: dk = [( τ + r τe) y ] dt + ( σey) dw (9). The household s optimiz ation An individual household maximizes its expeted overall utility by hoosing onsumption le vel and tax evasion rate e subjet to the resoure onstraint.

13 maxe, U = ln() exp( ρt) dt (0) 0 s.t. dk = [( τ + r τe) y ] dt + ( σey) dw () 0 ( τ + rτe) y, 0 k, k(0) = k () 0 e (3) This problem is transformed to the Bellman equation as it is explained earlier: ρi( kt, ) = max ln( t ( )) + I'( k) ( τ + rτe) y + I"( k)( σey) (4) { 0 } where I( k) = maxe, E0 ln( ) exp( ρt) dt s.t. (), (),(), (3). In other words, I( k ) is the value funtion, is the onditional expetation operator for the given initial E 0 value of apital, k(0) = k0. The FOC of the Bellman equation (4) leads to t () = I '( k) (5) I'( k) r τ et () = (6) I ''( k) σ y( t) 0 Inserting bak (6) and (5) into (4) we get I () k rτ () ln () ρik = I k τ rτ y + + I () k I () k σ y() t I () k I ( k) rτ I ()( k σ y) I ( k) σ y( t) + = ln I ( k)( ) A( g τ ) I () k k = + [ I ( k)] ( rτ) I ( k) σ (7) A general solution of this differential equation an be expressed in the following ln( k) + C form, Ik ( ) =. A substitution for I( k ) in (7) leads to: ρ ( ) ( g ) τ A ( r Ik ( ) ln( k) k τ ) = ρ + + ρ ρ ρσ (8) However, we know that due to tax evasion the publi input g is given by 3

14 g = τ( re) y = ετy (9) Taking into aount the prodution funtion given by (3), we write By rearranging (30), we obtain By inserting (3) into (8), we get g = τε Ak g (30) g = Aτ ε (3) k ( τ) AA ( τε) ( rτ) Ik ( ) = ln( ρk) + + ρ ρ ρσ Then we an find from (5) and ( 6) that, t () = ρk (3) (33) et) ( rτ = (34) σ A g ( k ) Substituting (33) and (34) into () we obtain the equation of the apital evolution: Dividing ( 35) by k we get ( rτ) rτk dk = ([ τ + ] Ak g ρk) dt + dw (35) σ A g σ k ( ) dk ( ) (( ) g r τ r τ = τ A( k ) + ρ ) dt + dw (36) k σ σ Then the per apita growth rate equals γ e dk ( r ) E ( ) A g τ = = τ ( k ) = + k ρ (37) σ By substituting for g from (3) we get k ( r τ) σ e = ( ) AA ( ) + ρ (38) γ τ τε By analogy, we an show that in the absene of tax evasion the growth rate would be given by 4

15 γ 0 = ( τ) AA ( τ) ρ (39) Comparing (38) and (39), we see that tax evasion does not inrease the growth rate straightforwardly. There is a trade-off between the gain from tax evasion to disposable inome and the loss beause of the lower produtivity due to lower publi input. Therefore, the overall effet is ambiguous and depends on the on what effet is dominant..3 Welfare optimization Assume that government hooses tax rate so that the soial welfare is maximized. Sine, the eonomy populated with the idential individuals, this problem redues to the maximization of the utility of the representative agent. It an be shown that for the urrent setup the welfare maximization is equivalent to maximization of the growth rate of the individual s onsumption. This result is formulated as the following orollary. The proof is given in Appendix 5. Corollary. For the government of the eonomy with idential individuals maximizing log utility funtion and produing with CRTS tehnology, welfare maximization is equivalent to growth maximization. Based on this rationale we onsider growth maximization by government by setting the optimal tax rate for the individuals. Let us start with the Cobb-Douglas ase given by (3). By substituting for σ = eφ( eφ) θ τ, r = eφθ, and p = φe we state the growth maximization problem as: ( eφθ) max γe = ( τ) AA ( τ) ( ( eφθ) e) + ρ τ eφ( eφ) θ (40) Sine the growth rate here is a result of the optimal reation of the taxpayer to the given tax system, tax evasion, e, is not a variable but is a speifi value and thus independent of tax rate. The FOC of (40) gives us + ( τ) A dγ e = A ( ( eφθ ) e) ( τ ) dτ ( ( eφθ) e) ( τ) = 0 (4) Solving this yields 5

