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Tax evasion and the optimal inflation tax Juan Pablo Nicolini a

a,b,)

UniÕersidad Torcuato Di Tella, Minones 2159, (1428) Buenos Aires, Argentina b UniÕersitat Pompeu Fabra, Barcelona, Spain Accepted 11 September 1997

Abstract We developed a simple monetary model to study the effects of tax evasion on the optimal inflation tax. The model is constructed so that inflation might be an indirect way of taxing the underground sector of the economy. We show that while there are theoretical reasons for positive optimal inflation rates, the effects are quantitatively small, even in countries with large underground sectors. We calculate the optimal nominal interest rate for Peru to be between 7% and 19%, despite the fact that its underground sector is close to 40% of measured GNP. According to our calculations, the welfare gain of using inflation to tax the underground sector is also very small. q 1998 Elsevier Science B.V. JEL classification: E31; E52; H26 Keywords: Inflation; Underground economy; Friedman rule

1. Introduction In this paper, we explore the relationship between tax evasion and the optimal inflation tax. We construct a model in which, following Chari et al. Ž1993., in the absence of an underground sector the Friedman rule, i.e., a zero nominal interest rate, is optimal. We then show that if there is an underground sector where cash is used for transactions, the optimal nominal interest rate is positive. A nice feature of our model is that we can easily match the key parameters that determine the

)

Corresponding author.

0304-3878r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 0 4 - 3 8 7 8 Ž 9 7 . 0 0 0 6 3 - 1

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optimal nominal interest rate to observations on real economies. Thus, we can quantify the effect of tax evasion on the optimal inflation tax. The main conclusion we draw from the numerical work is that the effect seems to be very small, even in countries like Peru where, according to the work of de Soto Ž1987., the underground sector accounts for roughly 40% of GDP. The question of the optimum inflation rate has been a central issue in monetary theory since the seminal contribution of Friedman Ž1969.. He argued that the optimal monetary policy, defined as the one that maximizes consumer surplus, is characterized by a nominal interest rate equal to zero. The conditions under which this policy, known as the Friedman rule, is optimal, has been the subject of many studies in the last twenty years. 1 For example, it has been shown that if the tax system is inefficient, in the sense that ordinary taxes are costly to implement, the Friedman rule is not optimal—see Vegh Ž1989. and Aizenman Ž1983.. In these models, the inefficiencies were introduced through ad hoc collection costs functions. A problem with that modeling strategy is that, as there is no clear mapping from the ad hoc cost function to real data, it does not offer a reasonable framework for quantitative evaluations. Thus, these models are unable to say how far apart from the Friedman rule the nominal interest rate should be. In addition, by assuming a single cost function, these models cannot isolate the effect of tax evasion from the effects of, say, tax collection or tax enforcement costs. Our model considers tax evasion only and offers a natural way to quantify its effects on the optimal inflation rates. In our model, there is a continuum of markets with locations indexed by the unit interval. We model tax evasion by assuming that the government cannot be present in a subset of the markets, and therefore cannot enforce the tax laws in those markets. To the extent that cash is used in the underground markets, inflation may be an indirect way of taxing the underground goods. The idea of using inflation to tax the underground economy has already been used by Canzoneri and Rogers Ž1991.. They focused on the cost and benefits of building currency unions, and did not address the issue of the optimal inflation rate. There is no interesting trade-off between inflation and alternative taxes in their model. We do explore more deeply the relationship between tax evasion and the optimal inflation tax. The analysis of the paper is normative; it does not aim at explaining why inflation rates in, say, Peru, were very high during the eighties. Rather, its aim is to determine whether the high degree of tax evasion should be considered in the design of optimal monetary policy when there are alternative ways to finance expenditures. Along the paper, to make the Ramsey problem interesting, we

1

Phelps Ž1973.; Lucas and Stokey Ž1983.; Woodford Ž1990.; Chari et al. Ž1993. and Correia and Teles Ž1994, 1996. are examples of this.

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maintain the assumption that there always exists an alternative tax, and we study the optimal combination between inflation and that alternative tax. According to our computations, the effect of a large underground sector on the optimal inflation rate is rather small. In the most extreme scenario, the optimal yearly nominal interest rate for Peru, with an underground sector of 40%, is 19%. The welfare effects of following Friedman’s rule is, for the highest elasticity considered, slightly above 1% of GNP. The paper proceeds as follows. In Section 2 we briefly discuss the literature on underground economies to motivate the model. In Section 3 we present the model and study efficient allocations. In Section 4 we present the numerical work and discuss its implications. Section 5 contains some concluding remarks.

