tax evasion inequality and progressive taxes a … · 2012-11-08 · tax evasion, inequality, and...

TAX EVASION, INEQUALITY, AND

PROGRESSIVE TAXES: A POLITICAL

ECONOMY PERSPECTIVE

Mark Phoon

Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Economics and Finance

QUT Business School

Queensland University of Technology

2012

i

Abstract

The standard approach to tax compliance applies the economics-of-crime

methodology pioneered by Becker (1968): in its first application, due to Allingham

and Sandmo (1972) it models the behaviour of agents as a decision involving a

choice of the extent of their income to report to tax authorities, given a certain

institutional environment, represented by parameters such as the probability of

detection and penalties in the event the agent is caught. While this basic framework

yields important insights on tax compliance behavior, it has some critical limitations.

Specifically, it indicates a level of compliance that is significantly below what is

observed in the data.

This thesis revisits the original framework with a view towards addressing this issue,

and examining the political economy implications of tax evasion for progressivity in

the tax structure. The approach followed involves building a macroeconomic,

dynamic equilibrium model for the purpose of examining these issues, by using a

step-wise model building procedure starting with some very simple variations of the

basic Allingham and Sandmo construct, which are eventually integrated to a

dynamic general equilibrium overlapping generations framework with heterogeneous

agents. One of the variations involves incorporating the Allingham and Sandmo

construct into a two-period model of a small open economy of the type originally

attributed to Fisher (1930). A further variation of this simple construct involves

allowing agents to initially decide whether to evade taxes or not. In the event they

decide to evade, the agents then have to decide the extent of income or wealth they

wish to under-report. We find that the ‘evade or not’ assumption has strikingly

different and more realistic implications for the extent of evasion, and demonstrate

that it is a more appropriate modeling strategy in the context of macroeconomic

models, which are essentially dynamic in nature, and involve consumption

smoothing across time and across various states of nature. Specifically, since

deciding to undertake tax evasion impacts on the consumption smoothing ability of

the agent by creating two states of nature in which the agent is ‘caught’ or ‘not-

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caught’, there is a possibility that their utility under certainty, when they choose not

to evade, is higher than the expected utility obtained when they choose to evade.

Furthermore, the simple two-period model incorporating an ‘evade or not’ choice

can be used to demonstrate some strikingly different political economy implications

relative to its Allingham and Sandmo counterpart. In variations of the two models

that allow for voting on the tax parameter, we find that agents typically choose to

vote for a high degree of progressivity by choosing the highest available tax rate

from the menu of choices available to them. There is, however, a small range of

inequality levels for which agents in the ‘evade or not’ model vote for a relatively

low value of the tax rate.

The final steps in the model building procedure involve grafting the two-period

models with a political economy choice into a dynamic overlapping generations

setting with more general, non-linear tax schedules and a ‘cost-of evasion’ function

that is increasing in the extent of evasion. Results based on numerical simulations of

these models show further improvement in the model’s ability to match empirically

plausible levels of tax evasion. In addition, the differences between the political

economy implications of the ‘evade or not’ version of the model and its Allingham

and Sandmo counterpart are now very striking; there is now a large range of values

of the inequality parameter for which agents in the ‘evade or not’ model vote for a

low degree of progressivity. This is because, in the ‘evade or not’ version of the

model, low values of the tax rate encourages a large number of agents to choose the

‘not-evade’ option, so that the redistributive mechanism is more ‘efficient’ relative to

the situations in which tax rates are high.

Some further implications of the models of this thesis relate to whether variations in

the level of inequality, and parameters such as the probability of detection and

penalties for tax evasion matter for the political economy results. We find that (i) the

political economy outcomes for the tax rate are quite insensitive to changes in

inequality, and (ii) the voting outcomes change in non-monotonic ways in response

to changes in the probability of detection and penalty rates. Specifically, the model

suggests that changes in inequality should not matter, although the political outcome

for the tax rate for a given level of inequality is conditional on whether there is a

large or small or large extent of evasion in the economy. We conclude that further

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theoretical research into macroeconomic models of tax evasion is required to identify

the structural relationships underpinning the link between inequality and

redistribution in the presence of tax evasion. The models of this thesis provide a

necessary first step in that direction.

v

Abstract ................................................................................................................................ i

List of Tables ................................................................................................................... viii

List of Figures .................................................................................................................... ix

Statement of Original Authorship ................................................................................... xi

CHAPTER 1 ............................................................................................................................... 1

Introduction .......................................................................................................................... 1

CHAPTER 2 ............................................................................................................................. 11

Background and Motivation .............................................................................................. 11

2.1 Introduction ........................................................................................................ 11

2.2 The Theory of Tax Evasion ............................................................................... 12

2.3 A Review of Empirical Models of Tax Evasion ................................................ 16

2.4 Models of Tax Evasion in Macroeconomics ...................................................... 19

2.5 Political Economy/Voting Models of Taxation and Tax Evasion ...................... 21

2.6 Conclusion ......................................................................................................... 24

CHAPTER 3 ............................................................................................................................. 26

The Benchmark Model and Some Simple Extensions ....................................................... 26

3.1 Introduction ............................................................................................................ 26

3.2 Revisiting Allingham and Sandmo Model ............................................................. 34

3.2.1 Theoretical Analysis ...................................................................................... 34

A: The Basic Allingham and Sandmo Model ................................................................. 34

B: The Allingham and Sandmo Model with ‘Evade or Not’ Choice .............................. 38

3.2.2 Numerical Experiments with the Basic AS model and the ‘Evade or Not’

Model 40

3.3 Towards a Macroeconomic Model of Tax Evasion: A Step-by-Step Approach .... 43

3.3.1 AS Model with Two-Periods and its ‘Evade or Not’ Counterpart ................. 43

3.3.1.1 Theoretical Analysis .................................................................................. 43

A: The Allingham and Sandmo Two-Period Model ....................................................... 43

B: The Allingham and Sandmo Two-Period Model with ‘Evade or Not’ Choice .......... 45

3.3.1.2 A Simple Numerical Experiment Two-Period AS Model and ‘Evade or

Not’ Model ................................................................................................................. 47

3.3.2 Two-Period Model with Heterogeneous Agents, Redistributive Transfers and

Vote on θ ........................................................................................................................ 48

3.3.2.1 Theoretical Analysis .......................................................................................... 48

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A: The Allingham and Sandmo Two-Period with Model Heterogeneous Agents and

Redistributive Transfers ................................................................................................. 48

B: The ‘Evade or Not’ Choice Model with Heterogeneous Agents, Redistributive

Transfers and Vote on θ ................................................................................................. 50

3.3.2.2 Numerical Experiments for Inequality ....................................................... 52

3.3.2.3 Political Economy Extensions ................................................................... 57

3.3.2.4 Numerical Experiments for the Political Economy Extension ................... 59

3.3.3 Two-Period Model and Political Economy with Cost of Evasion ................. 67

3.3.3.1 Theoretical Analysis .................................................................................. 67

A: The Allingham and Sandmo Model with Cost of Evasion ......................................... 67

B: ‘Evade or Not’ Choice Model with Cost of Evasion ................................................. 68

3.3.3.2 Numerical Experiments .............................................................................. 69

3.4 Conclusion ............................................................................................................. 78

CHAPTER 4 ............................................................................................................................. 82

On Inequality, Tax Evasion and Progressive Taxes........................................................... 82

4.1 Introduction ............................................................................................................ 82

4.2 The Economic Environment ................................................................................. 85

4.2.1 The Benchmark Economy .............................................................................. 85

4.2.2 The Model with the ‘Evade or Not’ Choice .................................................. 92

4.2.3 Political economy Extensions ....................................................................... 94

4.3 A Further Discussion of Some Theoretical Issues. ............................................... 95

4.4 Choice of Parameters for Numerical Experiments ............................................... 100

4.5 Results of Quantitative Experiments .................................................................... 102

4.6 Brief Discussion on Wealth Dynamics ................................................................ 119

4.7 Concluding Remarks ............................................................................................ 121

CHAPTER 5 ...........................................................................................................................125

Concluding Remarks ........................................................................................................ 125

BIBLIOGRAPHY ..................................................................................................................132

APPENDIX ............................................................................................................................147

Appendix for Chapter 3 ..........................................................................................................147

Appendix 3.1: Comparison of Indirect Utility IUFAS and IUFNE. ................................ 147

Appendix 3.2: Derivation of Conditions for an Interior Solution. ................................... 149

Appendix 3.3: Comparison of Indirect Utility IUFAS and IUFNE Two-Period Model. . 150

Appendix for Chapter 4 ..........................................................................................................151

vii

Appendix 4.1: Derivation of Variables for Expression in terms of Wt and α ................... 151

Appendix 4.2: Proof of Proposition 1 .............................................................................. 154

Appendix 4.3: Results of Experiments - Basic AS Model ............................................... 156

viii

List of Tables

Table 3.1: Number of Evaders for Different Levels of Inequality ............................................. 62

Table 3.2: Vote on θ for Different Levels of Inequality ............................................................ 74

Table 3.3: Number of Evaders for Different Levels of Inequality with Cost of Evasion ............ 79

Table 3.4: Vote on θ for Different Levels of Inequality with Cost of Evasion ........................... 84

Table 4.1: Number of Evaders for Different Levels of Inequality and θ ................................. 111

Table 4.2: Number of Evaders for Different Levels of Inequality and γ .................................. 113

Table 4.3: Vote on or γ: AS Model ........................................................................................ 116

Table 4.4: Vote on or γ : ‘Evade or Not’ Model ..................................................................... 117

Table 4.5: Sensitivity Analysis for p: AS Model ....................................................................... 118

Table 4.6: Sensitivity Analysis for d0: AS Model ..................................................................... 119

Table 4.7: Sensitivity Analysis for ϕ: AS Model ...................................................................... 119

Table 4.8: Sensitivity Analysis for p: ‘Evade or Not’ Model .................................................... 119

Table 4.9: Sensitivity Analysis for d0: ‘Evade or Not’ Model................................................... 120

Table 4.10: Sensitivity Analysis for ϕ: ‘Evade or Not’ Model ................................................. 121

ix

List of Figures

Figure 3.1: Lower bound for Probability of Detection on Tax Rate when π=2θ and π=1.5θ. .. 48

Figure 3.2: Proportion of Unreported Income as the Tax Rate Increases. ............................... 52

Figure 3.3: Comparison of Indirect Utility Functions of AS Model and ‘Evade or Not’ Model. 53

Figure 3.4: Comparison of Indirect Utility Functions of AS Model and ‘Evade or Not’ Model. 58

Figure 3.5: Timeline for the basic model. ................................................................................. 67

Figure 3.6: Timeline for model with ‘evade or not’ choice. ...................................................... 67

Figure 3.7: Agents’ preferences over θ in the AS economy. ..................................................... 68

Figure 3.8: Agents’ preferences over θ in the ‘Evade or Not’ economy. .................................. 69

Figure 3.9: Percentage of votes for various values of θ in the AS economy. ........................... 72

Figure 3.10: Percentage of votes various values of θ in the ‘evade or not’ economy. ............ 72

Figure 3.11: Proportion of unreported income (α) as a function of wealth for θ=0.15 and

Gini=0.3439 ............................................................................................................................. 78

Figure 3.12: Agents’ preferences over θ in the AS economy. ................................................... 81

Figure 3.13: Agents’ preferences over θ in the ‘Evade or Not’ economy. ................................ 82

Figure 3.14: Percentage of votes for various values of θ in the AS economy .......................... 83

Figure 3.15: Percentage of votes in favour of various values of θ in the ‘evade or not’

economy. ............................................................................................................................... 83

Figure 4.1: Timeline for the basic model. ................................................................................. 98

Figure 4.2: Timeline for model with ‘evade or not’ choice. ...................................................... 99

Figure 4.3: Plot of Unreported Income and Indirect Utility Functions ................................... 102

Figure 4.4: Extent of Evasion: Basic Model v/s ‘Evade or Not’ Variant for θ=0.10. ............... 107

Figure 4.5: Experiments with cost function parameter d0 ..................................................... 109

x

Figure 4.6: Experiment with p, the probability of detection. ................................................. 109

Figure 4.7: Experiments with ‘penalty rate’ ϕ. ...................................................................... 110

Figure 4.8: Inequality and the Extent of Evasion.................................................................... 115

Figure 4.9: Agents’ preferences over θ in the ‘evade or not’ economy.................................. 122

Figure 4.10: Wealth Dynamics of ‘Evade or Not’ Model. ....................................................... 123

Figure 4.11: Wealth Dynamics of AS Model.. ........................................................................ 124

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Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet

requirements for an award at this or any other higher education institution. To the

best of my knowledge and belief, the thesis contains no material previously

published or written by another person except where due reference is made.

Signature: _________________________

Date: _________________________

Introduction Chapter 1

1

CHAPTER 1

Introduction

History is replete with examples that suggest that policies and institutions are

endogenous, determined by the preferences of individuals, or groups of individuals,

in an economy. A catalyst for the eventual fall of the Roman Empire in 476 AD, for

example, is said to have been a universal poll tax levied by Constantine. (See for

example Gibbons, 1776-89). David Hume, in the chapter titled ‘Of Taxes’ in

Political Discourses (1752), remarks:

Historians inform us, that one of the chief causes of the destruction of the

Roman State was the alteration, which Constantine introduced into the

finances, by substituting a universal poll-tax, in lieu of almost all the tithes,

customs, and excises, which formerly composed the revenue of the empire.

The people, in all the provinces, were so grinded and oppressed by the

publicans, that they were glad to take refuge under the conquering arms of

the barbarians; whose dominion, as they had fewer necessities and less art,

was found preferable to the refined tyranny of the Romans.

While Hume refers to a case in which public opinion against taxes that were

too high was the catalyst for a change of regime, there have been several instances,

both historical and contemporary, in which public opinion has been in favour of

raising taxes for purposes such as redistribution or the financing of debt. A case in

point is the recent debt crisis in the United States, and various other economies of the

world.


2

The political economy of tax determination is also inextricably linked to

historical resistance towards taxation in the form of deliberate and overt evasion of

taxes. Typically referred to as ‘tax resistance’, tax evasion in these instances is based

on ethical or ideological beliefs, and takes place as a form of protest or rebellion

against policies of the extant regime. Several interesting historical cases are

presented, for instance, in Burg (2004). During the Russian Revolution of 1905-06,

for example, various anti-government coalitions advocated non-payment of taxes to

facilitate the removal of the Tsarist regime. In a more recent case of tax resistance,

the ‘poll-tax’ instituted in 1989 by Margaret Thatcher, believed to be highly

inequitable, resulted in civil unrest and refusal to pay such taxes by 30% of the

population in some councils.

There is also tax evasion of a more covert nature, in which an agent breaks

the law by underreporting the amount of his or her income or wealth that is eligible

for taxation. In this case the ‘breaking of the law’ is a personal or individual

decision, even though the underlying reasons may be numerous. That such

behaviour will have political economy implications for the tax structure should be

intuitively obvious. For example, in the presence of widespread tax evasion and

corruption, agents may have less faith in the tax system as a form of redistribution.

If that is the case, they may not choose to voice opinions in favour of a highly

progressive tax system, simply because they believe the redistributive mechanisms in

place to be ineffective. Furthermore, such a system would penalise the honest

taxpayers.

There is also substantial indirect evidence to support the idea that tax

structures are determined differently in the presence of corruption and tax evasion.


3

Bearse, Glomm, and Janeba (2000) suggest that developing economies facing such

problems prefer redistribution ‘in-kind’, through the provision of various public

goods, rather than through direct monetary transfers. They present a model in which

the crucial distinction between rich and poor countries is that rich countries have

access to a more productive tax collection technology than governments in poor

countries. As a result, because the quality of the public service is low and individuals

on the high end of the income distribution opt out, the median voter takes this into

consideration and allocates a larger share of the public budget to redistribution in-

kind.

The aim of this thesis, in essence, is to explore the above mentioned issues.

The scope of the thesis, however, is narrower than the preceding statement implies.

The key objective of this study is to provide the necessary first steps in the modelling

of such issues within a macroeconomic framework. This involves the construction of

a model with heterogeneous agents, so that the distributional implications of taxes

and tax evasion may be considered. The political economy angle is then modelled in

a simple way by allowing the agents to vote on their desired tax structure. More

importantly, the framework we construct incorporates the idea, hitherto unexplored

in the literature, that agents typically face various trade-offs that can only be

realistically modelled within a macroeconomic framework. A contribution of this

thesis, then, is to establish whether and to what extent such trade-offs are relevant to

the issues discussed above.

Of immediate motivational relevance is the lack of such models in the

macroeconomic literature on tax evasion. Existing models are typically of the

‘representative agent’ variety and look at the implications of tax evasion on growth.


4

Roubini and Sala-i Martin (1995), for example, study the relation between policies of

financial repression, inflation and economic growth. They set up a model which

shows that governments might want to repress the financial sector as it is viewed as

an ‘easy’ source of resources for the public budget (the inflation tax). Their findings

suggest that in countries where tax evasion is large, the government will optimally

choose to repress the financial sector in order to increase seigniorage taxation. Chen

(2003) integrates tax evasion into a standard AK growth model with public capital.

In his model the government optimizes the tax rate while individuals optimize tax

evasion. The author finds that an increase in both unit cost of tax evasion and

punishment/fines reduces tax evasion, whereas an increase in tax auditing reduces

tax evasion only if the cost of tax enforcement is not too high. All three policies have

ambiguous effects on economic growth, due mainly to their indirect effects upon tax

compliance and tax rate.

As the presence of inequality in some form is essential for the purpose of

addressing political economy issues, we feel that existing macroeconomic models of

tax evasion have shortcomings that need to be remedied, before they can be used for

addressing the issues of interest in this study. However, rather than start with an

extant macroeconomic model as a benchmark, our approach is to ‘start from scratch’

and build on the seminal work of Allingham and Sandmo (1972), which is

recognized in the literature as the very first approach in modelling tax evasion.

Again, we emphasise that the end result of this exercise is not the development of a

‘fully realised’ and ‘calibrated’ macroeconomic model in the dynamic-stochastic-

general-equilibrium (DSGE) tradition, but rather to provide a framework amenable

to such an extension. Even so, we believe that there are several insights to be gained


5

from such a step-by-step approach, which will be of substantial relevance to future

model building in the DSGE tradition.