16 * τ = (4) This result for the optimal tax rate is the same as we get from the optimization for the non-evasion ase * max γ 0 ( τ) AA ( τ) ρ τ 0 = =, whih is known as the Barro s optimal tax rate. In order to draw general onlusions we need to onsider the problem in a more general setting. For this purpose, we investigate the optimal tax problem for an eonomy with a more general form of the prodution funtion. In presentation, we follow Barro s (990) speifiation..3. CRTS prodution ase Assume that the tehnology is given by a funtion with onstant returns to sale (CRTS), y = F( k, g). Then, using the onstant returns to sale property, we an write y = F( k, g) = kϕ ( g/ k) (43) The government expenditure is given by g = τ y = τkϕ ( g/ k) (44) It is known that the growth rate for suh eonomies is given by γ = [ r ρ], where υ ρ > 0 is a onstant rate of time preferene, > 0 is a onstant intertemporal elastiity of υ substitution. However, the equilibrium interest rate should equal the private return to y apital r = ( τ ). For the prodution funtion given by ( 43) this return is then written k as ( τ y ) = ( ) ( g/ k)[ ( g/ y) ] k τ ϕ ϕ ( 45) Therefore, the growth rate in this environment an be stated as γ = [ ( τϕ ) ( η) ρ ], (46) υ 6

17 where η = τϕ is the elastiity of y with respet to g for given k. The steady-state growth implies that if the tax rate is onstant, then g/ y, g/ k and η are onstant. The optimization proedure leads to the ondition: dγ d = { [ ( τϕ ) ( η ) ρ] dτ dτ υ } d = ( η) ϕ + ( τ) ( ϕ) υ dτ ( τϕϕ ) = ( η) ϕ + υ η = ϕ( η) ( τ) ϕ ϕ υ + = ϕϕ ( ) = 0 υ d ( ϕ) is found by making use of the fat that g/ k = τϕ( g/ k): dτ d dτ d dτ ( ϕ ) = ϕ ' ϕ + τ ( ϕ ) (47) (48) By simplifying, we get d ( ϕ )( τϕ ') = ϕ ' ϕ dτ d ϕ ' ϕ ϕ' ϕ ( ϕ ) = = dτ ( τϕ ') ( η) (49) We inf er from the result in (49) that the growth maximization is attained when ϕ =. Let us repeat this optimization proedure for the ase with tax evasion. Assume that * the equilibrium tax evasion is determined by e = ( ε), where ε is the equilibrium inome-reporting rate. Then we re-write the growth rate as Here g γ e = [ ( ετ ) ϕ ( εη ) ρ] (50) υ = ετyand η = τϕ. The FOC of (50) with respet to τ is found as 7