2. On underground economies In the last decade there has been a renewed interest in the study of underground economies Žsee Tanzi, 1982; Feige, 1989; Dallago, 1990; Hardings and Jenkings, 1989.. These studies provide definitions, estimation methods and estimates of the size of the underground economies for a wide set of countries. There is not a single definition of the underground economy, since there are many different but related concepts attached to it. The term is used to refer to either illegal, unmeasured or unreported activities, depending on the interest of the researcher. As we are interested in the public finance aspect of the phenomenon, we define the underground sector of an economy as all income generating activities which do not comply with the tax obligations. Thus, the object we are interested in includes all illegal activities Žlike drugs and prostitution. and all legal activities that do not pay the corresponding taxes to the fiscal authority. Many indirect estimation methods have been proposed in the literature, each of them stressing a particular feature of the underground sector. The estimates vary considerably as different methods are used. For instance, for the US economy, estimates for the late seventies vary from 4% of GNP ŽPark, 1979. to 27% of GNP ŽFeige, 1989., and for Italy, they vary from 7.5% of GNP ŽContini, 1989. to 30.8% of GNP ŽSaba, 1980.. A very interesting and detailed study of the Peruvian underground sector was done by de Soto Ž1987.. He found that the income generated in the underground sector accounts for 38.9% of registered GNP. Several key patterns of the underground sector emerged from the studies. The first one, is that the range of goods traded in the underground sector varies a lot across countries. For instance, in Austria, the underground activity is concentrated in retail trade, restaurants and hotels; in Germany it is concentrated in construction and car repair; and in the US it is concentrated in home repairs and additions, food, child care and domestic service ŽSkolka, 1987.. In addition, illegal activities contribute a small percentage of the income generated in the underground sector. Feige Ž1989. estimates that unreported income from illegal activities is less that

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15% of total unreported income for the US in 1980. White Ž1987. reports an estimate slightly higher than 10% for 1981. The second pattern of the underground economy, is that the most important reason for tax evasion is the enforcement ability of the fiscal authority, given the institutional constraints it faces, and the technological constraints that the tax evaders face. It is illustrative to mention the existence of whole buildings in Naples, Italy, occupied by a large number of small underground textile factories, which can be quickly hidden when the fiscal inspectors arrive. 2 The third one is that both the formal and the underground sector use cash as well as other means of payment Žsee Feige, 1982., while it is consensual that the underground sector uses cash intensively. Given the above discussion, our model does not identify the underground sector with a specific set of goods in the utility function, as it has been done in the literature Žsee Canzoneri and Rogers, 1991.. In fact, in our model, all goods are identical from the viewpoint of technology, and enter symmetrically on preferences. The underground markets exist because of technological constraints the government faces, which make it impossible to collect taxes. We assume that there is a subset of the markets that the government cannot reach. 3 That subset is identified with the underground sector. We also impose a cash-in-advance constraint to the consumer’s optimal problem, such that there are cash and credit goods in both the underground and the official sectors of the economy.

3. The model There is a representative consumer with preferences over leisure and a continuum of consumption goods indexed by the interval w0, 1x. The utility function is of the form `

Ws Ý b t ts0

1

½H

0

U Ž ct Ž z . . d z y V Ž n t .

5

Ž 1.

where U is increasing and concave, and V is increasing and convex; c t Ž z . is consumption of the good z at time t, and n t is time allocated to work at time t.

2 Skolka Ž1987. makes a more extensive argument about the importance of technological and institutional constraints. Del Boca and Forte Ž1982. provide a detailed discussion of the Italian labor market explaining the technological and institutional characteristics that make clandestine employment attractive. 3 ŽNicolini, J. Unpublished PhD dissertation, University of Chicago, 1991. provides a model that generates monetary equilibria and underground markets by imposing location constraints on agents and the government.