Further motivation comes from the substantial body of literature in the areas

of microeconomics and public-finance which is relevant to the issues mentioned

above. Public finance models of voting on public goods, such as Epple and Romano

(1996a) and Borck (2009) for example, provide interesting insights to the issues

mentioned above. In Epple and Romano (1996a), the authors determine public

service provisions with private alternatives and find that the political outcome is

determined by agents at the top or low end of the distribution. This ‘ends against the

middle’ feature is often observed in models that preferences over policy dimensions

that are not ‘single-peaked’, and this is sometimes also typical of political economy

microeconomic models of tax evasion. In Borck (2009), for example, the author

analyses voting on linear income tax with redistributive lump-sum transfers in the

presence of tax evasion and finds such a feature is relevant to the determination of

outcomes. Again, these papers look at the political economy determination of

redistribution in a microeconomic context and, to our knowledge there are no extant

studies looking at the tax-evasion and redistribution within the framework of a

macroeconomic model.

We believe, however that these issues are of an intrinsically macroeconomic

nature, and could benefit from further development in a macroeconomic context.

Agents face decisions over different goods and across time, and political economy

outcomes are the result of decisions of interacting agents in the economy who have

different wealth and income levels, as well as different preferences in relation to the


6

tax structure or other government policies. Such aspects are more appropriately

modelled in a macroeconomic framework.

The success of political macroeconomic models in explaining various related

issues in the determination of policies provides further inspiration for our research.

The political mechanism of this line of literature focuses on redistribution of income

through a political process. The agents can either vote over a preferred tax rate or a

preferred level of government expenditure to redistribute resources (see for example,

Alesina and Rodrik 1994, and Persson and Tabellini 1994). It becomes obvious,

then, that the initial income distribution is vitally important to economic growth. The

political economy considerations in conjunction with tax evasion, however, have

only been looked at in a microeconomic context. The diversity and richness of

insights that have emerged in the inequality and growth literature motivate the

exploration along parallel lines in the tax evasion context.

The results of our theoretical exploration produce some interesting insights.

In Chapter 3, we find that the introduction of the ‘evade or not’ feature reduces the

extent of evasion even in the context of a very simple macroeconomic model of tax

evasion. We find that the extent of evasion in the ‘evade or not’ alternative is much

lower and more consistent with the empirical evidence. Another realistic outcome

that emerges is that the extent of evasion is increasing in wealth. This is achieved

while still maintaining CRRA preferences which are important in the

macroeconomic context if one is interested in building models capable of replicating

features of business cycles and economic growth. See for example, Cooley and

Prescott (1995), who elaborate on the reasons why CRRA preferences are needed to

match certain stylized facts of growth and business cycles.


7

For a range of values of the tax rate, the ‘evade or not’ model always produce

a lower amount of evasion in comparison to the AS model, and this is an important

contribution in the sense that that the standard AS model has been critiqued for

predicting unrealistically large amount of tax evasion for economic agents. In

addition, we find that within this range, the extent of evasion increases with

inequality. Furthermore, an interesting outcome emerges in relation to the mix of

evaders in the distribution. For low levels of the tax rate, evasion is concentrated at

the bottom end of the income distribution and this tendency is exacerbated when

inequality rises. The introduction of a cost-of-evasion function, however, switches

the identity of evaders in the distribution. It is now the richer agents rather that the

poor agents who evade from the payment of taxes. The results also show that the

effect of inequality seems to be non-monotonic in relation to the number of evaders

in the economy.

The political economy outcomes of the models in Chapter 3 suggest that, in

the vast majority of cases, redistribution is favoured in both the AS model and the

‘evade or not’ model in the presence of inequality. The only exception is for one

special case of the ‘evade or not’ construct without a cost-of-evasion function for a

very low level of inequality. In this instance, we find that the agents prefer

‘efficiency over equity’ and vote on a low level of progressivity. A low tax rate in

this model is ‘efficient’ in the sense that it is associated with a low level of tax

evasion. Higher taxes, on the other hand, are in principle associated with higher

redistribution, but the fact that they are associated with a higher degree of tax

evasion nullifies this effect. This feature re-emerges in the ‘evade or not’ model of

Chapter 4.


8

Finally, we find that the level of inequality does not seem to matter in

relation to the political economy determination of the tax structure. In our numerical

simulations, we report results for a wide range of inequality, as measured by the Gini

coefficient of income, and find that variations in this parameter do not qualitatively

alter the political economy results.

While the two-period models of Chapter 3 produce improvements in the

modelling of tax evasion, they still have some shortcomings. Specifically, there are

still a large number of values of the tax rate for which the models predict unrealistic

levels of tax evasion. There is, in fact, a large range of values for which the ‘evade

or not’ construct is identical to its AS counterpart. Basically, for a tax rate

approximately equal to or above 30%, all agents in the economy evade taxes, and for

any given tax rate evade a fairly high proportion of their incomes. To address these

issues, we turn to the model of the next chapter, which takes several steps in the

direct of a more realistic modelling of the issues addressed in Chapter 3.

In Chapter 4, we take the last step of the model-building process by grafting

the two-period models of the previous chapter in the framework of the well-known

overlapping generations model used extensively in macroeconomics as a workhorse

for addressing issues in relation to inequality and growth. We further extend the

model by incorporating a non-linear tax schedule and an alternative modelling of the

penalty structure. We find that these extensions produce more realistic results in

relation the number of evaders in the economy. In this model, we do not get a

scenario in which 100% of the population is evading taxes across the range of tax

rates considered.


9

In addition, this extension preserves several of the results obtained earlier in

Chapter 3, some of which we had found to be consistent with outcomes of empirical

analyses in the literature on tax evasion. We find, again, a non-monotonic

relationship between the number of evaders and inequality, a result consistent with

the fact that the empirical analysis of the link between inequality and tax evasion is

inconclusive.1 Also, for a given level of inequality, as was the case in the models of

Chapter 3, we find that the number of evaders is increasing in the tax rate, a feature

of the model that is supported by empirical studies, as, for example, that of Fisman

(2001). In relation to the non-linear tax structure, we find that a higher degree of tax

progressivity increases the number of evaders in the economy. For a given level of

inequality, however, we find that the relationship between the number of evaders and

tax progressivity is non-monotonic.

The political economy outcomes of the models produce the most interesting

results in this chapter. We find that the voting outcome in the ‘evade or not’ model is

in favour of the lowest possible tax rate available to the agents. In addition, the

agents also vote for the lowest degree of tax progressivity presented to them. This is

in contrast to the ‘evade or not’ models in Chapter 3 where, apart from a very low

level of inequality, agents in the economy vote for the highest tax rate presented to

them. The results of the AS model in this chapter are, however, similar to those in

Chapter 3. In these models, the agents vote for the highest tax rate and degree of

progressivity presented to them. This is due to the fact that since all agents are

evading taxes in this economy, the transfers received by the agents is maximised

when the tax structure is at its most progressive. In the ‘evade or not’ model,

however, agents vote for low tax rates as they typically encourage a lower amount

1 See for example, Christian (2004).


10

of tax evasion, making the redistributive mechanism more efficient. A policy

implication of this model, then, is that low taxes are better taxes from the point of

view of discouraging tax evasion.

The remaining chapters are organised as follows. Chapter 2 rationalises the

motivation for the thesis by providing a background of the relevant literature.

Chapter 3 revisits the original Allingham and Sandmo model, ‘building from

scratch’ the original context in a macroeconomic framework. Chapter 4 proposes a

novel approach to modelling tax evasion in a macroeconomic framework and

analyses the extent of tax evasion in those models. It provides an extension and

analysis of the political economy outcomes of the theoretical models established in

Chapter 3. Chapter 5 concludes.

Background and Motivation Chapter 2

11

CHAPTER 2

Background and Motivation

2.1 Introduction

This chapter seeks to provide a background of the relevant literature on tax

evasion from a theoretical, empirical, and political economy perspective. We first

include a review of the theory of tax evasion beginning with the seminal Allingham

and Sandmo (1972) paper and its extensions thereafter. We then proceed by

highlighting the empirical models of tax evasion, noting that the empirical literature

of tax evasion and progressive taxes is scant. This is followed by a summary of tax

evasion in macroeconomic models. Finally, we discuss the political economy and

voting models involving taxation and tax evasion noting that these models tend to be

analysed within a microeconomic framework.

The remaining sections are organised as follows. Section 2.2 reviews the

literature on the theoretical aspects of modelling tax evasion. Section 2.3 reviews the

empirical literature on tax evasion. Section 2.4 discusses the existing models of tax

evasion in the literature. Section 2.5 looks at the relevant political economy and

voting models dealing with the tax structure. Section 2.6 concludes.


12

2.2 The Theory of Tax Evasion

The formal economic theory of tax evasion is of relatively recent origin and

started to develop only a little over 30 years ago. Its beginning can be dated back to

1972 with the publication of the article “Income Tax evasion: A Theoretical

Analysis" by Michael Allingham and Agnar Sandmo. This was followed by a large

number of contributions to the literature which extended the original model in a

number of directions.

The standard approach to tax evasion is based on the economics of decision

making under risk. In particular, declaring income is viewed as being analogous to

purchasing a safe asset, while concealing income is viewed as being analogous to

purchasing a risky asset. As such, the tax evasion problem facing an individual then

essentially becomes a portfolio selection issue.

In the original Allingham and Sandmo (AS) (1972) model, the individual has

to decide how much income to report and how much to evade. The model makes no

account of the taxpayer's `real' decisions; labour supply and therefore gross earnings

are taken as given, and the same is true of income from capital. The agent then

chooses the amount to evade so as to maximise expected utility subject to some

probability of detection.

The model assumes that a taxpayer with an exogenous income of y is subject

to a tax rate of τ on this income. The decision of the taxpayer is to report an income

of x ≤ y, or to hide a proportion of income, α = y − x. There is a probability of p that

the taxpayer will be audited. Upon audit the tax authorities learn the true income of

the taxpayer, and in that case the taxpayer pays a penalty at the rate of π on the

unreported income in addition to the tax due. There are two states that the taxpayer


13

faces: one if he is audited (caught) and another if not. When he is not caught, his

income is ync

= (1−τ)x+α. When he is caught, his income is yc = (1−τ)x−(τ+π)α. The

taxpayers problem is given by: E(U) = (1 − p)U(ync

) + pU(yc) where E(U) denotes

the expected utility of the taxpayer, and the utility function is assumed to be concave

which implies that the taxpayer is risk averse.

The implications of the model suggest that a higher penalty rate or a higher

probability of detection always tends to discourage tax evasion. However, a notable

feature of the model is that an increase in the tax rate has an ambiguous effect on tax

evasion. There is an income effect which is negative; higher taxes make the

individual poorer and, therefore, less willing to take risks. On the other hand, there is

also a substitution effect that works in the direction of increased evasion; the “price”

paid by a non-evader is high, and therefore one substitutes out of non-evasion

towards evasion.

This ambiguous effect on tax evasion as noted by Yitzhaki (1974), however,

depends crucially on the assumption that the penalty is imposed on the amount of

income evaded. If, instead, the fine is imposed on the evaded tax, as in the cases of

the American and Israeli tax laws, there would be no substitution effect and,

accordingly, no ambiguity. This leads to an unambiguous result that an increase in

tax rates reduces tax evasion. Following Yitzhaki's (1974) modification of the AS

model, most models of tax evasion now assume that the penalty is levied on the

evaded tax, as it is under most tax laws.

There have been several extensions to the AS model since. One such

extension was provided by including a labour supply decision in the model so as to

make income endogenous. This extension has been investigated by Cowell (1981),


14

and Sandmo (1981), among others. In Sandmo (1981), the author develops a model

in which an increase in the penalty rate causes a decrease in the supply of hours

worked in the underground economy, implying that the portion of income not

reported will decrease. One drawback, however, is that the model does not generate a

clear-cut comparative statics result regarding the relationship between tax evasion

and the tax rate or the probability of being caught.

Another extension to the original AS model is to make the probability of

audit endogenous. Andreoni, Erard and Feinstein (1998) develop such a model

where the government can pre-commit to the level of audit it makes. Alternatively,

Graetz, Reinganum and Wilde (1986) model the probability of audit in a game-

theoretic framework with strategic interactions between the tax payer and the tax

authorities.

It is intuitively appealing, however, to speculate that higher tax rates will

encourage rather than repress evasion. There have been indeed, although not

unanimous, much evidence in support of the intuition that higher tax rates encourage

rather than repress evasion. Crane and Nourzad (1986), for example, find that

aggregate evasion rises in both absolute and relative terms with increases in the

marginal tax rate, but falls with increases in the detection probability, the penalty

rate, and the wage share of income. In addition, the authors also find that tax evasion

in both absolute and relative terms is positively related to the inflation rate.

In the social norm literature, the existence of a social norm suggests that an

individual will comply as long as he or she believes that compliance is the norm.

According to Elster (1989), a social norm can be distinguished by the feature that it

is process-oriented, unlike the outcome-orientation of individual rationality. This


15

perspective also suggests that, if the government can affect the social norm of

compliance, then such government policies represent another tool in the

government's battle with tax evaders.

Cowell and Gordon (1989), for example, introduce public goods into the

model and thus link public provisions and tax payments in the evasion decision.

Although this modification can explain the observed relation of evasion to the tax

rate, it does not capture the reasons why non-evasion is so prevalent. To capture

these aspects, Gordon (1989), which builds on the work of Akerlof (1980) and

Naylor (1989) among others, introduced a ‘psychic cost’ of evasion that increased as

evasion increased allowing a generation of the population into evaders and non-

evaders.

In addition, Myles and Naylor (1996) set out a model of tax evasion that has

included a return from conforming to the set of non-evaders and from adhering to the

social custom of non-evasion. They show that this can remain consistent with the

standard model of the evasion decision as a choice with risk once the decision to

evade has been taken. The authors then conclude that the method of incorporation of

conformity and social customs appear to provide a model of tax evasion that

successfully incorporates the standard decision with risk whilst providing aggregate

predictions that are capable of being consistent with observed data. There seems to

be strong evidence of the influence of social norms in tax compliance behaviour in

the empirical literature which we now turn to (see for example, Alm and Martinez-

Vazquez 2003).


16

2.3 A Review of Empirical Models of Tax Evasion

Turning to the empirical models of tax evasion, Clotfelter (1983) was the first

to empirically test the AS model. Using data from the 1969 Tax Compliance

Measurement Program (TCMP), he tests the effects of tax payers' after-tax income,

the tax rate, and other demographic indicators on the level of tax evasion by

estimating a standard Tobit model. He finds that the coefficients for both after-tax

income and tax rates are positive and significant.

A comprehensive investigation into the relationship between government

policy parameters and tax evasion and avoidance in a developing country context is

provided by Alm, Bahl, and Murray (1990). The authors develop and test a model to

examine the effect of government policy on tax evasion and avoidance decisions of

taxpayers. The study includes such factors as the tax rate, the payroll tax

contributions and benefits, the probability of audit, and the penalty rates. Using 1983

Jamaican individual level data, they estimate share equations for three dependent

variables: avoidance, evasion, and reported income. The results show that the tax

base rises with higher benefits for payroll tax contributions and falls with higher

marginal tax rates, more severe penalties, and a higher probability of detection, as

individuals substitute towards avoidance income.

In addition, Alm et al. (1991), in their analysis on Jamaica, find that the

avoidance and evasion of income tax have cost the government of Jamaica what is

equivalent to 84 per cent of actual collections. They suggest two major lessons for

government policy. First, incentives matter, and the tax base will likely increase

systematically and predictably to reductions in marginal tax rates. Second, a central


17

component of tax policy and tax reform in all developing countries should involve

administrative improvements that attack non-filing by self-employed individuals.

Feinstein (1991) uses pooled 1982 and 1985 TCMP data in order to decipher

the independent effect of the tax rate and income on tax evasion in light of the usual

strong positive relationship between tax rates and income. Because marginal tax

rates changed over this period for a given level of income, it is easier to identify the

separate effects of the two variables. The results from the pooled data show that the

coefficient on income in the evasion equation is insignificant; contrary to Clotfelters

finding, the results show a negative relationship between marginal tax rates and tax

evasion.

Using Swiss data, Pommerehne and Weck-Hannemann (1996) conduct an

empirical analysis of income tax non-compliance in Switzerland based on the

standard model of tax evasion. Non-compliance is found to be positively related to

the marginal tax burden and negatively to the probability of audit, though the latter

impact is only weak. Their extended model reveals that non-compliance is positively

related to inflation. Interestingly, the authors find that the penalty tax is not a

significant deterrent of tax evasion.

More recently, using China as a case study, Fisman and Wei (2004) look at

the effect of tax rates on tax evasion. The authors find that the ‘evasion gap’ is

highly correlated with the tax rate and that China is already on the wrong side of the

Laffer curve. This implies that any further increase in the tax rate is likely to reduce

rather than increase the tax revenue collected by the authorities.

In terms of the net tax gap, the United States (US) net tax gap was estimated

to be 13.7% with 57% in the non-farm proprietor income not reported (Slemrod


18

(2007)). In the United Kingdom (UK), it is estimated that about one third of self-

employment income is not reported to the tax authorities (Pissarides and Weber

(1989)). The magnitude of tax evasion is smaller in Sweden at around 8% (Slemrod

(2007)) but is widespread in Italy. According to Fiorio and d'Amuri (2005), the

underground economy is estimated to be around 27-48% of official Gross Domestic

Product (GDP).

In addition, developing and developed economies have different tax

structures. A stylised fact, for example, associated with the tax structure of

developing economies is their greater reliance on indirect as opposed to direct

taxation. According to Avi-Yonah and Margalioth (2006), the structure of taxation in

developing countries is radically different from that of developed countries. About

two thirds of the tax revenue in developed countries is obtained from direct taxes,

mostly personal income tax and social security contributions. The remaining one-

third comes primarily from domestic sales tax. The situation is exactly reversed in

developing countries: about two-thirds of the tax revenue comes from indirect taxes,

mostly value added taxes (VAT), sales tax, excises and taxes on trade. The latter

characteristic is driven by the practical implications of tax evasion for revenue

collection by governments. Specifically, in the presence of tax evasion, direct taxes

are harder to collect and administer, leading to a shift towards indirect taxation as a

source of revenue (Avi-Yona and Margalioth 2006).

In summary, a common feature between the theoretical models in the public

finance literature and the empirical evidence is that the theoretical models predict too


19

much evasion, which is inconsistent with the empirical data.2 We next turn to the

literature of tax evasion in macroeconomic models.