18 dγe d = { [ ( ετ ) ϕ ( εη ) ρ] dτ dτ υ } d = ( εη) εϕ + ( ετ) ( ϕ) υ dτ ( ετ) εϕ ϕ = ( εη) εϕ + υ εη = ϕε( εη) ( ετ) εϕ ϕ υ + = ϕε( ϕ ) = 0 υ (5) Again, we obtain an optimality ondition given by ϕ = as in the ase with noevasion. Hene, we onlude that the optimality ondition for the publi input is not affeted by tax evasion, and formulate the following proposition: Proposition : Given that all tax revenue is used to produe publi input to private prodution and the effiieny of the publi setor is independent of tax evasion, and the prodution funtion exhibits CRTS, tax evasion has no effet on the optimality ondition for the publi input. Impliation of this proposition is that in order to ahieve the optimal level of the publi input to prodution, the government hooses tax rate in a way that it takes into aount the rate of tax evasion, and thus targets the effetive tax rate. The government s task then beomes to equate the effetive tax rate to the optimal tax rate, as it stems from the optimality ondition we obtained earlier. In both ases, we have the optimality ondition ϕ ' = ϕ ' =, where subsripts n an e stand for no-tax evasion and tax evasion ases respetively. Denoting by g and g the amount of publi input in the environment with no-evasion and with evasion orrespondingly, we write down the equality of optimality onditions ϕ' n gn ge = ϕ' e k k. This leads to the equality of the variables gn = ge, for the given amount of k on the optimal growth path. However, under the assumption of the balaned government budget, we an substitute the publi output with the tax revenue and write n e n e τ y = ετ y (5) n e 8

19 where τ n is the tax rate in no evasion environment, τ e is the tax rate for the environment with evasion. Then we an find the optimal tax rate for the environment with tax evasion. * * τ n τ e = (53) ε As ε < in the presene of tax evasion, the optimal tax rate for the environment with tax evasion must be adjusted upwards in order to equalize the effetive tax rate with the optimal tax rate for the environment without tax evasion. The result is formulated in the following orollary of Proposition. Corollary : Welfare maximizing government sets the optimal tax rate in order to aount for tax evasion, so that the effetive tax rate is equal to the optimal tax rate for the no tax evasion ase. This result is valid under the assumption that tax evasion does not ause any ost exept the unertainty it reates..4 Deentralised equilibrium Based on Proposition, we an also show that tax evasion is not growth enhaning in a deentralised setting. Suppose the government ignores tax evasion and sets the tax rate equal to the optimal rate for the no-evasion environment, then the resulting growth path would be suboptimal. Note that in the presene of tax evasion the effetive tax rate is found as τ = τ( re) (as in(9)). For the sake of simpliity of exposition, we denote e the equilibrium inome-reporting rate as ε = ( re) and write τ e = τ e. Next, we note that the disposable inome in the presene of tax (54) evasion given by y de ( rτ ) = ( τ) A( Aτε) + an also be expressed as σ γ = ( τε ) AA ( τε ) ρ (55) e As both expressions of the growth rate are equivalent, we write the following identity: AA τε τε ρ τ A( Aτε) ρ ( ) ( ) ( ) ( r τ) + (56) σ 9

20 From (56) we infer that growth effet of tax evasion is two-prong: namely, there is a ( r τ) positive impat, Λ=, and a negative impat through lower publi input provision. σ Some manipulation of (56) leads us to Λ= A τ ε τ( ε) (57) The loss is found as a differene of the first terms of (56) and (57) Ζ= ( τ) AA ( τ) ( τ) AA ( τε) = ( τ) A τ ( ε ) (58) Now we an state the overall growth effet of the tax evasion as a sum of these two effets γ τ ε τ ε τ τ ε Δ = Λ Ζ = A ( ) ( ) A ( ) = A τ [ ε τ( ε) ( τ)( ε )] (59) = A τ [ ε ( τε) ( τ)] With all other things being equal, only tax evasion an make the growth rate different for the eonomy with the possibility of tax evasion. Given this ondition what is the tax evasion hoie that maximizes the growth effet? This gives rise to the following: max Δ γ = A τ [ ε ( τε) ( τ)] (60) The FOC of (60) leads to dδγ A τ [ ε = τε ( + )] = 0 (6) dε * A solution of whih by the virtue of the fat that τ = yields * ε = (6) This result tells us that for the given speifiation of the eonomy, when the tax rate is set to the optimal level that is when τ * =, the soially optimal tax evasion rate must be zero. This is not surprising, as any deviation from the optimal government size would lessen the growth rate. 0