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Consumption goods and government expenditures are produced with labor, according to the resources constraint nt s gt q

1

H0 c Ž z . d z t

Ž 2.

where g t is total exogenous government consumption at time t. A Pareto optimum allocation is a pair c t Ž z ., n t 4`ts0 , that maximizes Eq. Ž1. subject to Eq. Ž2.. It is straightforward to show that consumption must be constant across z’s at the level where the marginal utility of consumption is equal to the marginal utility of leisure. As it is well known, this allocation can be implemented as a competitive equilibrium if lump sum taxes are available. 3.1. CompetitiÕe equilibrium with taxes We assume away lump-sum taxation. Instead, only consumption taxes are available, and leisure cannot be taxed. 4 Thus, government expenditures cannot be financed without distorting the economy. As we mentioned above, we assume that the government cannot enforce the tax laws on some markets, the ones that belong to the underground sector of the economy. We also assume that some of the goods both in the official and the underground sector of the economy can only be traded using cash. We do so by imposing a Clower constraint to the problem of the consumer. We partition the unit interval in the following way:

w 0,1 x s w 0,a x j Ž a,a q b x j Ž a q b,a q b q c x j Ž a q b q c,1 x where a, b and c are positive real numbers such that their sum is lower than one. We assume that all goods in the interval Ž a,a q b q c x must be traded using cash, and that all goods in the interval w0,a q b . are traded in underground markets. Thus, goods in w0,ax, that we call credit-underground goods, pay no tax at all and goods in Ž a,a q b x, or cash-underground goods, pay the inflation tax. Similarly, goods in Ž a q b,a q b q c x or cash–official goods pay both inflation and consumption taxes while goods in Ž a q b q c,1x or credit–official goods only pay consumption taxes. Note that a, b, c and d s 1 y a y b y c represent the relative sizes of the four different types of goods. We also assume that the government issues bonds which pay a nominal interest rate R. This gives the government the possibility to run deficits or surpluses.

4

In this model, consumption taxes and income taxes are equivalent.

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The problem of the consumer is to maximize the utility function ŽEq. Ž1.. subject to Mtq1 q Btq1 q

1

H0 p Ž z . c Ž z . Ž1 q t D t

t

t

U

Ž z . . d z s p t n t q M t q Bt Ž 1 q R t . Ž 3.

for t G 0, M 0 q B0 s W0 and Mt s

aqbqc

Ha

pt Ž z . c t Ž z . d z

Ž 4.

where W0 is the initial stock of nominal assets held by the consumer, t t is the consumption tax rate at time t, pt Ž z . is the money price of good z at time t, pt is the money price of labor at t, Mtq1 is the demand for nominal money to be held in period t q 1, Bt is the demand for bonds at t q 1, and DU Ž z . is an index variable, equal to 0 if z g w0,a q b . and equal to 1 if z g w a q b,1x. This variable is used to impose the restriction that the underground markets cannot be taxed. Eq. Ž3. is the budget constraint and Eq. Ž4. is the cash-in-advance constraint. Note that we are assuming that the consumption tax rate is uniform across types of goods, except for the underground goods that do not pay taxes. In particular, we assume that the government cannot discriminate between cash and credit goods from a fiscal point of view. This seems to be the most realistic assumption, given that the distinction between cash and credit goods, as it is found in the literature, is not based on physical characteristics of the goods, but rather on credit technologies available to consumers which may be unobservable for the government. We will briefly discuss at the end how the solution would be if the government could tax differently cash from credit goods. A Competitive Equilibrium for given sequences of government expenditures, taxes and bonds that satisfy government’s budget constraint is a set c t Ž z .,n t , pt , R t , Mt 4`ts0 such that quantities maximize Eq. Ž1. subject to Eq. Ž3. and Eq. Ž4. and such that the market clearing condition ŽEq. Ž2.. is satisfied. Note that given the technology, it must be true that at any equilibrium pt Ž z . s pt for all z g w0,1x. The first order conditions of the consumer’s problems are U X Ž ct Ž z . . V X Ž nt .

s 1 q DU Ž z . t t q D C Ž z . R t

Ž 5.

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for all t, z g w0,1x, where D C Ž z . is an index variable equal to one when z g w a,a q b q c x, Žcash goods. and equal to zero otherwise Žcredit goods., and V X Ž nt . X

b V Ž n t q 1.

s Ž 1 q R tq1 .

pt pt q 1

Ž 6.