2.4 Models of Tax Evasion in Macroeconomics

There have been very few articles in the macroeconomics literature, however,

that incorporates tax evasion in the context of a dynamic general equilibrium

framework. Notable exceptions include Roubini and Sala-i-Martin (1995), Lin and

Yang (2001), and Chen (2003). Most of these models, however focus on growth-

related aspects of the tax evasion problem, rather than the distribution-related aspects

that form the subject matter of this thesis.

Among the first to analyse the effect of inflation on tax evasion using a

macroeconomic framework was Fishburn (1981). In his model, the author finds that

inflation can affect an agent's decision to evade taxes is by eroding the real value of a

given level of nominal disposable income. This provides an incentive for the

taxpayer to restore his purchasing power through evasion. Fishburn's results show

that a risk-neutral individual's evasion decision is independent of the price level but a

risk-averse individual's evasion decision depends on the properties of the relative

risk-aversion function.

Roubini and Sala-i Martin (1995) also incorporate tax evasion in their study

of the relation between policies of financial repression, inflation and economic

growth. The authors set up a model which shows that governments might want to

repress the financial sector as it is viewed as an ‘easy’ source of resources for the

public budget (the inflation tax). Their findings suggest that in countries where tax

evasion is large, the government will optimally choose to repress the financial sector

2 See Slemrod and Yitzhaki (2000) for a good survey on tax evasion, avoidance and administration.


20

in order to increase seigniorage taxation. This policy will then reduce the efficiency

of the financial sector, which will in turn reduce the growth rate of the economy.

Financial repression will therefore be associated with high tax evasion, low growth,

and high inflation. This model, however, is of a representative-agent type which is

not suitable for analysing models with inequality.

Similarly, Gupta (2008) analyses the relationship between tax evasion and

financial repression using a simple overlapping generations framework, calibrated to

four Southern European countries. In his model, tax evasion is determined

endogenously. He concludes that increases in the penalty rates of evading taxes

would induce agents to report a greater fraction of their income, while, increases in

the income-tax rates would cause them to evade a greater fraction of their income. In

addition, a higher fraction of reported income, resulting from lower level of

corruption or higher penalty rates, causes the government to inflate the economy at a

higher rate. Inflation, though, tends to fall when an increase in the fraction of

reported income originates from a fall in the tax rate. The author concludes that there

exits asymmetries in optimal monetary policy decisions, depending on what is

causing a change in the degree of tax evasion.

Extending the static model in a dynamic setup, Lin and Yang (2001) show

that while higher tax rates repress tax evasion in a static model, they encourage tax

evasion in a dynamic model. The novel implication of their result is that while

growth is decreasing in tax rates in the absence of tax evasion, it is U-shaped in tax

rates in the presence of tax evasion. More recently, Chen (2003) integrates tax

evasion into a standard AK growth model with public capital. In his model the

government optimises the tax rate while individuals optimise tax evasion. The author


21

finds that an increase in both unit cost of tax evasion and punishment/fines reduces

tax evasion, whereas an increase in tax auditing reduces tax evasion only if the cost

of tax enforcement is not too high. All three policies have ambiguous effects on

economic growth, due mainly to their indirect effects upon tax compliance and tax

rate.

While we believe that the impact of tax evasion on economic growth is a

fruitful area of research, any analysis of tax evasion that abstracts from distributional

issues is incomplete, given that tax evasion has important implications for

redistributive policy of any kind. This suggests, to us, that any political economy

model of policy determination needs to be examined in relation to its abstraction

from the tax evasion problem, especially in the context of models that are supposed

to be representative of developing economies. However, while there is a large

literature in microeconomics that looks at the political economy link between tax

evasion and redistribution, to our knowledge there are no macroeconomic models

that study this issue.

The microeconomics literature provides several insights into the political

economy link between tax evasion and policy determination, and to that end

provides a useful backdrop to our macroeconomic analysis of the problem. We

therefore provide a brief review of such literature in the subsequent section, along

with a discussion of some of the political economy models in macroeconomics,

which are also of relevance to our research.

2.5 Political Economy/Voting Models of Taxation and Tax Evasion

The success of political macroeconomic models in explaining various related

issues in the determination of policies provides further inspiration for our research.


22

The political mechanism of this line of literature focuses on the redistribution of

income through a political process. The standard theory of redistribution and

inequality is based on the seminal paper by Meltzer and Richard (1981). The

argument is based on the fact that the income gap between the median voter and the

mean voter is greater in more unequal societies. The median voter is therefore

expected to exert political pressure for redistributive government intervention

because the benefit he or she derives from redistributive transfers from the

government outweighs the cost of taxation needed to finance redistribution. For this

to be true, it is assumed that median voter preferences are taken into account in the

political process under majority voting and, also, that taxation is progressive.

In Alesina and Rodrik (1994), the agents can either vote over a preferred tax

rate or a preferred level of government expenditure to redistribute resources. It

becomes obvious, then, that the initial income distribution is vitally important to

economic growth. They find that the greater the inequality of wealth and income, the

higher the rate of taxation, and the lower the growth. Persson and Tabellini (1994),

also look at the effect of political decisions on growth. In a society where

distributional conflict is important, political decisions produce economic policies

that tax investment and growth-producing activities in order to redistribute income.

These political economy considerations in conjunction with tax evasion, however,

have only been looked at in a microeconomic context. Such diversity and richness of

insights that have emerged in the inequality and growth literature motivate the

exploration along parallel lines in the tax evasion context.

There is also substantial indirect evidence to support the idea that tax

structures are determined differently in the presence of corruption and tax evasion.


23

Bearse, Glomm, and Janeba (2000) suggest that developing economies facing such

problems prefer redistribution ‘in-kind’, through the provision of various public

goods, rather than through direct monetary transfers. They present a model in which

the crucial distinction between rich and poor countries is that rich countries have

access to a more productive tax collection technology than governments in poor

countries. As a result, because the quality of the public service is low and individuals

on the high end of the income distribution opt out, the median voter takes this into

consideration and allocates a larger share of the public budget to redistribution in-

kind.

Further motivation comes from the substantial body of literature in the areas

of microeconomics and public-finance which is relevant to the issues mentioned

above. In the public finance literature, personal income tax structures contain a trade-

off between efficiency and equity. It is commonly believed that a flat tax structure

(lump sum tax) produces efficiency as it does not distort the choices that individual

agents make. Proponents argue that a flat tax rate with a very broad base would both

alleviate distortion, and reduce the quantity of tax arbitrage options open to tax

payers in the current system.

Progressive taxes, on the other hand, are often designed to serve as a

redistributive mechanism in an economy. That is, to collect a greater proportion of

income from the richer agents relative to the poor, thus reducing the inequality of

disposable income relative to taxable income. Underlying this trade-off is the

presumption that a higher level of tax progressivity reduces income inequality.

Public finance models of voting on public goods, such as Epple and Romano

(1996a) and Borck (2009) for example, provide interesting insights to the issues


24

mentioned above. In Epple and Romano (1996a), the authors determine public

service provisions with private alternatives and find that the political outcome is

determined by agents at the top or low end of the distribution. This ‘ends against the

middle’ feature is often observed in models that preferences over policy dimensions

that are not ‘single-peaked’, and this is sometimes also typical of political economy

microeconomic models of tax evasion.

In Borck (2009) the author analyses voting on linear income tax with

redistributive lump-sum transfers in the presence of tax evasion and finds such a

feature is relevant to the determination of outcomes. Again, these papers look at the

political economy determination of redistribution in a microeconomic context and, to

our knowledge there are no extant studies looking at the tax-evasion and

redistribution within the framework of a macroeconomic model. These political

economy considerations in conjunction with tax evasion, however, have only been

looked at in a microeconomic context.

2.6 Conclusion

This chapter provided the background and motivation that will guide the

theoretical analysis to follow in the next chapter. We provided a summary of the

theory of tax evasion starting with the seminal Allingham and Sandmo (1972) article

and the various extensions of their model thereafter. A key limitation of this model,

which spawned several of the studies mentioned in Sections 2.2 and 2.3, was in

relation to the empirical plausibility of the levels of tax evasion predicted in the

model. Our approach to addressing this issues, presented in forthcoming chapters, is

however, very different; we believe that the solutions lie in the macroeconomic


25

modeling of the problem of tax evasion, for reasons elaborated upon in the

introductory chapter of this thesis.

Furthermore, as highlighted in section 2.4, extant macroeconomic models of

tax evasion abstract from distributional issues, thereby being inadequate for the

purpose of addressing several of the issues we are interested in. In particular, the

political economy determination of the tax structure, the subject of a large body of

microeconomics literature, some of which has been discussed in Section 2.5, has not

received much attention in the macroeconomic context. Given the richness of

insights that emerged in macroeconomics following the analysis of political

economy models of policy, such as those discussed in Section 2.5, it is of interest to

examine similar issues in the context of tax evasion.

To this end, there is a need to develop political economy macroeconomic

models capable of identifying the true structural relationships between tax evasion,

inequality and progressive taxes. The models presented in Chapters 3 and 4 of this

thesis, while being relatively simple in relation to contemporary macroeconomic

models provide insights that will be of relevance to future model building in this

area.

The Benchmark Model and Some Simple Extensions Chapter 3

26

CHAPTER 3

The Benchmark Model and Some Simple Extensions

3.1 Introduction

The standard approach to tax compliance applies the economics-of-crime

methodology pioneered by Becker (1968); in its first application, due to Allingham

and Sandmo (1972) it models the behavior of agents as a decision involving choice

of the extent of their income to report to tax authorities, given a certain institutional

environment, represented by parameters such as the probability of detection and

penalties in the event the agent is caught. This issue, however, has not been fully

explored in a macroeconomic context. In a macroeconomic context agents make

decisions about many goods and in different time periods, and therefore face many

intertemporal and intratemporal trade-offs that are intrinsically linked to the tax

evasion decision. For example, the decision to evade will have an impact on the

choice of consumption not only on this date, but also on a future date in time. It is

then reasonable to speculate that the state contingent planning, and forward looking

behaviour typical of macroeconomic models could yield different results. One of the

aims of this thesis is to examine whether these trade-offs are indeed relevant.

The model of this chapter pursues the above mentioned aim by ‘starting from

scratch’ and by revisiting the very first conceptual formalisation of the tax evasion

problem by Allingham and Sandmo (1972). In an elegant model using the

‘economics of crime’ idea pioneered by Becker (1968) they model the tax evasion


27

problem as a portfolio decision in which agents choose the amount of income to

report to the tax authorities in the presence of uncertainty. The uncertainty stems

from the fact that they may be audited have to pay a fine proportional to the

underreported amount of income in the event that they are caught.

This model has spawned a large amount of literature aimed at addressing a

key limitation of this particular construct. A critical issue pointed out in Sandmo

(2005) and previous literature is that it takes a cynical view of the evasion decision -

it assumes that the taxpayer does in fact engage in tax evasion given a restriction on

the parameters of the model. The question then, is that is this a reasonable

assumption? According to Sandmo (p 649, 2005) “While it is difficult to ascertain

the exact number of people who evade taxes it is clear that there are several who

don’t even though they have the opportunity to do so.”

The behaviour of such agents can only be explained by a reversal of the

restriction in question, which is that the tax rate (denoted θ) is greater than the

expected penalty rate (which is the probability of detection p multiplied by π, the

penalty rate). This however presents an empirical puzzle, and here we present

Sandmo’s (2005) discussion of it. According to a numerical example presented in

Sandmo (2005), if the penalty rate is twice the regular tax rate the condition in

question implies that the probability of detection which would be high enough to

deter tax evasion is greater than 0.5. Sandmo then states: “This number is far in

excess of most empirical estimates and raises the question of whether the model

depicts people as either too rational or too cynical compared to what we believe that

we know about their actual behaviour” (Sandmo, p 649, 2005).


28

In the first section of this chapter we further explore the point made by

Sandmo by examining a specific parameterisation of the model. In addition, our

numerical exploration highlights several aspects of the model that complement some

of the theoretical analysis in the original model. Our object is to develop intuition

regarding various aspects that are difficult to obtain in situations where theoretical

analyses provide ambiguous results. For instance, the impact of the tax rate on the

proportion of income that is reported is known to be ambiguous (see Allingham and

Sandmo 1972, pages 329-330). Within the context of our research, however, which

examines the link between tax evasion and progressivity, it is important to get an

idea of how tax evasion changes as progressivity increases. Furthermore, it is

important to do so within the context of a parameterisation of preferences that is

commonly used in macroeconomic models. We therefore restrict our analysis to the

log utility case, which in turn is a special case of the constant-relative-risk-aversion

(CRRA) style of preferences that are typical in macroeconomic models.

Some key insights emerge from this analysis. Firstly, while our results are

simply ‘illustrations’ of the more general theory, they provide information that is of

value in the parameterisation and development of a macroeconomic model of tax

evasion. Specifically, we find that the basic AS model ‘works’ for a very narrow and

unrealistic range of parameters, thereby reinforcing and further clarifying the point

made by Sandmo (2005).

Secondly, since we use a CRRA assumption, we find, as theoretical work in

the AS paper suggests, that the proportion of income reported is invariant to their

income or wealth. Behavioural research and empirical evidence, however, suggests

that proportion of (reported/unreported) income should (decrease/increase) as


29

income increases (see for example, Bloomquist 2003 and references therein). This is

a somewhat problematic issue, since to get this relationship in the standard model we

would need a decreasing-relative-risk-aversion (DRRA) assumption for preferences.

Macroeconomic models, on the other hand are restricted to use the CRRA

assumption in order to produce results that are consistent with steady state growth

and certain business cycle features.3 This feature of the model motivates some of the

extensions of the basic model that are discussed in this chapter. In a subsequent

section we add a ‘cost of evasion’ function similar to that of Chen (2003), and show

that it is possible to generate the desired relationship between the extent of evasion

and income while keeping the CRRA assumption intact.

In our model, and in Chen (2003), the ‘cost of evasion’ is a function of the

extent of evasion. We interpret our cost of evasion function to consist of both

pecuniary and non-pecuniary elements. The pecuniary component may consist of

bribes, while the non-pecuniary element may consist of a sense of guilt from non-

payment of taxes. For example, a false income declaration may induce anxiety or

guilt. An alternative interpretation of non-pecuniary costs is as damage to reputation

suffered upon detection. This is similar to that of Gordon (1989) where the menu of

costs incurred by evaders is expanded to include certain non-pecuniary

considerations. There are some other constructs such as that of Borck (2009) that

include a fixed cost of evasion. This type of feature typically entails agents deciding

whether or not to evade based on the size of this fixed cost. Those that evade then

have to decide how much to evade. While a ‘fixed cost’ invariably reduces the

extent of evasion, it is an unappealing way of dealing with the problems discussed

3 Cooley and Prescott (1995) restrict the growth economy by using the general parametric class of

preferences in the CRRA form. These preferences are tied to basic growth observations for the U.S

economy for factors of production such as capital and labour shares of output.


30

above, as one is effectively introducing a non-convexity and ‘forcing the issue’, so to

speak. However, it is of interest to ask whether an ‘evade or not’ decision can lead

to different outcomes without the introduction of a fixed cost.

A key contribution of this thesis results from asking this question. Basically,

this involves an agent comparing his or her utility in the case of certainty – i.e. when

he/she does not evade with the utility in the case in which he/she undertakes evasion.

We find that while this construct does not make a difference in the case of the basic

AS model, it makes a substantial difference in the case of other simple extensions of

the basic model that are considered in this chapter. Specifically, when we extend the

basic model to include consumption across two time periods, we find that there is a

range of parameters for which an agent chooses not to evade. In the extension

presented in Section 3.3.2 of this chapter, when we introduce a distribution of

income with heterogeneous agents, this feature of the model manifests itself in the

form of a substantial number of agents choosing not to evade for a reasonably

realistic range of values of the tax rate θ. We emphasise that this result emerges

without the introduction of a cost-of-evasion function.

Our intuition for this result is as follows: A key feature of macroeconomic

models is the desire to smooth consumption over time and across states, and also

across different goods. This is a result based on the ‘convexity’ assumption in

relation to preferences, which causes ‘balance’ in consumption to be desirable.4

Now this assumption is present in the basic AS model as well. However in the case

of macroeconomic models, the dimensions along which consumption smoothing

takes place are more varied relative to microeconomic models. The two-period

model is an extension which introduces consumption smoothing along a time

4 See Nicholson (2002) for this interpretation of convexity.


31

dimension. Tax evasion, on the other hand, within a state contingent framework of

the type we consider acts contrary to consumption smoothing. In a state contingent

model, the agent has to choose a consumption plan which specifies how much to

consume in the ‘good’ state in which he/she is ‘not caught’, and how much to

consume in the ‘bad’ state in which evasion is detected. The more he/she evades, the

more is the disparity across consumption in different time periods and across states.

This feature of a macroeconomic model, combined with consumption smoothing will

then enhance the desire to not evade.

The next step in the modelling process is to introduce political economy

considerations. In order to do so the models in Section 3.3.2 introduces

heterogeneity among agents in the form of a wealth distribution, and a lump sum

redistributive transfer that is given to all agents in the economy. Our numerical

simulations are based on a lognormal distribution with mean 3.2 and variance 0.8,

and we consider mean-preserving spreads of this distribution in order to assess the

implications of inequality on various outcomes of the model.5 We find that the

extent of evasion increases with inequality, but for a range of values for the tax rate,

the ‘evade or not’ model always produces a lower amount of evasion in comparison

to the AS version of the model. Furthermore, an interesting outcome emerges in

relation to the mix of evaders in the distribution. Typically for low values of the tax

rate evasion seems to be concentrated at the bottom end of the income distribution,

and this tendency is exacerbated when inequality increases. As this feature of the

model seems a little counterintuitive, in a later extension we incorporate a ‘cost of

5 The parameterization of the income distribution is similar to the models in Bhattacharya et al.

(2005) and Bearse et al. (2005).


32

evasion function’ that is increasing in the proportion of unreported income, which

has the effect of reversing this result.6

We find that the political economy outcome of this model is driven by equity

considerations, in spite of the presence of tax evasion. This is, in part due to the fact

that (a) redistribution occurs even in the presence of evasion, and (b) there are no

administrative costs so that revenue collected from penalties and fines can be used

for redistribution. While this feature of the model may be unrealistic, the alternative

of incorporating administration costs would involve a complicated extension of the

model. We choose to leave that as a future direction of research.