21 We an see that if the tax rate were higher than the optimal rate then some positive evasion would be growth enhaning. On the other hand, if the tax rate were less than optimal than tax evasion would only harm growth. However, the private agent takes the amount of the publi servies as given, he ne does not take into aount the adverse effet of his tax evasion on the publi good prodution. Sine the equilibrium tax evasion is non-zero, the deentralized outome should be suboptimal. For the CRTS ase, instead of maximizing the differene in the growth rates, as we did for the Cobb-Douglas ase, we just arry out a growth maximization exerise with regards to tax evasion (whih is the inome reporting rate ε = re tehnially). The optimization leads us to the FOC: γe d = τϕ( εη) + ( ετ) [ ϕ( εη)] = 0 ε υ de (63) Noting that, d dϕ [ ϕ( εη)] = ϕη+ ( εη). (64) dε dε dϕ dϕ and taking into aount that g/ k = ετϕ ( g/ k), we obtain = ϕ'( τϕ+ ετ ). dε dε This leads to d ( εη) [ ϕ( εη)] = ϕη + ϕ, then substituting it into (63) we get dε ( ετ ) γ e ( εη) = τϕ( εη) + ( ετ)[ ϕη + ϕ] ε υ ( ετ) = [ τϕ( εη) ( ετ) ϕη + ( εη) ϕ] ] υ ϕ = [( τ)( εη) ( ετ) η] = 0 υ (65) We reall that for the optimal tax ase ϕ = holds. This means η = τϕ = τ. So, we transform (65) to γe ϕ ϕ = [( τ)( εη) ( ετ) η] = ( ετ)( τ) = 0 ε υ υ The solution satisfying the FOC is ε > as the tax rate is always τ <. However, ε > is not possible by definition, thus we have a orner solution here with optimal inome

22 reporting rate ε =. This is equivalent to saying that tax evasion rate is zero. Therefore, when the tax rate is set to the optimal level the soially optimal tax evasion rate is zero. Again, the atomisti private agent does not take into aount his adverse effet of his tax evasion on prodution and maximizes his private gain from tax evasion, whih leads to positive tax evasion. That is to say that the private agent s probl em is: max Λ = ( eφθ) τekϕ (66) e where we make use of Λ= rτ ey = rτekϕ = (- eφθ) τekϕ. Then the equilibrium value for the tax evasion rate is positive and given by e = φθ < (67) * Therefore, providing that the tax rate is set without aounting for tax evasion, the deentralized equilibrium is soially suboptimal in the environment with more general CRTS tehnologies. By taking aount of tax evasion the government at best ahieves the same growth rate as in the absene of tax evasion. 3 THE ECONOMY WITH CORRUPTION Sine tax evasion may take plae even in the absene of orruption, we need to inorporate orruption into the model speified above. The at of orruption happens when publi offiials attempt to apture rents by abusing the power entrusted to them from the government or the private agents. Corruption an take plae ex ante or ex post. In the ex ante ase, the orrupt bureauray reates a situation, when the publi good is sold at a prie higher than its marginal ost. For that, the orrupt bureaurat either dereases supply of the publi good or bundles it with exessive red tape. In the either ase, the ost of publi good is higher for the private agent. In ase of shortage of the publi good, the private agent pays higher prie for the good as it is desribed by Shleifer and Vishny (993). The other situations desribed by Barreto (000) involves red tape in publi good provision. Note, that in Barreto (000) red tape generally is not a produt of orruption, but rather a feature of any bureauray. Guriev (004) also analyses red tape and orruption, and shows that when the bureauray is orrupt the level of red tape is above