for all t. Eq. Ž5. equalizes the marginal rate of substitution between leisure and all consumption goods to its corresponding relative price and Eq. Ž6. is the standard Fisher. In addition, in any equilibrium, the nominal interest rate must be greater or equal to zero; otherwise, the consumers could make infinite profits by buying money, that pays a zero nominal interest rate, and selling bonds. To avoid competitive equilibria with Ponzi schemes, we follow Chari et al. Ž1993. and impose very large upper and lower bounds on the stock of government debt. As it is clear from the equilibrium conditions, the equilibrium quantities depend on fiscal and monetary policies. Let p t s t t , Mt 4 be the policy at time t, and let p s p t 4`ts0 be the policy plan at time zero. Also, let c t Ž z Žp .. and n t Žp . be the optimal allocation rules for consumers at time t. The optimal policy problem is to choose a policy plan that maximizes the utility of the representative consumer subject to the constraint that the resulting allocation must be a competitive equilibrium. The solution to the optimal policy problem, or Ramsey Equilibrium can be formally stated as follows. A Ramsey Equilibrium is a policy plan p and a sequence of allocation rules c t Ž z Žp .., n t Žp .4`ts0 such that: 1. The policy rule p maximizes Eq. Ž1. subject to Eq. Ž2., the non-negativity constraint on nominal interest rates and the bounds on government debt, given the sequence of allocation rules. 2. The allocation rules are the solutions to the representative consumer’s problem. Formally, the problem is to maximize Eq. Ž1. subject to Eqs. Ž2. – Ž6. plus a non-negativity constraint on nominal interest rates and the bounds on Bt . To verify that this is the case, note that any competitive equilibrium allocation must satisfy Eq. Ž2. to Eq. Ž6. together with the constraint on nominal interest rates and the bounds on government debt. Also, any allocation that satisfies Eqs. Ž2. – Ž6. and the constraints on nominal interest rates and government debt can be decentralized as a competitive equilibrium by properly choosing taxes and money supplies according to the first order conditions ŽEq. Ž5. and Eq. Ž6... Following the literature, the solution concept we use assumes that the government has a commitment technology, such that it can bind itself to the optimal plan at time zero. However, if the initial stock of nominal assets is not zero, the government has incentives to repudiate it, making the price level arbitrarily large or arbitrarily low. To avoid such a solution, we assume, as in Lucas and Stokey Ž1983. and Chari et al. Ž1993. that W0 s 0.

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3.2. The Ramsey equilibrium In order to reduce the dimension of the problem we follow Lucas and Stokey Ž1983., and write the sequence of budget constraints ŽEq. Ž3.. as a single life-time budget constraint `

Ý Qt ts0

1

H0 c Ž z . Ž1 q D

U

t

Ž z . t t q DC Ž z . R t .d z y nt s 0

Ž 7.

where Q t is the real discount factor. If we use Eq. Ž5. and Eq. Ž6. to eliminate prices, we obtain `

Ýbt ts0

1

X

X

H0 c Ž z . U Ž c Ž z . . d z y V Ž n . n t

t

t

t

s 0.

Ž 8.

As we mentioned before, in a competitive equilibrium the nominal interest rate must be larger or equal to zero. Let x u and y u be the generic name for credit and cash goods in the underground sector. Likewise, let x o and y o be the generic name for cash and credit goods in the official sector. From consumer’s first order conditions U X Ž ytu . s U X Ž x tu . Ž 1 q R t . , thus, the non-negativity constraint on nominal interest rates can be written as U X Ž ytu . G U X Ž x tu . .

Ž 9.

Finally, we must also consider the restrictions on fiscal policy, namely, the condition that the credit goods in the underground sector cannot be taxed and the condition that the consumption tax rate levied is the same across all consumption goods. Using the consumer’s first order conditions, these two constraints can be written as V X Ž n t . y U X Ž x tu . s 0,

Ž 10 .

U X Ž ytu . y U X Ž x tu . q U X Ž x to . y U X Ž yto . s 0.

Ž 11 .

Restriction ŽEq. Ž11.. establishes an interesting trade-off between the consumption tax and the inflation tax. The objective of the government is to spread distortions across all possible goods. However, if inflation is used to tax the cash-underground goods, then the cash-official goods will carry a larger fiscal burden relative to the credit-official goods. As the government increases the inflation tax to reduce the distortion between cash-underground goods and credit-official goods, it will be increasing the distortion between cash-official and credit-official goods. The relative importance of the sectors will play a crucial role in the solution.

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Once the optimal quantities have been obtained we can use the equilibrium conditions to construct the supporting policies that decentralize the equilibrium. The monetary policy that supports the sequence of equilibrium interest rates can be obtained from the Fisher Eq. Ž6. and the cash-in-advance constraint ŽEq. Ž4... The Ramsey Problem is then to maximize Eq. Ž1. subject to Eqs. Ž8., Ž2., Ž9. – Ž11.. If we let the corresponding multipliers be l, d t , u t , g t and f t , and we assume, following the literature, that the optimum is interior, the solution must satisfy the following first order conditions U X Ž x tu . Ž 1 q l . q U Y Ž x tu . l x tu q Ž f t y u t y g t . ra s d t ,

Ž 12 .