In the last section, we introduce a cost-of-evasion function, and this feature

has the effect, as described above, to switch the identity of the evaders in the

distribution – it is now the rich rather than the poor who evade. The political

economy and other outcomes, however, remain unchanged: agents at the lower end

of the distribution form a majority and their desired tax rate is the highest in the

menu of choices available to them.

While the enforcement parameters such as the probability of detection and

penalties for evasion do not alter the overall political economy outcomes in the

stylised models presented in this chapter, they do cause significant shifts in the

preference profiles of the voters, leading to situations in which non-majority

outcomes can occur. In these cases we apply the plurality rule and the winning

outcome remains unchanged. One can see from the voter distribution of preferences,

however, that if one applied a majority runoff procedure between two of the

6 Christian (1994) for example, finds that higher-income people evade less than those with low

incomes relative to the size of their incomes. See also Slemrod (2007) for a review about the

magnitude, nature, and determinants of tax evasion with an emphasis on U.S income tax.


33

outcomes with the highest votes, then an alternative outcome with a low level of

progressivity could be chosen. Alternatively, models with lobbies or other complex

voting structures, and those which model an equity-efficiency trade-off by

incorporating work-effort could produce a diverse set of outcomes. We leave these

extensions as a future direction of research.

Furthermore, we also find in the models of this chapter that increasing the

inequality in the distribution does not impact on the voting outcome. Specifically,

the model suggests that changes in inequality should not matter, although the

political outcome for the tax rate for a given level of inequality is conditional on

whether there is a large or small or large extent of evasion in the economy.

The remaining sections are organised as follows. Section 3.2 revisits the

basic original Allingham and Sandmo model, and introduces the model with an

‘evade or not’ choice. A numerical analysis of the two models is investigated and

analysed. Section 3.3 extends the models with a view towards building a

macroeconomic framework, which is presented in the next chapter. This is done

systematically, by firstly introducing time dimensions in the models followed by

incorporating heterogeneous agents and redistributive transfers. We then proceed by

introducing a cost-of-evasion function in the models. Finally, we analyse the

political economy implications of the models by allowing the agents to vote on a

range of tax rates presented to them. Section 3.4 concludes and motivates the

extension to follow in Chapter 4 of the thesis.


34

3.2 Revisiting Allingham and Sandmo Model

3.2.1 Theoretical Analysis

A: The Basic Allingham and Sandmo Model

We first begin, in this section, by revisiting the original Allingham and

Sandmo (1972) model and studying a simple parameterisation of it. The aim is to use

a type of parameterisation of preferences that is common to macroeconomic models,

and thereby develop some intuition regarding the conditions in which tax evasion

would emerge in such models. Specifically, we explore the point made by Sandmo

(2005), discussed in the previous section, by looking at the case in which utility is

logarithmic, which in turn is a special case of the constant-relative-risk-aversion

(CRRA) style of preferences that are typical in macroeconomic models.

In the original AS model, the agent’s labour supply is taken as a given (this

includes the agent’s gross earnings and income gained from capital). The agent

makes his/her decision of the amount of income to report or evade at the moment of

filling in his/her tax returns. According to Sandmo (2005), this may be an advantage

because it leads to clear and reasonably unambiguous hypotheses.

Assuming log utility, an agent’s preferences are given by the following:

[ ] ( ) ( )

{ ( )} (1)

where is the probability of being caught, is actual income or wealth, is the tax

rate, is the fine paid from evading taxes, and is the amount of income reported.

Given that log utility is assumed, U is increasing and concave, so that the tax payer

is risk averse. Such an agent chooses the amount of income to report, to maximise


35

equation (1). Maximising over the choice of yields the following first-order

condition for an optimum:

( )

( )

( ) (2)

Solving for the optimal we get:

( )

(3)

where is the optimal level of income reported to the tax authorities. This implies

that for the log utility case, the proportion of income reported is constant. Intuitively,

this result seems a little unreasonable. One would expect, for example, to see a

smaller proportion of income reported by the richer agents in comparison with the

poorer agents. In the AS framework this can only be achieved by assuming DRRA

preferences.

For an interior solution, the following conditions need to be satisfied:

( )

|

( )

( )

(4)

and

( )

|

( )

( )

( )

( ) (5)

Equation (4) is satisfied iff:

( )

( ) (6)

And equation (5) is satisfied iff:

(7)


36

which implies the following restriction to ensure an interior solution:

( )

( )

(8)

The condition ( )

( ) ensures that amount of reported income, , is

always some positive amount. This is what we have termed the lower bound on X.

The upper bound on X, given by the condition

, ensures that the amount on

reported income cannot be greater than the income itself. The implications of the

latter condition were discussed in the previous section, and we briefly reiterate them

here. As suggested by Sandmo (2005), if the penalty rate is twice the regular tax rate

the condition in question implies that the probability of detection which would be

high enough to deter tax evasion is greater than 0.5. This is far in excess of most

empirical estimates and raises the question of whether the model depicts a much

greater degree of evasion than is observed in the data.

In the case of lower bound, too, there are some implications of an analogous

and intuitively unappealing nature. In Figure 3.1 we plot the lower bound for two

cases, one in which π = 2θ – the case discussed by Sandmo (2005) in relation to the

upper bound, and one in which π=(1.5)θ.7 This is illustrated in Figure 3.1 below:

7 Although we also plot negative values on the y-axis, as probabilities are between 0 and 1 only, only

the graph above the 0 on the y axis is meaningful.


37

Figure 3.1: Lower bound for Probability of Detection on Tax Rate when π=2θ and

π=1.5θ.

Now, when π=2θ, if the tax rate is set at θ=0.2, then the probability of

detection would have to be set at an implausibly high value of p=0.4 to prevent

100% evasion – i.e to prevent the case in which agent chooses to not report any of

his income. Likewise, for π=1.5θ, if for example, the tax rate is set at θ=0.4, then the

probability of detection would have to be p=0.55 to satisfy the lower bound

condition.

However, within the range in question, the basic Allingahm and Sandmo

results are quite intuitive. In the comparative-statics presented in their paper, they

show that the extent of income reported is increasing in the penalty rate and


38

probability of detection. The effect of the tax rate θ is, however, ambiguous. We

therefore present a numerical analysis of this relationship in Section 3.2.2.

Furthermore, we return to the question asked in the previous section: Given

that the parameters for an interior solution hold, is it possible for the agent’s utility in

the ‘certainty scenario’ whereby he chooses not to evade any of his income, to be

higher than the expected utility under evasion? Specifically, would the outcome of

the model be any different if we allowed the agent to first decide whether or not to

evade, and then if he or she decides to evade, choose the amount to evade. We

consider the ‘evade or not’ choice in the model presented below.

B: The Allingham and Sandmo Model with ‘Evade or Not’ Choice

In that case, we would compare the indirect utility obtained from substituting

(3) into (1) with the indirect utility obtained by solving the model below:

( ) (9)

subject to

(10)

Here variables are analogously defined with ‘ne’ representing ‘not-evading’.

Assuming log utility, the indirect utility function for non-evasion (IUFNE) is given

by:

( ) ( ) (11)

Likewise, we can derive the indirect utility function in the Allingham and Sandmo

case by substituting for X* derived in equation (3) into the utility function (1).


39

Comparing the indirect utilities of the AS model (labeled IUFAS) and the model

without evasion (labeled IUFNE) gives the following:8

iff

( ) ( ( ))

( ) ( )

( ) ( ( )) ( ) ( ) (12)

where ( ( )

). We can see from equation (12) that the first term on the

left-hand side (LHS) is always going to be greater than the first term on the right-

hand side (RHS) as 0 < δ <1 if the conditions for an interior solution are satisfied.

For the indirect utility in the AS model to always be greater than the indirect utility

of the not-evade alternative, we would also require the second term on the LHS to

greater than the second term on the RHS. Again, we can show that if the conditions

for an interior solution are satisfied, the second term on the LHS is less than the

second term on the RHS, making it difficult to compare the expressions of both side

of the inequality.9

While our numerical simulations for this case show that IUFAS > IUFNE for

a large range of parameters compatible with condition (8), it is not possible to prove

this analytically. However, as will become clear in the subsequent sections, it is easy

to disprove an analogous proposition via numerical example in the two-period

extension of the basic model, which is considered in the next section. Specifically, in

8 For a derivation of this inequality see Appendix 3.1.

9 That the second term of the LHS is smaller than the second term of the RHS, provided pπ<θ, is

shown in Appendix 3.1.


40

the next model we consider in Section 3.3, there is a range of parameters for which

the interior solutions are satisfied, and the utility under certainty with no evasion is

higher.

3.2.2 Numerical Experiments with the Basic AS model and the ‘Evade or Not’

Model

We now conduct numerical experiments on the proportion of unreported

income, α, and the tax rate. The ‘benchmark’ set of parameters of the models in this

chapter are: θ=0.35; p=0.3; π=θ+0.32.10

We conduct experiments only on the range

of parameter values in which the conditions for an interior solution are satisfied.

When conducting numerical experiments on specific parameters, the rest of the

parameters are held at their benchmark rate given above. For the purpose of sections

3.2 and 3.3.1 we also assume a wealth level W=25.

We first consider how, in the basic AS model, the extent of evasion varies

with the tax rate θ, a feature that will be of relevance when interpreting the results of

the extensions that follow in this chapter. Note that we can write the amount of

reported income as ( ) , where α is the proportion of unreported wealth.

Figure 3.2 shows that the relationship between the proportion of unreported income

(α) and the tax rate (θ) is non-monotonic. The proportion of unreported income is

increasing for lower levels of the tax rate but decreasing for relatively higher levels.

The highest proportion of unreported income, where α=0.88, occurs at a tax rate of

around θ=0.4. A possible interpretation for Figure 3.2 could be due to income and

substitution effects associated with the changes in the tax rate. The substitution effect

10

Our choice of benchmark parameters is also related to the condition for an interior solution – we

select them in a way that permits a reasonable range for the simulations presented below. At this

stage, given that we dealing with fairly simple extension of the basic AS model, a full-fledged

‘calibration’ exercise is not feasible.


41

captures the fact that as tax rates rise, the opportunity cost from not evading becomes

higher. The ‘income effect’, according to Allingham and Sandmo, should be zero

for the case of CRRA preferences. However, there is another aspect of the

substitution effect here: as the tax rate goes up and, the amount of expected penalties

and fines increase for a given proportion of income evaded, making the opportunity

cost of not evading lower. In the figure below, the former effect seems to dominate

in the range in which the tax rate increases from 0.15 to 0.4, and the proportion of

unreported income increases. This latter effect comes into play when the tax rate

increases from 0.4 to 0.65.

Figure 3.2: Proportion of Unreported Income as the Tax Rate Increases.

We also illustrate our earlier discussion in relation to an ‘evade or not’ option

for the agent by comparing the utility under certainty which is labelled as IUFNE


42

and the utility in the AS model labelled as IUFAS in Figure 3.3. Figure 3.3 shows

that the indirect utility function of the AS model (represented by the blue line) is

always higher than the indirect utility function of the ‘evade or not’ choice

(represented by the green line) for the range of parameters that satisfy the interior

solution conditions. In this case, therefore, and ‘evade or not’ extension is not

applicable. The reason for presenting this comparison at this point is motivated by

the fact that it makes it easier to present and discuss a reversal of this result in

subsequent sections.

Figure 3.3: Comparison of Indirect Utility Functions of AS Model and ‘Evade or Not’

Model.


43

3.3 Towards a Macroeconomic Model of Tax Evasion: A Step-by-Step

Approach

3.3.1 AS Model with Two-Periods and its ‘Evade or Not’ Counterpart

3.3.1.1 Theoretical Analysis

A: The Allingham and Sandmo Two-Period Model

We now focus our attention on the AS model with time dimensions by

introducing two periods of consumption in the agent’s utility function. This amounts

to grafting the tax evasion decision into a standard Fisher (1930) two period small

open economy model of the type analysed by Obstfeld and Rogoff (1996).

Assuming log utility, the preferences of an individual are:

(

)

( ) ( )

(13)

where the superscripts c and nc denote the states where the agents are caught and not

caught respectively, and the subscripts 1 and 2 denote the different time periods, and

the variable C refers to consumption. Such an individual maximises equation (13)

subject to the following period 1 and 2 budget constraints depending on whether or

not his evasion has been detected. Equations (14) and (15) below refer to the budget

constraints for an individual in periods 1 and 2 in the state that he is caught. When

the state is ‘not caught’ his budget constraints are given by equations (16) and (17).

( ) (14)

( )

(15)

(16)


44

( )

(17)

As is obvious from the equations above, the agent has no wealth endowment in the

second period of his life and therefore must save to finance consumption in period 2.

We assume that this is a small open economy, which takes the world interest rate r

given. In addition, consumption and saving (denoted S) in the first period must not

exceed his or her disposable wealth, which further depends on whether his or evasion

has been detected. In this case, if we substitute (14)-(17) into (13), the first order

conditions for

are given by:

( )

( ) (18)

( )( )

( ) (19)

( )

( )

( )

( )

(20)

It is then straightforward to manipulate equations (14)–(20) to express the variables

in terms of

[ ( )]

(21)

( )

[ ( )] (22)

[ ]

(23)

( )

[ ] (24)

Substituting for ncc CC 11 , into (14) we can solve for X as follows:


45

[ ( )]

[ ]

( )

( )

Solving for the equation above for X, we get:

( )

(25)

We can see that the proportion of reported income, , is identical to the AS model,

as evident by comparing equation (3) with equation (25). This implies that extending

AS model by incorporating two periods and state-contingent planning of

consumption and savings does not have any bearing on the proportion of income

evaded, or the comparative static analyses of how this proportion of income changes

with respect to the parameters π, θ or p.11

Likewise, it is easy to show that the

restriction for an interior solution is the same as equation (8).12

However, we will

shortly find that it has an interesting implication for the ‘evade or not’ choice

discussed earlier.

B: The Allingham and Sandmo Two-Period Model with ‘Evade or Not’ Choice

Once again, we can compare the indirect utility obtained from equation (14)

with the indirect utility obtained by solving the model below:

( ) (

) (26)

subject to

(27)

11

As noted above, these analyses have been performed in the seminal Allingham and Sandmo (1972)

paper, for a more general case of the utility function, so such a repetition is unnecessary here. 12

Again such a derivation is unnecessary given our extension has yielded the same expression for X*

that the basis AS model did, but for the sake of completeness a derivation of the conditions for an

interior solution in this particular case is presented in Appendix 3.2.


46

( )

(28)


Assuming log utility, and deriving the optimal consumption and saving plans we can

substitute them into (26) to derive the indirect utility function for non-evasion

(IUFNE), which is given by:

( ) ( ) (29)

Comparing the indirect utilities of the AS model (IUFAS) and the ‘evade or not’

model (IUFNE) it can be shown that:13

iff

[(

)

] ( )

( )

(30)

where X* is given by equation (25).

Comparing the utilities of the two models, it is again not possible to prove the

proposition that the utility with evasion (IUFAS) is higher than the certainty scenario

(IUFNE) given that the conditions for an interior solution are satisfied. However, in

this instance, we are able to disprove this proposition via numerical example.

Specifically, there is a range of parameters for which the interior solutions are

satisfied, and the utility under certainty with no evasion is higher. This result

suggests that the ‘evade or not’ formulation is the more appropriate construct in the

two-period context, and is presented in the following section.

13

See Appendix 3.3 for this derivation.


47

3.3.1.2 A Simple Numerical Experiment Two-Period AS Model and ‘Evade or

Not’ Model

We conduct the numerical experiments based on the parameters for an

interior solution. We do not present the comparative statics analysis showing how X

varies with the other parameters p, π, and θ as the outcomes in that respect are

identical to those of the basic model. We do, however, for reasons outlined above,

want to analyse the indirect utility function of the AS model and the ‘evade or not’

choice in this two-period dimension.

Figure 3.4 below illustrates the results. In this two-period model, we can see

that the indirect utility function of the AS model (IUFAS) is not higher than the

indirect utility function for the ‘evade or not’ model (IUFNE) for a range of values

of the tax rate that are consistent with restrictions for an interior solution for X, the

amount of reported income. For relatively low tax rates, i.e. for θ between 0.15-0.25,

we find that choosing not to evade gives the agents a higher utility. In this case

therefore, an appropriate modelling of the tax evasion decision should incorporate an

‘evade or not’ choice. In what follows, therefore we build on this model further by

introducing heterogeneous agents and redistributive transfers. However, for the sake

comparison, we also present corresponding outcomes in analogous versions of the

AS economy, in which agents do not have an ‘evade or not’ choice.


48

Figure 3.4: Comparison of Indirect Utility Functions of AS Model and ‘Evade or Not’

Model.

3.3.2 Two-Period Model with Heterogeneous Agents, Redistributive Transfers

and Vote on θ


A: The Allingham and Sandmo Two-Period with Model Heterogeneous Agents and

Redistributive Transfers

We now take a step towards developing the model so that political economy

aspects may be addressed. This involves the construction of a model with

heterogeneous agents, so that the distributional implications of taxes and tax evasion

may be considered. The benchmark distribution, for example, is lognormal with


49

mean 3.2 and variance 0.8. We consider a sample of 501 values from this

distribution, with a Gini coefficient of .4073. The political economy angle is then

modelled in a simple way by allowing the agents to vote on their desired tax

structure. The agent’s decision making process now involves redistributive transfers.

It assumed that the tax-authority maintains a balanced budget so that average

revenue collection is the lump-sum transfer given to all individuals. Individuals do

not pay taxes, or receive transfers in the second period of their lives. The expected

revenue for lump-sum transfers in the AS model is given by:

∑ ∑{ ( )}

(31)

In the above equation, the first term is total revenue collected for income

reported while the second term is the expected revenue of the fines collected from

agents who evade taxes and are caught, where the expected revenue collected from

these agents is the total revenue evaded multiplied by the probability of detection p.14

The preferences of an individual are the same as that in equation (13) but the budget

constraints are altered and given by the following:

( ) (32)

( )

(33)

(34)

( )

(35)

where TR represents redistributive transfers, which are computed by averaging the

total revenue collected in expression (31) over all agents in the economy. We can see

that the budget constraints are now different from previous models as a result of the

14

For large N it is not unreasonable to assume that the probability of detection p is also the proportion

of agents in the economy who have been detected evading their incomes.