23 the soial optimum. In general, red tape is a type of the publi servie that produes useful information about the private agents. I assume that red tape is just an unprodutive hurdle reated by orrupt offiials. It is assumed that all other useful properties of the publi setor inluding the informative red tape are embodied in the publi goods they provide. In other words, my red tape is only the exessive red tape indued by the orrupt offiials to oere the private agents to pay bribes. The ex post orruption happens only after the interation between the private and publi agents. It usually involves the situation when the publi agent obtains some information about the private agent s failure in law or regulation abidane. Then in order to avoid the penalty for the infringement the private agent is willing to pay bribes to the publi agent. A orrupt publi agent hooses bribes and oneals the infringement. Thus, a orrupt deal ours only if it is benefiial for the both agents. Based on the disussion above we speify the following forms of orruption:. orruption in tax olletion, whih is manifested by onealment of the fat of tax evasion in ase of detetion by a orrupt tax inspetor;. orruption in publi good prodution, whih is manifested by use of publi position to reate exessive red tape (transfer servies) to extort inome from the private individuals. Sine our modelling approah is to inorporate different types of orruption into the basi model we onsidered earlier, we need to disuss how eah suh an addition will ompliate the dynamis of the model. We start with a simple ase of orruption, when the tax inspetors assigned to audit the taxpayers engage in orruption and with some probability oneals the tax evasion for the bribe paid by the tax evader. Then we extend the mo del to inlude orruption in the publi good provision. 3. Corruption in tax olletion We assume the same tax system as in the basi model desribed above. In other words, the probability of detetion of the tax evader is given by p taxpayer should pay fine equal to θτ ey, where θ >. = φe and when deteted the 3

24 An individual taxpayer treats the tax rate, tax audit probability, and penalty rate as given. We are also introduing orruptibility of the tax inspetors onduting the tax audits. We assume that a tax inspetor an be orruptible with probability p. The extent of orruptibility depends on the quality of the institutions or speifially in their effetiveness in ontrolling orruption. Therefore, we an again write that the probability of the tax inspetor being orrupt is a funtion of the quality of the publi institutions: p = p ( φ) Due to orruption the penalty rate beomes random as when audited and deteted a taxpayer may pay bribe instead of tax penalty. In other words, θ in (8) should be adjusted to the following: Where b θ with probabil ity q p θ = = b with probability p In general the bribe rate depends on the bargaining power of the involved private agents, whih again depends on the institutional arrangements. The less the risk for the The expeted value of the random penalty rate then is given by (68) < θ is the bribe rate, so tax evasion osts the bribe paid instead of penalty, if the inspetor is orrupt. tax inspetor to be aught and punished the more bargaining power he wields. E[ θ ] = θ = pb+ qθ (69) Sine 0 p <, the expeted penalty rate is lower when the tax inspetors are orrupt. Given the ontext, for an individual taxpayer being audited and getting a orrupt deal is random. Thus, disposable inome after taxes and audit is also a random variable given by, y d ( τ) y+ ( θ ) τey, with probability p = ( τ) y+ τey, with probability p (70) 4

25 Then the random return on one unit of evaded tax is determined as r = with probability (-p) and r = ( θ ) with probability p. The expeted return to one unit of evaded tax is given by r = eφθ. We notie that if the derease in the effetive penalty ra te is dominates the inrease in tax evasion rate, or r r, the expeted return on evaded tax with orruption is higher than without orruption. In any ase, we onlude that tax evasion inreases with orruption in taxation, as the effetive penalty rate is stritly lower. The variane of the return on tax evasion is found as in the ase for the tax evasion without orruption: σ = pqθτ (7) With orruption in the tax olletion the level of the publi good is defined as g = ετ y (7) With orruption in taxation, ε < re, as a part of the amount paid by the taxpayer is the bribes paid to the tax inpetors. Sine r r and the tax evasion rate is greater, we onlude from (7) that with orruption the amount of publi good produed is less than in the environment without orruption. by For the setting with orruption in taxation, the private disposable inome is expressed y = ( τ + τ re) y+ w τ (73) d We notie that exept the rate of return to tax evasion r r and its variane σ, the disposable inome is found in similar way to the ase without orruption. Using the expression obtained for disposable inome, we an state the transition equation of this dynami system: ( rτ) rτk [ ] t ( dk = τ Ak g ρk + d + dw 4 A g (7 ) k) σ σ 5