U X Ž ytu . Ž 1 q l . q U Y Ž ytu . l ytu q Ž u t y f t . rb s d t ,

Ž 13 .

U X Ž yto . Ž 1 q l . q U Y Ž yto . l yto q f trc s d t ,

Ž 14 .

U X Ž x to . Ž 1 q l . q U Y Ž x to . l x to y f trd s d t ,

Ž 15 .

V X Ž n t . Ž 1 q l. q V Y Ž n t . w l n t y g t x s dt .

Ž 16 .

To find a solution, one can solve Eqs. Ž12. – Ž16. plus the last four constraints for consumption quantities, work effort and all time indexed multipliers as functions of l. Then, the value for l is obtained from the implementability constraint. Our focus is on the optimal mix between consumption taxes and the nominal interest rate, rather than on the evolution of taxes over time. 5 Thus, in order to simplify the analysis we assume that government expenditures are constant over time. In this case, the set of Eqs. Ž12. – Ž16. is the same for every t, and the optimal quantities, tax rates and nominal interest rates are constant over time. Thus, from now on, we get rid of the time subscripts. If there were no underground-cash markets, i.e., b s 0, constraint ŽEq. Ž11.. should not be imposed. Thus, the first order conditions of the optimal policy problem would be Eq. Ž14. and Eq. Ž15., but imposing f s 0. In this case, both conditions are the same, so at the optimum the tax on y o is equal to the tax on x o , which means that the optimal nominal interest rate is zero. Thus, in the absence of an underground sector, Friedman’s rule is optimal. This should not be surprising since under this assumption, our model is a special case of Proposition 2 in Chari et al. Ž1993.. Note that for symmetry in preferences, we assume that the utility function is homothetic in cash and credit goods, which is the sufficient condition for the optimality of the Friedman rule established in Chari et al. Ž1993.. In order to study the problem of the optimum quantity of money, or equivalently, the optimal nominal interest rate problem in the case in which the size of 5

The temporal structure of the model is the same as in Lucas and Stokey Ž1983.. No new results regarding temporal issues arise in this model.

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the underground-cash sector is not zero, the multiplier u will be of great use. Note that Kuhn–Tucker conditions imply

u G 0, U X Ž x u . y U X Ž y u . G 0, U X Ž x u . y U X Ž y u . u s 0. We will show that U X Ž x u . y U X Ž y u . ) 0 and u s 0 if government expenditures are positive and if there are cash–underground markets Ž b ) 0.. Thus, the optimal nominal interest rate is higher than the Friedman rule. Proposition 1. At a Ramsey Equilibrium, if goÕernment expenditures are strictly positiÕe (g ) 0) and there exists an underground sector which uses cash in transactions (b ) 0), then the optimal nominal interest rate is strictly positiÕe. Proof. Assume that the optimal nominal interest rate is zero. Then, from the competitive equilibrium conditions x u s y u and x o s y o . Then, using Eq. Ž14. and Eq. Ž15. we obtain that the multiplier f must be equal to zero. Using this in Eq. Ž12. and Eq. Ž13. we obtain

usy

b aqb

g

Using this result in Eq. Ž12. and Eq. Ž16. we obtain

ls

U Y Ž x u . x u y V Y Ž n. n Y

Y

u

u

V Ž n. Ž a q b . y U Ž x . b

.

Ž 17 .

Now, multiplying Eq. Ž12. to Eq. Ž16. by x u , y u , y o , x o , and yn respectively, adding up, and using Eq. Ž17., it is possible to obtain 6

lC s yd g

Ž 18 .

where 2

Cs Ž yo . Ž cqd.UY Ž yo . q

U Y Ž x u . V Y Ž n . Ž n y ax u y by u .

2

V Y Ž n. Ž a q b . y U Y Ž x u .

Note C - 0 because V Y ) 0 and U Y - 0. As g ) 0, condition ŽEq. Ž18.. implies that l ) 0. Therefore, condition ŽEq. Ž17.. implies that u - 0, which contradicts 6

This procedure is the one used by Lucas and Stokey Ž1983..