50

inclusion of redistributive transfers. The disposable income of the agent has now

increased which would alter his/her consumption and saving plans, and his or her

desired tax rate.15

One can also anticipate further differences to emerge in the ‘evade

or not’ formulation, given that transfers in a model with an ‘evade or not’ choice

would have to be calculated differently.

Note that the introduction of a lump sum transfer will not have an impact on the

first order conditions of the agent’s optimisation problem, and identical steps are

involved in deriving the optimal plans, which are now given by:

[ ( ) ]

(21’)

( )

[ ( ) ] (22’)

[ ]

(23’)

( )

[ ] (24’)

In addition, the proportion of reported income, X, is also identical to the expression

derived for the basic AS model (see equation (3)).

B: The ‘Evade or Not’ Choice Model with Heterogeneous Agents, Redistributive

Transfers and Vote on θ

Again the ‘evade or not’ model involves a comparison analogous to that

discussed in Section 3.3.1.1 above. Note, however, that there is a more complex

calculation involved for the agent in the ‘evade or not’ economy in relation to

redistributive transfers. In the ‘evade or not’ choice economy, the revenue collected

15

Note that transfers are taken as given by the agent, in the sense that he or she cannot individually

influence the vote on taxes.


51

for redistributive transfers is a combination of two parts. The first part is the revenue

collected from agents who do not evade and pay taxes on their actual income. The

second part involves the revenue collected from the taxes paid by agents who evade,

and the fines collected from the agents who evade and are caught. The revenue

collected from agents who do not evade (TNE) is given by the following:

∑

(36)

The expected revenue collected from agents who evade taxes (TE) is given by

the following:

∑

∑{ ( )}

(37)

The total revenue collected in the ‘evade or not’ model is then the sum of TNE

and TE. This revenue is averaged over all agents and distributed as a lump sum

transfer TR.16

Once the transfer for the ‘evade or not’ model is computed, it is easy to

describe the optimisation problem. Those who do not evade solve an analogous

version of the certainty problem in Section 3.3.1.1 B, with the transfer term

appearing on the RHS of the period 1 budget constraint. Those who evade face a

problem analogous to the one described in Section 3.3.1.1 A, with the term for

transfers appearing in counterparts of equations (21)-(24). An agent decides whether

or not to evade by comparing the respective utilities. Again, we cannot make an

analytical comparison of the utility functions and have to resort to numerical

16

Again not that agent’s regard this as exogenous as they cannot individually affect the aggregate

transfer in the economy. However, there is full and perfect information in this environment – each

agent knows the distribution of wealth and the preferences of the other agents and is therefore able to

compute the size of this transfer.


52

simulations, which are reported in the next section. Also note that for agents who

evade the amount of evasion is still determined by equation (3).

We have so far not described the political economy aspect of this model.

Before we do so, it is instructive to gain some intuition and insight regarding the

introduction of redistribution by means of some numerical simulations. These

simulations are presented below.

3.3.2.2 Numerical Experiments for Inequality

We first start by presenting numerical experiments on the number of evaders

in the economy when income inequality varies for different levels of the tax rate. For

ease of reference we remind the reader that the benchmark parameter values are:

θ=0.35; p=0.3; r=0.06; π=θ+0.32. In addition, the benchmark distribution is

lognormal with mean 3.2 and variance 0.8. We consider a sample of 501 values

from this distribution. This is close to the values chosen by Bearse, Glomm and

Janeba (2000), who argue that such a choice does a good job of capturing the actual

U.S household distribution in 1992 if income is measure in thousands of dollars. As

we are going to analyse implications for increasing inequality for the outcomes of

our model, we also consider several mean-preserving spreads of this distribution.

Table 3.1 below illustrates these results.


53

Table 3.1: Number of Evaders for Different Levels of Inequality

θ Gini=0.2735

No. of

Evaders

Evade or

Not Model

Gini=0.3439

No. of

Evaders

Evade or

Not Model

Gini=0.3807

No. of

Evaders

Evade or

Not Model

Gini=0.4073

No. of

Evaders

Evade or

Not Model

Gini=0.5895

No. of

Evaders

Evade or

Not Model

0.15 0 0 8 17 189

0.20 0 0 0 0 0

0.25 0 0 0 0 0

0.30 501 501 501 501 501

0.35 501 501 501 501 501

0.40 501 501 501 501 501

0.45 501 501 501 501 501

0.50 501 501 501 501 501

0.55 501 501 501 501 501

θ Gini=0.5975

No. of

Evaders

Evade or

Not Model

Gini=0.6736

No. of

Evaders

Evade or

Not Model

Gini=0.7143

No. of

Evaders

Evade or

Not Model

Gini=0.8346

No. of

Evaders

Evade or

Not Model

Gini=0.8388

No. of

Evaders

Evade or

Not Model

0.15 262 303 305 389 393

0.20 0 0 0 282 299

0.25 0 0 0 0 0


54

0.30 501 501 501 501 501

0.35 501 501 501 501 501

0.40 501 501 501 501 501

0.45 501 501 501 501 501

0.50 501 501 501 501 501

0.55 501 501 501 501 501

Note that the ‘evade or not’ economy is identical to the AS economy for

values of the tax rate greater that equal to 0.30. For the range of values below θ=0.3,

we note the following:

(a) We can see that different levels of inequality give rise to different

outcomes for tax evasion; for the range of parameters considered

here, the number of evaders increase as inequality increases.

When θ=0.15 for example, the number of evaders increases from

8 to 17 when inequality rises from 0.3807 to 0.4073, and

increases further to 189 when the Gini-coefficient is raised again

to 0.5895. This means that the number of non-evaders in this

economy is around 62.28%. This is consistent to the results found

in Christian (1994) where 60% of U.S taxpayers do not understate

their income. For relatively high levels of tax rates, however, all

agents in the economy evade taxes, regardless of the inequality

levels considered here. In such cases, then, the ‘evade or not’

model is identical to its AS counterpart.


55

(b) For a given level of inequality, the relationship between the

number of evaders and tax rates is a little less clear. From the

results there seems to be a non-monotonic relationship between

the number of evaders and the tax rates. For example, when the

Gini-coefficient is at 0.4073, the number of evaders tallies at 17

for θ=0.15. When θ rises to 0.20 and 0.25, however, the economy

has no evaders. For all other values of θ, ie. 0.30-0.55, the model

is similar to the AS model where all agents in the economy evade

to some extent from the payment of taxes. This non-monotonicity

is hard to explain, but intuition suggests that the non-monotonic

relationship between tax rates and the proportion of unreported

income, discussed earlier, may have something to do with it.

Specifically, we saw in Figure 3.2, once the tax rate

increases to beyond θ=0.4, the proportion of unreported income

decreases as θ increases. This means that while all agents in the

economy are evading, they evade smaller proportions of their

income as the tax rate increases beyond 0.4. Within this range, the

choice is between not evading at all or evading a relatively small

proportion of their income. The ‘evade or not’ choice prior to

that point, however, entails a comparison of utility from ‘not

evading’ with evading a proportion of income that increases as θ

increases. This proportion increases steeply for low values of θ,

before the substitution effect of expected penalties and fines

associated with higher tax rates makes the evasion decision more


56

costly. These differences in the two ranges of θ could be the

reason underlying the results we see in Table 3.1

In addition, an interesting feature of this model relates to the identity of the

evaders in the ‘evade or not’ variant; it is the poorest agents in this model that

engage in tax evasion. The benefit from evading in this model is more pronounced at

the lower end of the income distribution, as the consumption gains from evading are

higher when the level of consumption is low. This could, in part, also explain why

the number of evaders increases as inequality increases in the first two rows of Table

3.1; as inequality increases, there is a greater mass of agents at the lower end of the

income distribution, and all these agents experience higher marginal consumption

gains from evasion.

As mentioned before, while there is evidence to suggest that poorer agents do

evade more, we also consider an alternative construct in Section 3.3.4, in which the

richer agents evade more. This is achieved by incorporating a cost-of-evasion

function which is increasing in the proportion of income evaded. This distribution of

evaders will change, however, when we extend the model further to include a cost-

of-evasion function in Section 3.3.4. According to Cox (1984), middle income

taxpayers find it harder to evade since a larger share of their income derives from

wages and salaries that employers report to the authorities. This implies that the

probability of detection for the middle income agents is different than the high

income and low income agents. However, as the probability of detection is identical


57

for all agents in our economy, one may interpret our model as consisting of only the

poor and the rich with no middle class groups.17

3.3.2.3 Political Economy Extensions

Here we consider extensions of the models presented in Section 3.3.2.1 above

to include a political economy determination of one of the parameters of the tax

system. Essentially, we assume that voting takes place at the beginning of the period

and agents are allowed to vote on θ. After the vote, in economy A (the AS model)

agents make their evasion decision and state contingent plans, followed by the

auditing by tax authorities, after which transfers are made and the state contingent

plans are carried out. In economy B (‘evade or not’ model), the only difference is

that after the vote agents decide whether or not to evade, and if they choose to evade,

they decide how much to evade. Subsequently, auditing takes place, transfers are

made, and consumption and saving plans are carried out. The timing of events of the

political economy versions of the two economies is described in Figures 3.5 and 3.6

below.

17

Data suggests that it is the richest and poorest agents who evade more, see for example, Cox

(1984) and Bloomquist (2003).


58

Figure 3.5: Timeline for the basic model.

Figure 3.6: Timeline for model with ‘evade or not’ choice.

Period 1

Voting by

agents

Voting

outcome

revealed Auditing

Transfers

given

Period 2

Period

1state

contingent

plans

carried out

Agents

carry out

state

contingent

plans

period 2.

Period 1

Voting by

agents

Voting

outcome

revealed

Auditing

Transfers

given

Period 2

Period 1 state

contingent

plans/

deterministic

plans carried

out

Agents carry

out state

contingent /

deterministic

plans.

Agents decide

whether to

evade. Those

evading make

state

contingent

plans. Others

make

deterministic

plans

Agents

make

evasion

decision,

state

contingent

plans


59

3.3.2.4 Numerical Experiments for the Political Economy Extension

We now present some numerical results of the two models regarding the

political economy outcomes. Before we do so, however, it is instructive to look at

the indirect utility as a function of the tax rate, for various agents at different

positions in the income distribution. Figure 3.7 plots does this for the case of the AS

version of the model. Here, Agent 1 represents the poorest agent, while Agent 501

represents the richest agent in the economy, and agents are arranged in ascending

order of their income or wealth. Therefore Agent 251, for example, is the median

agent in the sample income distribution considered here.

Figure 3.7: Agents’ preferences over θ in the AS economy. (Simulation based on a 501 agent

economy, indexed in ascending order of their initial wealth, with benchmark parameters).


60

We can see that the preferences of Agent 1 and Agent 251, over a range of

tax rates, are non-single peaked. These agents prefer lower and higher levels of tax

rates in relation to some of the ‘in-between’ values. In addition, the results also show

that, as expected, agents on the higher end of the income distribution prefer

relatively low tax rates whereas, agents on the low to middle end of the income

distribution prefer relatively higher tax rates. A likely interpretation for the ‘non-

single peakedness’ could be due to the trade-off between the payment of taxes and

the expected benefits obtained from redistribution and transfers. It is possible that at

high levels of taxation, the revenue collected (and hence transfers) will be relatively

higher even though there is significantly greater evasion in the economy. In the

interests of getting a large transfer, an agent might therefore prefer a higher tax rate

over a ‘middle-level’ tax rate. In this instance then, the agent benefits from a higher

tax rate as he/she will not be paying the full amount of taxes (an evader) but receive

relatively high transfers. As we have seen from the results of Table 3.1 earlier, a

significant number of agents effectively ‘switch’ from not-evading to evading when

θ increases. Note that the ‘non-single peakedness’ observed in some cases will have

interesting implications for the political economy outcome of the vote on θ, the tax

rate, as the standard median voter theorem due to Black (1948) no longer applies in

this instance. In Figure 3.8 below, we present the utility functions of the agents in the

‘evade or not’ economy.


61

Figure 3.8: Agents’ preferences over θ in the ‘Evade or Not’ economy. (Simulation based on

a 501 agent economy, indexed in ascending order of their initial wealth, with benchmark

parameters).

The utility functions of the agents in the ‘evade or not’ economy differ

somewhat slightly than that of the AS model but are still non-single peaked.

Furthermore, a greater degree of non-monotonicity in the preferences has been

created by incorporating the ‘evade or not’ choice. In this model, agents on the

higher end of the income distribution still prefer relatively lower tax rates, and agents

on the very low end of the distribution still prefer high tax rates. In the case of the

middle income agents, however, the ‘non-single peakedness’ is more dramatic; they

prefer the relatively extreme levels of tax rates to the ‘in-between’ cases, and seem to


62

be indifferent between relatively low tax rates and high tax rates.18

We conjecture

that the reason for the ‘non-single peakedness’ characteristic in the ‘evade or not’

model is similar to that AS model described earlier. That is, the possible trade-off

between the benefits gained from transfers/redistribution and the costs of paying

taxes.

The results of the political economy outcomes for our benchmark parameters

are presented in Figures 3.9 and 3.10 below. These figures present the percentage of

votes in favour over different values of θ, for our benchmark parameters with the

Gini coefficient of the distribution set at 0.4073. Figure 3.9 presents the results for

the AS economy while Figure 3.10 represents the corresponding outcomes for the

‘evade or not’ economy. In relation to voting outcomes of the model, we find that in

both the AS model and the model with the ‘evade or not’ choice the winning value of

θ is 0.55, with 72.85% of the vote in the AS economy and 51.30% of the vote in the

‘evade or not’ economy. This is the highest choice available to the agents given the

interior condition discussed in previous sections. The second highest vote in the AS

model is for θ=0.2 with around 27% of the vote, while in the ‘evade or not’ model

the θ=0.15 received roughly 33% of votes. As mentioned before, we find that the

political economy outcome of this model is driven by equity considerations, in spite

of the presence of tax evasion. This is in part due to the fact that firstly,

redistribution occurs even in the presence of evasion, and secondly, there are no

administrative costs so that revenue collected from penalties and fines can be used

18

Note that the non-single peakedness of preferences is a feature common to several political

economy models with voting, such as Epple and Romano (1996a) and also in political economy

models of tax evasion, such as the model described in Borck (2009). Typically, outcomes in these

models can be determined by groups of individuals other than the median voter. Sometimes an ‘ends

against the middle’ feature appears so that agents at the bottom and top ends of the distribution

determine the outcome of the vote. As will be evident later, this can happen within the context of the

models discussed in this thesis. Even so, within the context of these models it is a relatively rare

occurrence and happens in some cases for a narrow range of parameters.


63

for redistribution. We see that in both models, the highest available tax rate is

preferred in the presence of inequality.

What is also interesting is that the distribution of votes in the two models

differs on the low-end of the tax spectrum. In the AS economy, the second highest

percentage of votes is for the tax rate θ=0.2. In the ‘evade or not’ economy, however,

the second highest percentage of votes is for the lowest value of the tax rate

available, θ=0.15. Note that in the presence of different voting procedures, such as a

majority run-off, the political economy outcomes for the tax rate could be very

different. In a majority run-off procedure, for example, there is a second stage to the

voting process in which the outcomes with the highest and second-highest votes are

pitted against each other. In such cases, if the outcome with the highest number of

votes does not have a majority, interesting outcomes can occur depending on how

voter preferences are distributed in relation to the remaining two alternatives.

Furthermore, given the nature of preferences described in the Figures 3.7and 3.8, it is

evident that richer models that include lobbies or power groups could lead to a

different political outcome, and would consequently be an interesting direction of

future research.


64

Figure 3.9: Percentage of votes for various values of θ in the AS economy.


65


economy.

Next we look at the voting outcomes on θ with various levels of inequality.

We can see that in general, increasing the level of inequality does not alter the voting

outcome of the tax rate in both models. Only for a very low level of inequality

(Gini=0.2735) in the ‘evade or not’ model does the voting outcome change. In this

instance, the highest percentage of votes is for the tax rate of θ=0.2. This is a

striking difference between the two models. On the other hand, in the AS model, for

a low level of inequality (Gini=0.2735), the highest percentage of votes is for the tax

rate θ=0.55. This result is perhaps due to the increased non-monotonicity in the

agent’s utility function that we saw in Figure 3.8; such features typically make

intuitive interpretations challenging. It is of interest to note now, that this feature

becomes more pronounced when we include a non-linear tax function and a variation

of the fines/penalty modelling structure which will be discussed in Chapter 4.


66

It is also interesting, that in the ‘evade or not’ model, for a range of

inequalities, the winning vote on θ is not a majority outcome but a plurality one. If

we allowed the voting outcome to be determined by a majority runoff in this case,

we would still end up with 0.20 as the winning outcome in the ‘evade or not’ case.

This is due to the fact that 29.94% of the votes was for θ=0.55 and the remaining

26.95% of the votes was for a tax rate of θ=0.15. A majority runoff in this scenario

would not alter the outcome of the tax rate as those agents who voted for θ=0.15

would now vote for θ=0.20 as opposed to θ=0.55. Referring back to Table 3.1, recall

that for low values of the tax rate, all agents typically chose not to evade. It would

seem, then, that preferences of the majority of agents are to choose a tax rate that

leads to less evasion, and in this case a tax rate of θ=0.20 leads to no evasion in the

economy.