26 Analogously with the basi model desribed earlier, we an find the growth rate and the ontrol funtions: ( r τ) = ( ) AA ( ) + ρ (75) σ γ τ τε r τ e = (76) σ A k g ( ) Examining (75), (76), and (77) we infer that the orruption in tax olletion only inreases tax evasion, but does not hange the struture of the model. Consequently, the dynamis of the model does not hange with the introdution of orruption in tax olletion. Therefore, we onlude that Proposition holds for the environment with orruption in tax olletion. = ρk 3. Exessive red tape and deterministi inome extortion It is logial to start the dynami analysis of orruption in publi good provision with deterministi ase, then inorporate the unertainty this type of orruption may give rise. As we disussed earlier, orruption in publi good provision ours through reation of red tape, whih is used to extrat rents from the private agents. Speifially, we suppose that the publi offiials reate exessive red tape and extort a part of private after-tax inome. Effetively, the tax revenue arued to the government is used to produe of the publi input into the private prodution and exessive red tape, whih enables the orrupt offiials to oere the private agents to pay bribes. One may argue that the ost of red tape reation is negligible so should be ignored. So we assume that red tape does not ost anything to produe. Also we should introdue a onept of tehnology for the publi good prodution. After all publi goods are also produed by the human beings and thus the way they organize this prodution proess should determine their output. Sine the publi setor is embodied in different publi institutions, the quality or organizational level of these institutions should play a ruial role in determining the produtivity or the effiieny of the publi setor. We assumed in (4) that the quality of (77) 6

27 the publi institutions an be measured by index φ (0,). The values of this index in the publi good provision ontext an be interpreted as follows: As all publi institutions are established for some purpose, then an institutions with φ = implies that this institution serves its purpose fully. In order to make the setup a bit more general, we an assume that the produtivity of the publi setor depends of the average quality index φ and given by ψ = ψ( φ ) Then depending on the quality (or effiieny) of the publi setor the same amount of tax revenue an be transformed to varying amount of publi goods. This ondition is formulated as the budget balane given by g= ψετ y (78) w here g is the produtive publi input, ψ is the produtivity oeffiient, while ετ y = T. It is also intuitive that the apability of the bureauray to reate exessive red tape depends on the quality of the institutions of the government. In ountries with weak publi institutions, the bureaurats may have plenty of opportunities to be engaged in the rent seeking through exessive red tape, as the possibility of detetion and punishment is low. At the same time, the larger publi setor size also should be assoiated with greater red tape. The reason for that is that the more interations between the government and the private setor the more hanes for the bureaurats to extort bribes from the private agents. In general, the inrease in extortions should not be proportional to the inrease in the government size. Nevertheless, for the simpliity we are assuming this linear dependene as an approximation. Suppose, as a result, the publi offiial by aepting bribes aptures a rent equal to R = R( φ, g, b) (79) Put another way, the rent aptured by the bureauray is related to the rate bribes in taxation, the size of the publi setor and its quality. It is reasonable to assume that an inrease in the size of the government should lead to a proportional inrease in rents 7