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the Kuhn–Tucker conditions. This contradiction shows that the optimum interest rate cannot be zero. As it cannot be negative, it must be strictly positive. QED. The fact that b is a very important parameter to determine the optimal nominal interest rate becomes also evident if we assume that b s d, i.e., that the size of the cash–underground sector is the same as the size of the credit–official sector. In this case, Eq. Ž13. and Eq. Ž15. are identical Žremember that u s 0., which means that the optimal values of y u and x o are the same. But this means that the tax rate and the nominal interest rate are the same also. Which is the intuition of this result? Consider the case where the interest rate is zero and all expenditures are financed with a consumption tax. Then, there is no distortion between y o and x o , but a large distortion between y u and y o . On the other extreme, assume that all the expenditure is financed with inflation. There will be no distortion between y u and y o but a large distortion between y o and x o . The government will prefer to put the higher distortion in the smaller sector. If both sectors have the same size, you distribute the distortion equally, which is the result mentioned above. Consider now the case in which the government could tax differently cash and credit goods. 7 Then, constraint ŽEq. Ž11.. should not be imposed, which amounts to setting f equal to zero. In this case, it is clear from the optimal conditions that the solution requires equal consumption for all goods in w a,1x, i.e., all goods except for credit goods in the underground sector. The Žunique. tax structure which supports this allocation is letting the nominal interest rate be equal to the tax rate on credit–official goods, and the tax rate on cash–official goods equal to zero. 8 In this way, the government taxes all goods it can Žall but the credit goods in the underground sector. at the same rate. The fact that there are cash goods implies that the government has the ability to tax a broader set of goods and better smooth the distortions, improving welfare. In particular, if all underground goods where cash, the government could obtain the same allocation as in an economy without underground markets. This is an interesting result, because it shows that imposing a cash-in-advance constraint on consumers increases Žin a weak sense. welfare by allowing the government to partially tax the underground sector, rather than decreasing it Žin a weak sense. like most cash-in-advance models found in the literature. Thus, this result Žwhich can also be found in Canzoneri and Rogers, 1991. could explain why legal restrictions theories of money can be based on a welfare maximizer government. 9 7

The case analyzed by Canzoneri and Rogers Ž1991. corresponds to this case, plus the additional assumption that there are no credit–underground goods and no leisure, so that the allocation is Pareto optimal. 8 Note that since government expenditures are positive, the nominal interest rate will be strictly positive so constraint ŽEq. Ž9.. will not be binding. 9 In ŽNicolini, J. Unpublished PhD dissertation, University of Chicago, 1991., the government finds it optimal to impose legal restrictions on the bonds it issues so that there are monetary equilibria and welfare is improved.

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4. Simulation exercise In this section we calibrate the model and solve it numerically to evaluate the quantitative relevance of the effects discussed in this paper. The model above is quite simple and so are the calibration and simulation exercises we perform. Despite their simplicity, we believe they capture the most important features of the problem we are studying. We calibrate the model for two countries, Peru and USA. Peru has a very large underground sector and we have some reasonable estimates of its size, so it is a natural candidate to study. On the other hand, the US seems to have a fairly small underground sector, and there are reasonable estimates of its size. We use annual data for 1982, which is the period in which the study of the underground sector in Peru was made. 10 The estimates for the underground sector in the USA we use were done around that year as well. We solved the model assuming that the utility function is of the form UŽ c. s

c Ž1y s .

Ž1ys .

, V Ž n. s n

which is a standard specification in applied macroeconomics. Thus, the parameters of the model are a, b, c, g and s . For the value of s in the US, we used 2, a number which is in line with the real business cycle literature. There is no similar tradition for Peru, so we used the value obtained for the USA, and replicated the exercise for alternative values. The results, reported in Table 1, were very robust to variations in s . The other four parameters we need to calibrate are the relative sizes of the four different sectors Ž a, b and c . and the relative size of the government. Thus, the natural observations to use are the size of the underground economy over GNP Ž u., that gives us a notion of a q b, the value of government expenditure over GDP Ž m ., that gives us a notion of g, and the real value of currency Žwe used M 1. over GDP Ž m., that gives us a notion of b q c. To be able to pin down the four parameters we needed to know the relative cash intensity Ž q . of the underground sector relative to the official sector, and there are no estimates for it, except a general consensus that the underground sector is cash intensive relative to the official sector Ži.e., q G 1.. We started by assuming that q s 1 Žthus finding a lower bound for the size of the cash–underground sector. and we repeated the exercise for higher values.

10

The choice of years as the unit period is arbitrary. Most of the estimates of underground sectors are based on yearly data so it seemed a reasonable choice.