Table 3.2: Vote on θ for Different Levels of Inequality

Gini Vote on θ

(Evade or

Not)

% in Favour

(Evade or

Not)

Vote on θ

(AS model)

% in Favour

(AS model)

0.2735 0.2000 43.1138 0.5500 72.2555

0.3439 0.5500 40.9182 0.5500 74.2515

0.3807 0.5500 48.1038 0.5500 71.8563

0.4073 0.5500 51.2974 0.5500 72.8543

0.5895 0.5500 67.2655 0.5500 78.0439

0.5975 0.5500 67.6647 0.5500 77.4451

0.6736 0.5500 73.0539 0.5500 78.6427

0.7143 0.5500 77.2455 0.5500 82.0359


67

0.8346 0.5500 86.6267 0.5500 89.0220

0.8388 0.5500 86.0279 0.5500 88.2236

3.3.3 Two-Period Model and Political Economy with Cost of Evasion


A: The Allingham and Sandmo Model with Cost of Evasion

We now analyse a two-period political economy model with a cost function

associated with evading taxes. The model is similar to the one described in the

earlier section with one exception, the decision to evade taxes, as in Chen (2003),

now involves a cost described by ( ), which is increasing in α, and where α is the

proportion of unreported income. As discussed earlier, we interpret our cost of

evasion function to consist of both pecuniary and non-pecuniary elements. The

pecuniary component may consist of bribes, while the non-pecuniary element may

consist of a sense of guilt from non-payment of taxes. In addition, agents are

heterogeneous in their wealth endowment. The preferences of an individual are given

by the following:

(

)

( ) ( )

(38)

such an individual maximises equation (52) subject to the following budget

constraints:

( ) ( (

) )

(39)


68

( )

(40)

( ) (41)

( )

(42)

here TR represents transfers, ( ) where α (the decision variable in this

instance) is the amount of unreported income, and is the cost function that

varies with α. The first-order conditions, with respect to

, and , for an

optimum are now given by the following:

( )

( ) (43)

( )( )

( ) (44)

{( ) } ( )

{ } (45)

We can see that the first-order condition with respect to α is no longer the same as in

the previous models. In addition, conditions for an interior solution have changed in

this case.19

B: ‘Evade or Not’ Choice Model with Cost of Evasion

Again the ‘evade or not’ model involves a comparison analogous to that

discussed in Section 3.3.1.1 above. For completeness, however, we state them briefly

here. The preferences of an individual in the ‘evade or not’ model are:

( ) (

) (46)

19

We cannot, however, derive these conditions analytically, and we no longer have an analytical

solution to the agents’ problem. While we can express all other variables in terms of W and α, the

solution for α can only be determined by numerically solving (45). The computational procedure for α

is done by setting up a grid that ranges from 0.00001 to 0.999 with increments of 0.001. The optimal

value of α is found by evaluating the first-order condition at different values of α to find the point at

which (45) holds with equality.


69

Such an individual maximises equation (46) subject to the following budget

constraints:

(47)

( )

(48)

Those who do not evade solve an analogous version of the certainty problem in

Section 3.3.1.1, with the budget constraints given by equations (47)-(48). Those

who evade face a problem analogous to the one described in Section 3.3.2.1, with the

budget constraints given by equations (39)-(42). Likewise, an agent decides whether

or not to evade by comparing the respective utilities. Again, since it is not possible to

do so analytically, we resort to numerical solutions to determine the agent’s decision.

The numerical experiments for both the AS model and the ‘evade or not’ model are

presented in the following section.

3.3.3.2 Numerical Experiments

We first start by considering the distribution of wealth and the proportion of

unreported income (α) in the two models. Figure 3.11 plots the proportion of

unreported income for individuals in the AS economy and ‘evade or not’ economy

for a value of θ=0.15 and a Gini-coefficient of 0.3439. The blue line represents the

AS model and the green dots represent the ‘evade or not’ model. An interesting

outcome that has emerged is that the proportion of unreported income (α) now varies

with wealth. This is in stark contrast to the models without a cost-of-evasion

function where the proportion of reported income X was a constant with respect to


70

wealth. This result seems more realistic as we would expect richer agents to evade a

higher proportion of their income.20

The distribution of evaders has also changed. It is now the richer agents in

the economy who evade taxes while the poorest agents do not evade and report their

full income. We can also see that the extent of evasion increases with wealth in both

models. First consider evasion in the AS model. Here the proportion of unreported

income by the wealthiest agents is around 0.055, whereas the proportion of

unreported income by the lower income agents is around 0.015. This result is

substantially lower than the standard AS formulation and the proportion of income

evaded is much more realistic in this instance. For example, according to Bloomquist

(2003), the IRS estimated that taxpayers who filed returns reported about 99% of all

wage income and those with non-farm proprietor income reported about 67.7%.

In the ‘evade or not’ economy, however, we can see that a large number of

agents choose not to evade taxes, reducing the extent of evasion in another sense,

and thereby taking the model a step further to what is observed empirically. In a

sample of 501 agents only 110 of the richest agents choose to evade. Once again, this

result is more consistent with the empirical evidence. As mentioned earlier,

according to Christian (1994), 60% of U.S taxpayers do not understate their income.

In our ‘evade or not’ model, about 78% of agents reported their true income and do

not evade from the payment of taxes.

20

This is the reason for which a large number of tax evasion studies assume DRRA preference (see

for example, Jung et al. 1994). For our purposes, as emphasised earlier, we wish to preserve the

CRRA assumption discussed in Section 3.1.


71

Figure 3.11: Proportion of unreported income (α) as a function of wealth for θ=0.15 and

Gini=0.3439.

We then turn to the effect of inequality on the number of evaders in the

‘evade or not’ model (recall that in the AS model all agents in the economy evade).

We can see from Table 3.3 that different levels of inequality give rise to different

outcomes for θ=0.15 and θ=0.2.21

The extent of evasion in the economy does indeed

change when inequality varies. However, the effect of inequality seems to be non-

monotonic with respect to the number of evaders in the economy. For example, the

number of evaders increases from 22 to 110 when inequality rises from 0.2735 to

0.3439 but falls to 20 when the Gini-coefficient is raised further to 0.4073.

Interestingly, the introduction of a cost-of-evasion function seems to have increased

21

Note that with the introduction of a cost-of-evasion function, the range of parameters is longer

restricted to the conditions for an interior solution given in the previous section. Our selection for the

range of θ in these experiments relate to realistic values of tax rates that is observed around the world.


72

the number of evaders in the economy in comparison to the model without a cost-of-

evasion function. Once again, there is a non-linear relationship between the extent of

evasion and tax rates, and changes in the distribution further interact with this non-

linearity. An intuitive explanation for this feature is hard to come by but we

speculate that this may have something to do with the trade-off between the payment

of taxes and the expected benefits obtained from redistributive transfers.

Table 3.3: Number of Evaders for Different Levels of Inequality with Cost of

Evasion

θ Gini=0.2735

No. of

Evaders

Evade or

Not Model

Gini=0.3439

No. of

Evaders

Evade or

Not Model

Gini=0.3807

No. of

Evaders

Evade or

Not Model

Gini=0.4073

No. of

Evaders

Evade or

Not Model

Gini=0.5895

No. of

Evaders

Evade or

Not Model

0.15 22 110 0 20 9

0.20 496 495 501 330 501

0.25 501 501 501 501 501

0.30 501 501 501 501 501

0.35 501 501 501 501 501

0.40 501 501 501 501 501

0.45 501 501 501 501 501

0.50 501 501 501 501 501

0.55 501 501 501 501 501

0.60 501 501 501 501 501

0.65 501 501 501 501 501


73

θ Gini=0.5975

No. of

Evaders

Evade or

Not Model

Gini=0.6736

No. of

Evaders

Evade or

Not Model

Gini=0.7143

No. of

Evaders

Evade or

Not Model

Gini=0.8346

No. of

Evaders

Evade or

Not Model

Gini=0.8388

No. of

Evaders

Evade or

Not Model

0.15 13 37 501 501 501

0.20 501 501 501 501 501

0.25 501 501 501 501 501

0.30 501 501 501 501 501

0.35 501 501 501 501 501

0.40 501 501 501 501 501

0.45 501 501 501 501 501

0.50 501 501 501 501 501

0.55 501 501 501 501 501

0.60 501 501 501 501 501

0.65 501 501 501 501 501

Next, we analyse the indirect utility functions of the agents over their

preferences of θ in the AS economy. The results are presented in Figures 3.12 and

3.13. Immediately, we can see that the utility functions of the agents are single-

peaked once the cost-of-evasion parameter is incorporated in the model. This is in

contrast to the previous model (the model without a cost of evasion) where the

indirect utility functions of the agents were non-single peaked. In the ‘evade or not’

model, we can see that the preferences of the agents over the tax rate are also single

peaked. As a result, the voting outcomes and distribution of votes may be different to


74

the model without a cost associated with tax evasion (which will be explored

shortly).

In addition, the indirect utility functions show that agents on the low and

middle end of the income distribution prefer relatively high tax rates, whereas the

richer agents prefer relatively low tax rates. This is due to the effect of redistributive

transfers discussed earlier. In addition, in the ‘evade or not’ model, Agent 351,

seems to prefer tax rates that are in the middle of the spectrum as opposed to the

ends. A tax rate of θ=0.4 gives Agent 351 the highest utility from the considered

range. This suggests that the pattern of votes, in terms of the percentage of agents in

favour of any particular tax rate, may be different relative to the previous model.

Figures 3.12 and 3.13 present the results.

Figure 3.12: Agents’ preferences over θ in the AS economy.


75

Figure 3.13: Agents’ preferences over θ in the ‘Evade or Not’ economy.

Our intuition is confirmed by the results presented in Figures 3.14 and 3.15,

which present the results of voting over different values of θ. As expected, the

inclusion of a cost of evasion has not altered the winning outcome of θ;

redistribution is favoured in both models in the presence of inequality. Furthermore

we get stronger results in favour of such outcomes given the single peakedness of the

preferences. In this case, we may simply look at the median agent’s (i.e. agent

251’s) preference to determine the voting outcome. The majority of votes in both

economies are in favour of the highest tax rate of θ=0.65. The second preferred tax

rate in both economies is θ=0.15. This is in contrast to the ‘evade or not’ model

without a cost-of-evasion function where the second preferred tax rate was θ=0.2.

The distribution of votes in both the AS model and the ‘evade or not’ alternative is

also almost identical. Figures 3.14 and 3.15 below present the results.


76

Figure 3.14: Percentage of votes for various values of θ in the AS economy.


economy.


77

Finally, in Table 3.4 below, we look at the voting outcomes on θ with various

levels of inequality with the introduction of a cost-of-evasion function. We can see

that in both the AS and ‘evade or not’ model, increasing the level of inequality does

not alter the voting outcome of the tax rate. The voting outcomes of the models are

identical, with agents in the economy voting for the highest tax rate available to them

of θ=0.65. This is in contrast to the model without cost of evasion presented in the

previous section where for an inequality level of Gini=0.2735, the vote in the ‘evade

or not’ model was θ=0.20. The percentage of votes in both the AS and ‘evade or not’

model are also identical, and in this instance we get a majority outcome where the

winning value of θ is always greater than fifty percent. This is in contrast to the

extensions without a cost-of-evasion function where the outcomes were the result of

a plurality vote.

Table 3.4: Vote on θ for Different Levels of Inequality with Cost of Evasion

Gini Vote on θ

(Evade or

Not)

% in Favour

(Evade or

Not)

Vote on θ

(AS model)

% in Favour

(AS model)

0.2735 0.6500 58.4830 0.6500 58.4830

0.3439 0.6500 66.6667 0.6500 66.6667

0.3807 0.6500 65.0699 0.6500 65.0699

0.4073 0.6500 68.0639 0.6500 68.0639

0.5895 0.6500 72.2555 0.6500 72.2555

0.5975 0.6500 72.2555 0.6500 72.2555

0.6736 0.6500 74.4511 0.6500 74.4511

0.7143 0.6500 79.0419 0.6500 79.0419


78

0.8346 0.6500 86.8263 0.6500 86.8263

0.8388 0.6500 86.0279 0.6500 86.0279

3.4 Conclusion

The key objective of this chapter has been to provide the necessary first steps

in the modelling of tax evasion within a macroeconomic framework. This involves

the construction of a model with heterogeneous agents and redistributive transfers,

so that the implications of tax evasion may be considered. The political economy

angle is then modelled in a simple way by allowing the agents to vote on their

desired tax structure. More importantly, the framework we construct incorporates the

idea that agents typically face various trade-offs that can only be realistically

modelled within a macroeconomic framework.

The results of our analysis lead to some interesting insights. The introduction

of the ‘Evade or Not’ feature of the model is a key contribution to the literature

because it reduces the extent of evasion even in the context of a very simple

macroeconomic model of tax evasion. We find that the extent of evasion in the

‘evade or not’ alternative is much lower and more consistent with the empirical

evidence.

Another realistic outcome that emerges is that the extent of evasion is

increasing in wealth. Typically tax evasion studies have to resort to DRRA

preferences to achieve levels of evasion that are increasing in wealth, a feature that

has some empirical support in the literature. 22

In the context of the model of this

chapter, this is achieved while still maintaining CRRA preferences. This is

22

See for example, Vogel (1974).


79

important in the sense that macroeconomic models require preferences to be

restricted to the CRRA class if they are to be consistent with some stylised facts

pertaining to business cycles and economic growth. There is also a reduction of the

extent of evasion in another sense: the percentage of evaders in the model economy

is also reduced to numbers that are more consistent with the empirical estimates in

the literature.

We find that, for a range of values of the tax rate, the ‘evade or not’ model

always produces a lower amount of evasion in comparison to the AS model. In

addition, we find that within this range, the extent of evasion increases with

inequality. Furthermore, an interesting outcome emerges in relation to the mix of

evaders in the distribution. For low levels of the tax rate evasion is concentrated at

the bottom end of the income distribution and this tendency is exacerbated when

inequality rises. When we introduce a cost-of-evasion function, we see that the

identity of evaders in the distribution have now switched. It is now the richer agents

rather that the poor agents who evade from the payment of taxes. The results also

show, that in this instance, the effect of inequality seems to be non-monotonic with

respect to the number of evaders in the economy.

The political economy outcomes of the models are also of interest. We find

that in the vast majority of cases, redistribution is favoured in both the AS model and

the ‘evade or not’ model in the presence of inequality. One notable exception is for

one special case of the ‘evade or not’ construct without a cost-of-evasion function

for a very low level on inequality. In this instance, we find that the agents prefer

‘efficiency over equity’ and vote on a low level of progressivity. This feature re-


80

emerges in the ‘evade or not’ model of the next chapter. Finally, we find that the

level of inequality does not seem to matter in relation to the tax structure.

The experiments of this chapter suggest some interesting directions for future

research. A two-period framework of the type studied in this chapter can be easily

extended to a two-period overlapping generations model, a construct commonly used

in macroeconomics to study issues that have a long-run, dynamic flavour, such as

economic growth and the persistence of inequality. While is not the intention of this

thesis to look at the economic growth related implications of the model, we continue

to focus on inequality, progressivity and tax evasion, and the interaction of these

elements in a macroeconomic environment. To that end, we extend the model of this

chapter to an overlapping generations model that includes intergenerational links due

to the presence of bequests left to the next generation. The purpose of this extension

is twofold: firstly, the addition of another variable introduces another dimension

along which the agent faces economic trade-off, which in turn interacts with the tax

evasion decision, taking the model a step closer to reality. Secondly, the presence of

these intergenerational links allows us to address another important issue, namely,

the implications of tax evasion for persistence in inequality.

We also explore a slight, and arguably more realistic variation of the

construct studied in this chapter, in relation to the fines imposed on the agent in the

state that he or she is caught evading taxes, which will be described and motivated in

the next chapter. Another issue of interest relates to the structure of taxation, and we

are interested in exploring whether results obtained in this chapter are robust to the

inclusion of more non-linear structure capable of generating greater progressivity in

taxes. The model of the next chapter incorporates these features, and we find that


81

several of the appealing implications of the models of this chapter are preserved in a

more detailed, realistic macroeconomic framework. We also find some interesting

results in the political economy context, and in relation to the persistence in

inequality.

On Inequality, Tax Evasion and Progressive Taxes Chapter 4

82

CHAPTER 4

On Inequality, Tax Evasion and Progressive Taxes

4.1 Introduction

In this chapter we take the final step in the construction of a macroeconomic

framework to analyze the political economy determination of progressivity in the

presence of tax evasion and inequality. Specifically, we graft the final version of

two-period model of the previous chapter within a standard two-period overlapping

generations construct with heterogeneous agents linked across generations due to the

presence of bequests. The political economy structure remains very similar to the

setting of the previous chapter, and now resembles several simple political economy

models in the macroeconomics literature which also incorporate voting in some form

over a policy parameter. (See for example Alesina and Rodrick 1994, Lahiri and

Ratnasiri 2010, and Dolmas, Huffman and Wynne 2000).

In addition, we also incorporate a non-linear taxation structure in the models

that can further enhance the degree of tax progressivity in the economy. Even in the

context of earlier models of tax evasion some conclusions of the standard AS model

have not been robust to the inclusion of non-linear tax schedules and we revisit these

issues here (see for example Pencavel 1979). Non-linear tax structures have been

widely used in macroeconomic models such as those in Reiter (2000) and Yamarik

(2001), and we introduce a non-linear tax function here that also nests the linear tax

case in our extensions.


83

We also consider an alternative specification of the penalties and fines. The

motivation for this stems from the fact that in the context of poorer agents, the fines

from evading may be excessively high in relation to their income, restricting them to

a very low level of consumption for the agent in the state where he/she is caught, to

the extent that the ‘punishment does not fit the crime’. We therefore, model the

penalties/fines as a fraction of the agent’s disposable income as opposed to a penalty

paid on evaded taxes. Of course, the amount paid in fines is in addition to amount of

unpaid taxes that are also collected from the agent when he or she is caught.

The results of our models produce some interesting insights. Qualitatively,

the results that emerge from this chapter are similar to the results of the last model in

the previous chapter, except in relation to the political economy results, which will

be discussed shortly. That is, the ‘evade or not’ model in comparison to the AS

model reduces the number of evaders in the economy. The results also show that,

when there is a cost associated with evasion, it is the wealthiest agents in the

economy that engage in tax evasion. Recall that this was also the case in the last

model of the previous chapter when the cost-of-evasion function was introduced.

On a quantitative level, however, the results differ in some respects.

Introducing a variation on penalties seems to increase the extent of evasion in the

models. In both the AS and ‘evade or not’ model, the proportion of unreported

income is much higher for the agents that choose to evade. There is also an increase

in the number of evaders for low levels of the tax rate but a very slight decrease for

higher levels of the tax rate. The latter result reflects the fact that a more progressive

tax schedule implies a higher tax burden relative to a linear tax schedule, which can

have negative impact on the extent of tax evasion if we are to follow the results and


84

intuition gained from the previous chapter. Recall, for example, that in the basic AS

model the relationship between tax rates and the proportion of income detected is a

non-monotonic one. We find from our numerical analysis that this feature,

combined with the fact that there is now greater progressivity in the tax schedule, has

an interesting implication for the model of this chapter: we find that at higher levels

of wealth the proportion of income evaded drops as wealth increases. This aspect is

in contrast to the last model of the previous chapter, which incorporated a cost-of

evasion function, which has been retained in the model of this chapter. In that

model, the proportion of income evaded was a monotonically increasing function of

wealth.