28 aptured by the bureauray. The quality of the publi institutions should be embodied in the effiieny of the extortive behaviour of the bureaurats. The bribe rates in all publi setors should be omparable, thus the bribes paid in taxation should affet bribes paid in other publi setor ativities. Let us assume that this effiieny parameter is given by χ= χφ ( ) The higher values of φ imply lower effiieny of the bureaurats in getting bribes. Under this rationale we then assume the following expliit funtional form for the rents of the orrupt bureauray: R = χψετ y (80) where ε = re. This is the effetive quotient for the tax burden, whih inludes the bribes paid by the taxpayers. In other words, we are saying that the rents aptured through extortions are proportional to the gross burden of the publi setor inluding the bribes taken at the taxation stage. Note, only if there is no bribes paid in taxation the gross publi burden equals to the tax revenue olleted by the government. With orruption there is always a wedge between the burden borne by the taxpayer and the revenue olleted by the government. Equivalently, due to orrupt extortions the taxpayer departs with a part of after-tax inome equal to R. That leaves him with the disposable inome given by y = ( τ ) y+ τ rey+ w R d So we see that if we aount for predation of the orrupt offiials then after-tax inome is lower than with tax evasion only. Realling the expression (80) for R we an write: τ y = ( τ ) y+ τrey+ w χψετy (8) d Taking into aount that ε = re and denoting τ τ = τ(+ χψ) (8) we re-write (8) as y = ( τ + τ re) y + w τ (83) d The transition equation by analogy with the previous ase is given by: 8

29 dk ( rτ ) = ([ τ + ] σ A k g ( ) k) dt Ak g ρ + rτ k dw σ (84) This result indiates that the optimization problem has not hanged mathematially as the transition equation is the same as in the basi model speified earlier. So that, when the private agents are subjet to deterministi inome extortions through exessive red tape, the inome unertainty still is generated by tax evasion only. Therefore, the dynami harateristi of the model does not hange. Then the poliy funtions are determined analogously to the tax evasion ase. t () = ρk (85) rτk et () = (86) σ y As the tax evasion proess has not hanged, the expeted returns to tax evasion and its volatility has not hanged too. However, τ = τ(+ χψ)> τ then from (86) we onlude that when the publi setor is predatory the optimal tax evasion rate inreases. The growth rate for this environment is obtained in a similar fashion to the tax evasion ase and given by ( ) [ r τ ] γ = ( τ ) A ψετa + ρ (87) σ Reall that we obtain g Ak = ψετ g, whih yields us g ψετa = k. Substituting it in (87), ( ) [ r τ ] γ = ( τ ) A ψετa + ρ (88) σ The growth rate in suh an environment is lower than if there was tax evasion only. Firstly, the burden imposed by the publi setor is heavier due to red tape. Seondly, less produtive publi input is offered due to greater tax evasion and the waste aused by red tape. In a more general speifiation, we write the prodution funtion as y = F( k, g) 9

30 If we adjust for the distortions aused by tax evasion and orruption, engendered red tape we obtain the following expression for the growth rate: [ r τ] γ = [( τ) Fk( k, g) ρ] υ + σ (89) where g = ψετ y is the government input in the orrupt environment, F ( k, g ) is the marginal produt of apital. k We know that for a well-behaved prodution funtion y = F( k, g ), the growth optimization yields a unique value for the optimal tax rate. The optimal tax rate then mathed with the only optimal publi input value for the given tehnology. The publi input satisfies the ondition g = ψε g0 < g0 as red tape and tax evasion stritly derease the publi input to private prodution. When τ < τ, the growth rate given by (89) is suboptimal, as both the tax rate and the publi input are ineffiient in this ase. If the effetive burden is greater than the statutory burden τ > τ, then private disposable inome is lower. Therefore, the grow th rate in suh a orrupt environment is again stritly lower than in the lean environment. Consequently, in either outome with orruption the growth rate is suboptimal or γ > γ (90) where γ is the optimal growth rate in the lean environment, γ is the growth rate for the orrupt environment. The finding is summarised as follows: Proposition. In the environment with tax evasion, misuse of publi funds, and extortion of private inome by orrupt publi offiials, private disposable inome is lower due to: lower produtivity aused by lower publi input, higher publi burden aused by extortions. The lower disposable inome leads to lower apital aumulation and that in turn - to lower growth. 3.3 Exessive red tape and stohasti inome extortion It is more realisti to assume that the extortion takes plae with some probability. This setting also allows us to see how an inrease in inome unertainty affets growth. With 30