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Table 1 Optimal yearly nominal interest rate for Peru

q s1 qs4 q s10

s s2

s s4

s s1r2

9.14% 15.84% 19.05%

9.44% 16.03% 19.17%

7.34% 14.49% 18.42%

The equations used to calibrate the parameters are thus us

ax u Ž t , R . q by u Ž t , R . cy o Ž t , R . q dx o Ž t , R . q g

,

where u is the size of the underground sector, relative to official GNP, m gs cy o Ž t , R . q dx o Ž t , R . , 1ym where m is the share of government expenditures on GDP, ms

by u Ž t , R . q cy o Ž t , R . cy o Ž t , R . q dx o Ž t , R . q g

,

where m is the real quantity of money as a fraction of GNP, b a

sq

c d

,

and finally, the identity 1saqbqcqd where the demand functions can easily be derived from consumer’s first order conditions, given values of R, t and s . The values for t , m, R and m were obtained from the International Financial Statistics. We calculated t by dividing total government revenues by GDP. The values for u were obtained from de Soto Ž1987. for Peru: and Tanzi Ž1982. for the US. As there is some controversy regarding this number for the US Žsee Feige, 1989., we reproduced the exercise using a value for u s 0.1. For both countries we also solved the model assuming q s 4 and q s 10. The base values we used are summarized in Table 2. The results for the optimal nominal interest rate for Peru are reported in Table 1. The base values case corresponds to the upper-left cell. The exercise was repeated changing the values of s and q. As Table 1 shows, the optimal nominal interest rate for Peru, despite having an underground sector which accounts for 40% of GDP is between 9% and 19% a year. The results are not sensitive to the value for s but the value of q seems more critical. As we mentioned before, we do not have quantitative information of

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Table 2 Base values COUNTRY

s

R

t

u

0

m

q

PERU USA

2 2

0.6 0.09

0.15 0.28

0.4 0.05

0.2 0.25

0.1 0.15

1 1

it, except that 1 is a lower bound. Thus, it seems that 10% a year is a lower bound for the optimal nominal interest rate. On the other hand, if one is willing to assume that q s 10 is an upper bound, then 20% is an upper bound on the optimal nominal interest rate per year for Peru. The quantitative effect of tax evasion on optimal inflation does not appear to be large. One of the reasons why this is so in our model is that despite having a large underground sector, and even assuming that it is much more cash intensive that the official sector, the value of money over GDP for Peru is only 10%. Thus, the ability to tax the underground goods through inflation is very limited, because the private sector is not using much cash for transactions. This fact is reflected in our calibration, where the value for the size of the cash–underground sector, b in our model, is only between 3% and 12% of the unit interval, depending on the assumed value for q. These results suggest that massive tax evasion does not justify the very high inflation rates observed in Latin American countries. Table 3 reports the optimal nominal interest rate for the US. As before, the base values case corresponds to the upper-left cell. The exercise was repeated changing u and q. If we use the estimate of Tanzi for the size of the underground sector in the USA Ž u s 0.05., the optimal nominal interest rate is lower than what it was in 1982 Ž0.09%, see Table 2. even if we assume that q s 10. Note that if the real interest rate is around 5% Ž b around 0.95. we need to assume that q is almost 4 to justify positive inflation on the basis of tax evasion. To justify the observed nominal interest rates for the USA in 1982, we need to assume that the underground sector is twice the size estimated by Tanzi, and that it is at least four times more cash intensive than the official sector. Several features of Tables 1 and 3 are interesting. The numbers for Peru may seem somehow small relative to the numbers for the USA, given that the

Table 3 Optimal yearly nominal interest rate for USA

us 0.05 us 0.10

q s1

qs4

q s10

2.50% 4.70%

5.80% 10.00%

8.00% 13.60%

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underground sector for Peru is four times larger than the largest number we used for the US. However, as Table 2 makes clear, there are two other numbers relevant for the problem. The first is total government expenditures. In this model, as government expenditures go up, both consumption taxes and inflation go up. As government expenditures over GDP is higher for the US, the optimal nominal interest rates is also higher. The second number is the value of money relative to GDP. The number for Peru Ž10%. is lower than for the US Ž15%., so the calibrated size of the cash–underground sector is relatively smaller for Peru, and as the theoretical analysis shows, the optimal nominal interest rate is smaller in that case. To conclude, we report the welfare effects of alternative policies. We calculate the percentage increase in income necessary to compensate the consumer for the loss in utility in three different cases. First, we calculate the cost of the tax system Žwe compare the lump-sum case with the Ramsey solution when there is an underground sector.. Second, we calculate the cost of tax evasion Žwe compare the second best assuming that there is no underground sector with the second best when there is underground sector.. Finally, we calculate the cost of following the Friedman rule when there is an underground sector Žwe compare the second best with the Friedman rule when there is an underground sector.. The results for Peru are reported in Table 4 and for the US in Table 5. For the calculations we used the same set of parameters used to compute the numbers in Table 3 and Table 1. The most interesting numbers are the ones in the last row. They measure the welfare cost of following the Friedman rule when there is an underground sector. For the case of Peru, with massive tax evasion, the gain in welfare of optimally using inflation to tax those transactions is fairly small. Only when s s 0.5 and