The political economy outcome of the ‘evade or not’ model also produces

some striking differences. In the ‘evade or not’ model, the agents vote for the lowest

possible tax rate available to them. Recall that in the previous section, this was only

the case for a very low level of inequality in the ‘evade or not’ model without a cost-

of-evasion function. The results for the AS model, however, are identical to the

results of the AS models in the previous chapter. That is, agents in the AS economy

prefer redistribution and vote for the highest possible tax structure presented to them.

The remaining sections of the chapter are organized as follows. In Section

4.2.1, we first describe a ‘benchmark model’, which integrates the approach

followed in the tax-compliance literature pioneered by Allingham and Sandmo

(1972) to a heterogeneous-agent dynamic equilibrium model. In Section 4.2.2 we

present an extension of the basic model which allows agents to initially decide

whether they wish to evade taxes or not. In Section 4.2.1 we consider political

economy extensions of the models presented in Sections 4.2.1 and 4.2.2. In Section


85

4.3, we reiterate some theoretical issues pertaining to our modeling choices.

Specifically, we motivate why incorporating an ‘evade or not’ decision into the basic

construct is necessary, given that the basic model is flexible enough to include the

corner solution in which no evasion takes place. In Section 4.4, we discuss the

parameterization of the model, while in Section 4.5 we analyze the results of some

quantitative experiments based on the models described in Section 4.2. Section 4.6

provides a brief discussion on the wealth dynamics of the models. Concluding

remarks are presented in Section 4.7.

4.2 The Economic Environment

4.2.1 The Benchmark Economy

We consider a small open economy of 2-period lived overlapping generations

of agents. Time is discrete, with ,...2,1,0t , and there is no population growth.

Agents are heterogeneous with respect to inheritance received from the previous

generation. The distribution of this inherited wealth for the generation born in period

t is given by )(WFt , which represents the fraction of the population with wealth less

than or equal to W. An agent therefore has a wealth endowment tW , and is expected

to pay taxes according to the function )( tWt , which satisfies the feasibility

conditions: (1) WWt )( , i.e., taxes paid cannot exceed the wealth endowment, and

(2) 1)( Wt , i.e., disposable income is non-decreasing in the initial pre-tax

endowment (see for example, Yamarik 2001, and Donder and Hindriks 2004). Note

that the tax structure considered in this chapter is general enough to nest the linear

tax as a special case, depending on the parameters of the functional form in question.


86

However, the individual does not necessarily report all of his or her wealth,

and therefore pays taxes on the amount W)1( , where is the proportion of

unreported income. This decision involves a cost described by )(d , which is

increasing in α. The decision regarding the proportion is taken prior to the audit

by the tax-authorities, as is the decision regarding the consumption-saving plan of

the agent, which will be described shortly. The probability of detection of this

evasion and the subsequent punishment after the audit is given by p , where

]1,0[p . The punishment involves payment of any unpaid taxes, and a proportion

of the income that is left over after all taxes have been paid which we define as ϕ

where ϕ ϵ[0,1]. Conventionally the modelling strategy would involve an imposition

of a penalty in proportion of the amount of unpaid taxes. However, it is possible that

this reduces the agent’s disposable income in the state when caught to a negative or a

very low amount. In that case one would have to impose inequality constraints on

the optimisation problem, which would complicate matters without impacting on the

results in a qualitative sense. Secondly, too low a consumption level in this state is

unappealing in the sense that the punishment from evasion would be excessive.

The tax-authority maintains a balanced budget so that average revenue

collection is equal the lump-sum transfer given to all individuals born in t.

Individuals do not pay taxes, or receive transfers in the second period of their lives.

The expected equilibrium lump-sum transfer is given by:

maxmaxmax

minˆˆ

)()()()()()()(

W

W

W

W

d

W

W

d WdFWdycpWdFWtWtpWdFWt .

In the above equation dyc represents ‘disposable income when caught’, and

t

d WW )1( , the proportion of wealth that is reported to the tax authority. Also,


87

Wmin and Wmax refer to the initial wealth levels of the poorest and richest agents in

the economy, so the integral in the first term is over the entire support of the

distribution , given by [Wmin Wmax]. The wealth level W refers to the critical level

of wealth beyond which agents choose to evade taxes.23

The second integral then

refers to the unpaid part of tax revenue which is collected from evaders that are

caught. This term is therefore pre-multiplied by the probability of detection, p, since

only a proportion p of non-evaders are caught. The last term, similarly refers to the

expected penalties collected from the evaders that are caught.

The preferences of an individual born in t are given by:

)1(.)()()()1()()()( 1111

nc

t

nc

t

nc

t

c

t

c

t

c

t bvcucupbvcucup

Here, we assume that u and ν are concave and twice continuously

differentiable. The superscripts c and nc represent the states “caught” and “not-

caught”, and individual chooses his/her state-contingent consumption, saving and

bequest plan nc

t

c

t

nc

t

c

t

nc

t

c

t

nc

t

c

t bbccsscc 1111 ,,,,,,, , and the proportion of W that is

unreported, , to maximize (1) subject to the following budget constraints:

],)()()[1( c

ttt

c

t sdWtWc (2)

)3(,)1( 11

c

t

c

tt

c

t bsrc

23

Recall that in the model of the previous chapter introducing a ‘cost-of-evasion’ function resulted in

the richer agents of the economy evading taxes. This feature is retained in the model of this chapter.


88

)4(,)()( nc

t

d

t

nc

t sdWtWc

)5(.)1( 11

nc

t

nc

tt

nc

t bsrc

In the above equations, tr is the exogenously determined world interest rate

faced by this small open economy. Substituting the constraints (2)-(5) in (1), and

maximizing over the choice of ,,,, 11

nc

t

c

t

nc

t

c

t bbss , yields the following first-order

conditions for an optimum:

)6(),()1()( 1

c

tt

c

t curcu

)7(),()1()( 1

nc

tt

nc

t curcu

)8(),()( 11

c

t

c

t bvcu

)9(),()( 11

nc

t

nc

t bvcu

)10(0)()()()1()()1)(( dWWtcupdcup t

dnc

t

c

t


89

Equations (6)-(9) are the Euler equations that are fairly standard in models of

this type and have the usual interpretations. Equation (10) equates the marginal

expected loss from evasion when caught, to the marginal benefit from evasion. This

interpretation is perhaps easier to see if one recognizes that (6), (7), (8), and (9)

imply

)(

)(

)(

)(

)(

)(

1

1

1

1

nc

t

c

t

nc

t

c

t

nc

t

c

t

bv

bv

cu

cu

cu

cu

.

Let this ratio be represented by , which is a function of W and parameters of the

model, but for given parameters and W can be regarded as a constant. Then, (10)

can be simplified to:

)()]1()1([)()1( dppWWtp t

d .

Assuming )log()( ccu , and )log()( bbv , it is straightforward to

manipulate equations (2)-(9) in order to express the variables

nc

t

c

t

nc

t

c

t

nc

t

c

t

nc

t

c

t bbccsscc 1111 ,,,,,,, in terms of tW and :24

)11(,)()()2(

)1(

dWtWc tt

c

t

)12(,)()()2(

1

dWtWc d

t

nc

t

24

See Appendix 4.1 for a detailed derivation.


90

)13(,)()()2(

)1)(1(1

dWtW

rc tt

c

t

)14(,)()()2(

)1(1

dWtW

rc d

t

nc

t

)15(,)()()2(

)1)(1(1

dWtW

rb tt

c

t

)16(,)()()2(

)1(1

dWtW

rb d

t

nc

t

)17(,)()()2(

)1)(1(

dWtWs tt

c

t

)18(.)()()2(

)1(

dWtWs d

t

nc

t

Substituting for c

tc and nc

tc in (10), we then have an implicit equation in

and tW . In the following section, we will assume that WWt )( , and that


91

2

0)( dd , )1,0( . The cost-of-evasion function and associated parameters are

interpreted as in Chen (2003), and similar to the function introduced at the end of the

previous chapter. In the tax function, the parameter represents the degree of

progression in taxes: 1 implies a non-linear progressive tax scheme in which

marginal tax rates are increasing in income and wealth, and 1 represents a

‘counter-factual’ regressive tax scheme with decreasing marginal tax rates. In the

1 case, the marginal tax rate is constant, represented by the parameter θ, in which

case the model has a tax structure similar to that of the previous chapter. Making

these further substitutions in equation (10), one can numerically solve for * .25

Once * is known equations (11)-(18) can be used to derive optimal values of other

variables. Furthermore since (15) and (16) respectively represent bequests (and

consequently the next generation’s wealth) in the cases evasion is detected and not

detected, we can represent the evolution of wealth by:

)19()1(),(

),()(

2

1

1

pyprobabilitwithW

pyprobabilitwithWWW

t

t

tt

where

)20(,)()()2(

)1)(1()( *

1

dWtW

rW ttt

25

The computational procedure for α is done by setting up a grid that ranges from 0.00001 to 0.999

with increments of 0.001. The optimal value of α is found by evaluating the first-order condition at

different values of α to find the point at which (10) holds with equality.


92

)21(.)())(()2(

)1()( **

2

dWtW

rW d

tt

4.2.2 The Model with the ‘Evade or Not’ Choice

In this variation we allow agents in the economy to choose whether or not to

evade taxes by comparing expected utilities from evading or not evading taxes. If

not evading taxes, agents born in t maximize

)22()()()( 11

ne

t

ne

t

ne

t bvcucu

subject to

)23(,)( ne

ttt

ne

t sWtWc

)24(.)1( 11

ne

t

ne

t

ne

t bsrc


In this case the optimal consumption and bequest plans are given by

)25(,)()2(

1tt

ne

t WtWc


93

)26(,)()2(

)1(1 tt

ne

t WtWr

c

)27(,)()2(

)1(1 tt

ne

t WtWr

b

)28(.)()2(

1tt

ne

t WtWs

Proposition 1 below implicitly describes a critical level of wealth above

which agents in the economy will decide to evade taxes on a proportion * of their

income.26

Proposition 1: Given * and W, an agent will evade iff

)(

)())(()()()1(1***

tt

pd

t

p

tt

WtW

dWtWdWtW

Basically, agents in this economy will choose to evade if and only if the

probability weighted geometric average of their disposable wealth in the states

‘caught’ and ‘not caught’ exceeds the disposable wealth when choosing not to evade.

For a proof of the above see the Appendix 4.2.

26

Note that * is the optimal proportion of under-reported income if the person is ‘forced’ to evade.


94

4.2.3 Political economy Extensions

Here we consider extensions of the models presented in Sections 4.2.1 and

4.2.2 above to include a political economy determination of one of the parameters of

the tax system. Essentially, we assume that voting takes place at the beginning of

the period and only young agents are allowed to vote on b or . After the vote,

agents in the AS economy make their evasion decision and state contingent plans,

followed by the auditing by tax authorities, after which transfers are made and the

state contingent plans are carried out. In the ‘evade or not’ economy, the only

difference is that after the vote agents decide whether or not to evade, and if they

choose to evade, they decide how much to evade. Subsequently, auditing takes

place, transfers are made, and consumption, saving and bequest plans are carried out.

The timing of events of the political economy versions of the two economies is

described in Figures 4.1 and 4.2 below.

Figure 4.1: Timeline for the basic model.


95

Figure 4.2: Timeline for model with ‘evade or not’ choice.

4.3 A Further Discussion of Some Theoretical Issues.

In this section we reiterate some theoretical issues pertaining to the modelling

of the tax evasion problem in a macroeconomic context. As mentioned in the

previous chapters, we considered modifying the basic AS construct by modelling the

tax evasion decision as a state-contingent plan, and in framework involving

consumption smoothing over time. Secondly, we introduce an ‘evade or not’

decision to our basic variation of the AS construct for reasons that have been made

clear earlier, but are worthwhile restating formally in the context of this model.

Consider the indirect utility function of a typical agent in the basic AS

variant, denoted VAS, evaluated at . Substituting equations (11) - (16) into (1),

we find, after some algebraic manipulation, that it is of the following form:

)29(.)1ln())(ln(32

)1ln(3 2

0

rWtWV ttAS


96

On the other hand, when we consider the indirect utility function of ‘not-

evade’ decision, denoted VNE, we get:

)30(.)1ln())(ln(3 2 rWtWV ttNE

Note that the additional term appearing in (29), ( )

, is negative since

, even in the case . This means that the basic AS variant involves a

lower indirect utility from the choice in comparison with the ‘evade or not’

choice, a feature that is unappealing from an intuitive point of view.27

Secondly, consider the indirect utility function of the basic AS variant with

. It is given by:

)31(.)1ln()]()([2

)1ln()1(3

)]()([2

)1ln(3

2

rdWtWp

dWtWp

V

d

t

ttAS

In the event that there is a corner solution in the problem, one would expect

this function to be declining over the range of . However, note that:

)32(

)]()([2

1

)](')')((')[1(3

)]()([2

1

)]('[3

dWt

dWWtp

dWtW

dpV

d

dd

tt

AS

27

Of course, this problem may be resolved by setting up a non-convex version of the problem in

which both and are step-wise functions of the choice of α, whereby and when

, and and when . The problem is more easily resolved, however, by simply

incorporating an ‘evade or not’ choice in the model.


97

First, note that this derivative, when evaluated at α = 0, is positive under our

assumptions about the functional forms for the tax and cost functions, so that the

conditions for an interior solution are satisfied. Secondly, as α increases, in equation

(32) the sign of the first term is negative (because ( ) ) but the sign of the

second term can be negative if ( )( ) ( ) .28

The sign of

is

therefore ambiguous. One cannot therefore rule out a situation in which it is an

increasing or non-monotonic function over the range of . Furthermore, in the event

that it is an increasing or non-monotonic function of , given that is a continuous

function of , there could be a substantial range of values of for which is less

than , and yet the agent chooses to evade. It then becomes obvious that the

problem is more appropriately modelled with an ‘evade or not’ decision.

To illustrate this issue, as we did in the case of previous chapters, we

consider a few numerical examples. In Figure 4.3, we plot the indirect utility

function of the agent in the basic AS model, as a function of . The peak of this

function, for wealth levels 20, 50, 100, and 200, occur respectively at ,

, , and . However, in Cases 1 and 2 the indirect utility

function for the ‘not evade’ decision is higher than that of the indirect utility function

in the case of the ‘optimal ’ in the AS variant. This means that the basic variant

would suggest a choice of , while in the ‘evade or not variant the agent

would correctly choose to not evade. In cases 3 and 4, though, both models would

give the same result.

In view of the above, and as in the case of the previous chapter, we believe

that the ‘evade or not’ formulation is more suitable and appropriate in the context of

28

Recall that ( ) so that Wd’

= -Wt <0, and that ( ) .


98

the issues addressed in this chapter. For the sake of comparison across the two

formulations, and also across the models in Chapter 3, we present results based on

the numerical simulations of both the AS model and the ‘evade or not’ alternative in

Section 4 of this chapter.

Figure 4.3: Case 1: Here the AS variant leads to choice of alpha=0.09, while the ‘evade or

not’ model leads to a ‘not evade’ choice.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

2.5

3

3.5

4

4.5

5

5.5

6

6.5

7

X: 0.09

Y: 6.924

VA

S

Basic AS variant

Wt = 20

VNE

= 7.21


99

Case 2: Here the AS variant leads to choice of alpha=0.29, while the ‘evade or not’ model

leads to a ‘not evade’ choice.

Case 3: Here both models correctly identify the extent of evasion.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 17

7.5

8

8.5

X: 0.29

Y: 8.316

VA

S

Basic AS variant

Wt = 50

VNE

= 8.48

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19.45

9.5

9.55

9.6

9.65

9.7

9.75

9.8

9.85

9.9

X: 0.55

Y: 9.875

VA

S

Basic AS variant

Wt =100

VNE

= 9.81


100

Case 4: Here both models correctly identify the extent of evasion.

4.4 Choice of Parameters for Numerical Experiments

In the next section we first compare the outcomes in relation to the extent of

evasion in the basic model and its variant with the ‘evade or not’ choice. For our

experiments, we choose a ‘benchmark’ set of parameters given by the following:

30;05.1;06.;1.0;2.0;1;2. odrp .

We first provide a brief description on the selection of the parameter values

for our model. While the degree of complexity of this model is not so high as to

warrant a full-blown calibration exercise, our aim is to choose the parameters more

carefully relative to the previous chapter, given that we would like to build a

framework that is amenable to such parameterization. The parameter values have

been selected with much care and when no suitable reference is available we have

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 111

11.1

11.2

11.3

11.4

11.5

11.6

11.7

11.8

11.9

12

X: 0.95

Y: 11.86

VA

S

Basic AS variant

Wt = 200

VNE

= 11.39


101

chosen what we think is the best possible set of values based on the available

literature, and performed sensitivity analysis for these ranges. Since there is a great

deal of ambiguity in the tax-evasion literature with respect to the measurement of

various parameters, we believe that such an approach is appropriate. Interestingly, as

was the case with the last model of the previous chapter, the range of values for

which interior solutions are available has expanded, making it easier to conduct such

sensitivity analyses.

In addition, with an overlapping-generations model, certain parameter values

that are available in the microeconomic literature may not be appropriate for our

model; we therefore modify those parameters to correspond to the assumption that a

‘period’ in a two-period overlapping generations model corresponds to

approximately 30 years. For example, the probability of tax detection (p) is set at 0.2.

This is the value chosen in Atolia (2009), in which a two-period overlapping

generations model of tax evasion is considered.29

However, we supplement our

analysis with a sensitivity check in the range 0.15≤ p ≤ 0.50.30

Likewise, our value

for the interest rate is chosen from that paper

The tax parameters are loosely calibrated to produce the type of tax

progressivity observed in tax progressivity schedules of most OECD countries. We

choose a calibration of θ close to that of Dzhumashev and Gahramanov (2010),

while γ is chosen to ensure that the tax burden of the top decile of the population

corresponds to what is observed on average in some OECD economies.

29

The model in Atolia (2009), however, differs in the sense that it does not consider heterogeneity in

intragenerational wealth. 30

The upper end of this range is obviously quite generous, given a 50% audit rate is never observed

in the data.