Table 4 Welfare calculations for Peru q s1

qs4

q s10

s s2 Cost of tax system Cost of tax evasion Cost of Rs 0

0.80% 0.24% 0.02%

0.78% 0.22% 0.05%

0.76% 0.20% 0.07%

s s4 Cost of tax system Cost of tax evasion Cost of Rs 0

0.39% 0.12% 0.01%

0.37% 0.10% 0.02%

0.38% 0.11% 0.03%

s s 0.5 Cost of tax system Cost of tax evasion Cost of Rs 0

3.83% 1.38% 0.21%

3.69% 1.23% 0.71%

3.56% 1.09% 1.22%

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Table 5 Welfare calculations for USA q s1

qs4

q s10

us 0.05 Cost of tax system Cost of tax evasion Cost of Rs 0

1.68% 0.09% 0.00%

1.67% 0.08% 0.01%

1.66% 0.07% 0.02%

us 0.10 Cost of tax system Cost of tax evasion Cost of Rs 0

1.64% 0.16% 0.01%

1.62% 0.14% 0.03%

1.60% 0.12% 0.06%

q s 4 or higher, the welfare effects are over half a percentage point of GDP. For the case of the US, even assuming that the underground sector is twice the estimate of Tanzi, and that it is ten times more cash intensive that the official sector the welfare effect is around one twentieth of a percentage point of GNP. The numbers in the first row of Tables 4 and 5 represent the total cost of the tax system. It measures the welfare loss of distortionary taxes. We report numbers that can be compared with other calculations performed in the literature. In our model, the cost of the tax system for USA is around one and a half percentage points of GDP, which is in line with some static welfare calculations ŽLucas, 1981. and a bit lower than some dynamic calculations ŽChamley, 1981; Lucas, 1990.. It should be noticed though that they are small relative to other dynamic calculations ŽJones et al., 1993; Cooley and Hansen, 1993.. This should not be surprising, since in our model fiscal policy has no growth effects. Finally, the second row calculates the welfare cost of tax evasion. For the case of Peru, the cost can be large when compared to the total cost of the tax system, accounting for roughly a third of it in all the cases we analyzed. For the US however, the cost is much smaller. With the estimates of Tanzi, the cost of tax evasion is less than 5% of the total cost of the tax system. Thus, while fighting evasion in Peru can have significant effects on welfare, it does not seem a rewarding policy in the US, to the extent that one abstracts from the distribution of the tax burden in the population. The model of the paper is a representative agent model and thus abstracts from any distributional issue, the conclusions regarding the effects of tax evasion should be interpreted accordingly. Overall, these numbers suggest that massive tax evasion can be very costly as in the case of Peru, but that it may represent a small cost in a country like the US. However, the ability to reduce that cost by means of the inflation tax seems very limited even for the case of Peru. Thus, fighting tax evasion directly can have an important impact on welfare, but the inflation tax does not seem to be the most convenient way of doing it.

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5. Conclusions We develop a simple monetary model to study the public finance effects of tax evasion. We show that if currency is used in transactions in the markets that evade taxes, the optimal inflation rate is higher than what would be optimal in the absence of tax evasion. We also show that the key parameter to determine the optimal inflation tax is the size of the cash–underground sector relative to the size of the official–credit sector. The key insight of the model is that inflation is an indirect way of taxing the underground sector. Since the model can easily be matched with data on real economies, it can provide quantitative answers to the questions addressed in the paper. We calibrate the model and solve for the optimal inflation tax for a high tax evasion country ŽPeru. and for a low tax evasion country ŽUS.. We also calculate the welfare effects of some policy experiments. The main conclusion we draw is that while there may be theoretical reasons to justify positive inflation rates, the quantitative effect of tax evasion on the optimal inflation rate, even in countries with large underground sectors, is small. In addition, the welfare gain of using inflation to tax the underground sector is also very small.

Acknowledgements This paper is based on chapter II of my PhD dissertation. I would like to thank Robert Lucas, Robert Townsend, Mike Woodford, Andy Atkenson, Santiago Levy, Jorge Roldos, Pedro Teles and two referees for useful comments. All remaining errors are mine.

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