102

Next, we discuss the parameterization of d0. There is no appropriate prior

reference for the selection of the parameter d0 which applies to our model. Therefore,

we work within a range of values that are feasible. We start with an initial value of

d0=30 and experiment with different values of d0 in the range of d0=10 to d0=50. It is

important to note that this does not impact on our results in a qualitative sense.31

It remains to specify the choice of the benchmark parameters for the

distribution of income. The benchmark distribution is lognormal with mean 3.2 and

variance 0.8. We consider a sample of 501 values from this distribution, with a Gini

coefficient of .4073. This is close to the values chosen by Bearse, Glomm and Janeba

(2000), who argue that such a choice does a good job of capturing the actual U.S

household distribution in 1992 if income is measure in thousands of dollars (see also

Bhattacharya et al. 2002). However, since we are also going to analyse implications

for increasing inequality for the outcomes of our model, we also consider several

mean-preserving spreads of this distribution, just as we did in the case of the

previous chapter.

4.5 Results of Quantitative Experiments

Figure 4.4 presents a comparison of the two models using the benchmark set

of parameters. The solid line represents the basic model. As one can observe from

the figure, all agents in this economy evade taxes, and the proportion of unreported

income is a smooth monotonic function of agents’ wealth. In the variant with the

‘evade or not’ choice (represented by the green dotted line), however, we can see

that a large number of agents choose not to evade taxes. In a sample of 501 agents

31

To the best of our knowledge, only Chen (2003) has a similar cost function, and its

parameterisation is related to a mathematical condition required to produce real roots as a solution to a

differential equation in the model.


103

only 91 of the richest agents choose to evade. The extent of evasion of the agents

who choose to evade, is, of course identical to that of the basic model.

In comparison with the models from Chapter 3, one striking difference

emerges. The extent of evasion in the models presented here is much higher than the

last model presented in previous chapter, which had incorporated a cost-of-evasion

function. Recall that the proportion of unreported income by the wealthiest agents in

the previous model was around 0.055, whereas the proportion of unreported income

by the wealthiest agents is rather high, ranging from 0.8-0.99. This result is much

higher that what is observed in the empirical literature.32

In a sense, then, the

formulation for the fines in this chapter maybe less satisfactory in comparison to the

previous chapter. Note, however, that we still get a lower extent of evasion (in the

sense of number of agents evading) in comparison to the AS counterpart of this

model.

32

See for example Bloomquist (2003).


104

Figure 4.4: Extent of Evasion: Basic Model v/s ‘Evade or Not’ Variant for θ=0.20.

To illustrate the differences further we consider some other experiments.

Figure 4.5, 4.6, and 4.7 present experiments that vary the parameters od , p , and

respectively. In these figures, we only present the ‘evade or not’ variant for the sake

of clearer graphical exposition – the basic model in all these cases involves evasion

by all households in the economy, with appearing as a smooth monotonic

function of wealth.33

It is obvious from these experiments that the insightful and

sensible aspects of the basic construct are preserved in the ‘evade or not’ variant –

higher values of the enforcement and cost parameters curtail the extent of evasion.

33

See Appendix 4.3for experiments with the parameters d0, p and ϕ of the AS model.


105

Figure 4.5: Experiments with cost function parameter od .

Figure 4.6: Experiment with p, the probability of detection.


106

Figure 4.7: Experiments with ‘penalty rate’ ϕ.

Table 4.1 presents experiments of the number of evaders in relation to the tax

rate and inequality. The results are similar to those observed in the previous chapter.

That is, that progressivity appears to increase the number of evaders in the economy.

For example, with the benchmark Gini-coefficient of 0.4073, when the θ rises from

0.15 to 0.20, the number of evaders increases from 247 to 355. The effect of

inequality, however, seems to be non-monotonic with respect to the number of

evaders in the economy. For low levels of the tax rate, increasing the level of

inequality seems to increase, then decrease and increase again the number of evaders

on the economy. Once again, this result is similar to those obtained in the model of

the previous chapter with a cost-of-evasion function.

Quantitatively speaking, our extension in this chapter is a step towards

reality. In the formulation of the previous chapter, high levels of taxes produced a


107

situation in which all agents in the economy were evading and the ‘evade or not’

version of the model was identical to its AS counterpart. In this case, however, there

is a larger rage of values for which all the agents in the economy do not evade,

although the proportion of agents evading is still unrealistic in comparison to the

data. For example, when the level of inequality is at Gini=0.4073 and the tax rate is

at θ=0.35, the number of evaders in the economy stands at 474. Although this is still

an unrealistic number of evaders in comparison to the data, it is nevertheless an

improvement on the models the models of previous chapter, where for similar levels

of inequality and tax rates, all agents in the economy evaded taxes.

Table 4.1: Number of Evaders for Different Levels of Inequality and θ

θ Gini=0.2735

No. of

Evaders

Evade or Not

Model

Gini=0.3439

No. of

Evaders

Evade or

Not Model

Gini=0.3807

No. of

Evaders

Evade or Not

Model

Gini=0.4073

No. of

Evaders

Evade or Not

Model

0.15 270 276 280 247

0.20 425 415 395 355

0.25 473 457 447 420

0.30 486 483 471 459

0.35 496 493 484 474

0.40 499 495 491 482

0.45 500 497 492 485

0.50 500 498 493 489

0.55 500 500 494 492


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0.60 500 500 496 494

0.65 500 500 497 496

θ Gini=0.5895

No. of

Evaders

Evade or Not

Model

Gini=0.5975

No. of

Evaders

Evade or Not

Model

Gini=0.6736

No. of

Evaders

Evade or Not

Model

Gini=0.8346

No. of

Evaders

Evade or Not

Model

0.15 259 261 230 223

0.20 326 339 290 257

0.25 375 369 339 289

0.30 406 399 372 309

0.35 423 423 390 322

0.40 434 436 404 334

0.45 440 449 418 356

0.50 450 454 428 368

0.55 458 462 440 376

0.60 461 464 444 384

0.65 468 467 446 390

Next we look at the effect of changes in the non-linear taxes parameter (γ) on

the number of evaders in the economy. Table 4.2 below presents experiments of the

number of evaders in relation to the degree of progressivity and inequality. For a

given level of inequality, we find that the number of evaders is increasing in the


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degree of tax progressivity, γ. For example, with a Gini of 0.4073, the number of

evaders is at 254 when γ=1 but increases to 355 when γ rises to 1.05. The effect of

inequality once again, however, seems to be non-monotonic with respect to the

number of evaders in the economy. For low levels of γ, increasing the level of

inequality seems to increase, then decrease and increase again the number of evaders

on the economy. This non-monotonic result is similar to those obtained with the tax

rate experiments in Table 4.1. Once again, it is complex to definitively ascertain the

reasons for the non-monotonic relationship but we conjecture that this could be due

to the interactions from the non-linear relationship between the cost of evasion, the

proportion of unreported income, and the tax rates. In addition, the trade-off between

tax payments and transfers, discussed in the previous chapter, could also be a factor

for the non-monotonic relationship between inequality and tax evasion.

Table 4.2: Number of Evaders for Different Levels of Inequality and γ

γ Gini=0.2735

No. of

Evaders

Evade or Not

Model

Gini=0.3439

No. of

Evaders

Evade or

Not Model

Gini=0.3807

No. of

Evaders

Evade or Not

Model

Gini=0.4073

No. of

Evaders

Evade or Not

Model

1.00 275 280 283 254

1.01 320 313 309 278

1.02 353 345 329 305

1.03 378 367 347 323

1.04 411 397 370 339

1.05 425 415 395 355


110

1.06 448 422 412 377

1.07 459 436 423 387

1.08 461 449 434 397

1.09 468 454 441 408

1.10 474 459 451 425

γ Gini=0.5895

No. of

Evaders

Evade or Not

Model

Gini=0.5975

No. of

Evaders

Evade or Not

Model

Gini=0.6736

No. of

Evaders

Evade or Not

Model

Gini=0.8346

No. of

Evaders

Evade or Not

Model

1.00 262 263 231 227

1.01 280 277 241 234

1.02 296 296 254 242

1.03 303 314 263 248

1.04 315 326 277 250

1.05 327 339 290 257

1.06 339 352 299 264

1.07 347 357 310 271

1.08 362 361 319 275

1.09 368 367 328 278

1.10 376 373 340 284


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The extent of evasion in relation to changing the initial distribution are a little

less clear-cut, but for some ranges of inequality levels, we get the outcome that

inequality typically encourages the extent of evasion. Figure 4.8 illustrates this

result, which is consistent with empirical evidence, presented, for example, in

Bloomquist (2003). Again, we can observe that the extent of evasion is significantly

lower in the ‘evade or not’ model. This feature suggests that voting outcomes in

relation to the tax parameters could be very different. This is indeed the case, as is

illustrated in Tables 4.3 and 4.4.

Figure 4.8: Inequality and the Extent of Evasion.

Turning to the political economy outcomes of the models we see that, in

Table 4.3, the agents in the AS model desire a very progressive tax structure, and the

outcome here is identical to the corresponding AS models of the previous chapter.


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In the simulations, we allowed for a vote on a discrete set of values for or .

Basically, in this model the agents prefer the most progressive value or or they

are presented with. We conjecture that this is primarily due to the fact that in the AS

model, all agents in the economy evade from the full payment of taxes (the degree of

evasion varies according the agent’s wealth). As a result, the redistribution

mechanism does not work efficiently and agents vote for the highest possible tax

structure in an attempt to generate higher tax revenues and hence redistributive

transfers. In Table 4.4, on the other hand, we can see that the vote on leads to a

choice of 0.15 in most cases. In the case of , agents choose the least progressive

value in the range presented to them. In this instance, the number of evaders in the

economy is significantly lower and therefore the tax mechanism is more ‘efficient’.

Agents do not have to vote for a relatively progressive tax structure as the revenue

lost through tax evasion is relatively small.

The results are rather interesting. In the ‘evade or not’ model we can see that

agents vote for a tax rate of θ=0.15, which is the lowest available choice presented to

them. This result differs from the models of the previous chapter where the agents

voted for the highest tax rate available. Recall that only, for a very low level of

inequality, the ‘evade or not’ model without a cost-of-evasion function produced a

voting outcome that resulted in relatively low tax rate. In addition, the agents in this

economy vote for the lowest degree of progressivity of γ=1 which is also

representative of a linear tax structure. For all levels of inequality, the outcome is

one where the majority vote prevails.

In the AS model, the agents vote for the highest tax rate as well as the highest

degree of progressivity available to them. The vote on θ is 0.65 and the vote on γ is


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1.1 for all levels of inequality. These results are similar to the results of the AS

models in Chapter 3.

Table 4.3: Vote on or : AS Model

Gini Vote on % in favour Vote on % in favour

.2735 .65 85.43 1.1 88.82

.3439 .65 96.21 1.1 97.00

.3807 .65 94.41 1.1 95.80

.4073 .65 90.62 1.1 92.61

.4939 .65 94.81 1.1 97.40

.5895 .65 98.60 1.1 99.60

.5975 .65 99.20 1.1 99.60

.6736 .65 99.40 1.1 99.60

.6748 .65 98.20 1.1 99.00

.8346 .65 99.20 1.1 99.20


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Table 4.4: Vote on or : ‘Evade or Not’ Model

Gini Vote on % in favour Vote on % in favour

.2735 .15 100 1 100

.3439 .15 100 1 100

.3807 .15 96.00 1 100

.4073 .15 94.61 1 100

.4939 .15 81.63 1 100

.5895 .15 71.65 1 100

.5975 .15 76.64 1 100

.6736 .15 95.01 1 100

.6748 .15 55.28 1 100

.8346 .65 48.70 1 100

We also conduct sensitivity analysis on the parameters p, d0, and . Tables

4.5 to 4.8 below present the results from the AS model while Tables 4.8 to 4.10

present the results from the ‘evade or not’ model. We can see that the voting

outcomes on both θ and γ are robust in the vast majority of cases for both the

models. In the AS model, the majority of voting outcomes are for θ=0.65 and γ=1.1.

This implies high levels of tax progressivity in the AS model even when we vary the

different parameters. Only for a very low level of inequality does the vote on θ

change in the AS model from θ=0.65 to θ=0.15. This is result is hard to explain but


115

could be due to the artifact of the intrinsic non-linearities in the model. In the ‘evade

or not’ construct, the results are also robust to changes in the parameters p, d0, and .

The voting outcomes are for low levels of tax progressivity and the majority of votes

are for θ=0.15 and γ=1. This outcome is only altered when we change the cost of

evasion (d0) or the penalty rate ( ) for very high levels of inequality. In this

scenario, the winning vote of θ is 0.65 but the degree of progressivity, γ, remains the

same at 1.

Table 4.5: Sensitivity Analysis for p: AS Model

Gini Vote

on

p=0.10

% in

favour

Vote

on

p=0.40

% in

favour

Vote

on

p=0.10

% in

favour

Vote

on

p=0.40

% in

favour

.2735 .15 78.64 .15 96.41 1 68.46 1.1 99.40

.3439 .65 93.61 .65 99.60 1.1 99.60 1.1 99.60

.4073 .65 87.62 .65 98.40 1.1 98.40 1.1 99.20

.5895 .65 98.40 .65 99.60 1.1 99.60 1.1 99.80

.6736 .65 99.00 .65 99.80 1.1 99.80 1.1 99.80


116

Table 4.6: Sensitivity Analysis for d0: AS Model

Gini Vote

on

d0=20

% in

favour

Vote

on

d0=50

% in

favour

Vote

on

d0=20

% in

favour

Vote

on

d0=50

% in

favour

.2735 .65 91.42 .65 87.03 1.1 98.00 1.1 93.01

.3439 .65 98.40 .65 97.21 1.1 99.60 1.1 99.60

.4073 .65 94.61 .65 93.61 1.1 98.80 1.1 98.80

.5895 .65 98.80 .65 98.80 1.1 99.60 1.1 99.60

.6736 .65 99.40 .65 99.40 1.1 99.80 1.1 99.80

Table 4.7: Sensitivity Analysis for : AS Model

Gini Vote

on

=0.05

% in

favour

Vote

on

=0.30

% in

favour

Vote

on

=0.05

% in

favour

Vote

on

=0.30

% in

favour

.2735 .65 89.02 .65 88.22 1.1 98.00 1.1 98.00

.3439 .65 97.41 .65 97.60 1.1 99.60 1.1 99.60

.4073 .65 93.81 .65 94.21 1.1 98.80 1.1 98.80

.5895 .65 98.80 .65 99.00 1.1 99.60 1.1 99.60


117

.6736 .65 99.40 .65 99.40 1.1 99.80 1.1 99.80

Table 4.8: Sensitivity Analysis for p: ‘Evade or Not’ Model

Gini Vote

on

p=0.10

% in

favour

Vote

on

p=0.40

% in

favour

Vote

on

p=0.10

% in

favour

Vote

on

p=0.40

% in

favour

.2735 .15 100 .15 100 1 100 1 91.42

.3439 .15 100 .15 100 1 100 1 100

.4073 .15 99.00 .15 98.60 1 100 1 99.80

.5895 .15 92.02 .15 87.62 1 96.01 1 100

.6736 .15 77.84 .15 53.89 1 80.44 1 77.45

Table 4.9: Sensitivity Analysis for d0: ‘Evade or Not’ Model

Gini Vote

on

d0=20

% in

favour

Vote

on

d0=50

% in

favour

Vote

on

d0=20

% in

favour

Vote

on

d0=50

% in

favour

.2735 .15 96.81 .15 100 1 100 1 100

.3439 .15 97.60 .15 100 1 100 1 100


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.4073 .15 96.21 .15 100 1 100 1 100

.5895 .15 61.48 .15 87.62 1 87.03 1 100

.6736 .65 56.29 .15 74.45 1.1 69.06 1 78.44

Table 4.10: Sensitivity Analysis for : ‘Evade or Not’ Model

Gini Vote

on

=0.05

% in

favour

Vote

on

=0.20

% in

favour

Vote

on

=0.05

% in

favour

Vote

on

=0.20

% in

favour

.2735 .15 95.01 .15 100 1 100 1 99.60

.3439 .15 95.81 .15 100 1 100 1 99.40

.4073 .15 93.41 .15 100 1 100 1 99.80

.5895 .15 70.46 .15 97.21 1 77.25 1 100

.6736 .65 52.30 .15 85.23 1 64.27 1 99.40

Recall that in the previous chapter, before introducing the cost-of-evasion

function, we had a similar outcome for a special case of θ. Such a result was possible

due to the non-singlepeakedness which vanished upon the introduction of cost-of-

evasion in Section 3.3.3 of that chapter. Here, however, the non-singlepeakedness in

the agents utility function returns, see Figure 4.9 below, in the ‘evade or not’ model.

In Figure 4.9, Agent 1 represents the poorest agent, while Agent 501 represents the


119

richest agent in the economy, and agents are arranged in ascending order of their

income or wealth. Therefore Agent 251, for example, is the median agent in the

sample income distribution considered here. The non-singlepeakedness emerges

again in the utility function of Agent 1 (and agents with relatively low wealth

distributions). This is because lower taxes produce fewer evaders, and thereby a

reasonable amount of redistribution relative to higher tax rates.

Figure 4.9: Agents’ preferences over θ in the ‘evade or not’ economy.

4.6 Brief Discussion on Wealth Dynamics

Finally, we briefly discuss the dynamics in the ‘evade or not’ model. To

illustrate the intuition regarding the long-run outcomes of the two models, we

consider the bequests functions for agents in the two economies. This is presented in


120

Figures 4.10 and 4.11 Figure 4.10 represents the wealth dynamics that is typical of

the model with the ‘evade or not’ choice.34

Figure 4.10: Wealth Dynamics of ‘Evade or Not’ Model.

Here, below a certain critical wealth level, implicitly defined by Proposition

1, agents do not evade. Below this wealth level the bequest function is characterised

by the line labelled bne

. To the right of this wealth level agents have state contingent

plans for bequests – Agents evade, and those who are caught leave bequests

characterised by the line bc, while those who are ‘not caught’ leave bequests

characterised by the line bnc

. It is clear from the graph that there is a unique steady

state of W - the long-run distribution of this economy is degenerate since everyone

in this economy ends up with W . There is no inequality in the long-run, and no one

evades taxes.

34

That is, based on our numerical simulations, we get a graph that typically looks like the one

presented in this figure.

Wt+11

11

Wt

W* W

450

bc

bnc

bne


121

We now turn to the distribution of the wealth dynamics of the AS economy.

This is shown in Figure 4.11 below.

Figure 4.11: Wealth Dynamics of AS Model.

In the basic A-S model it is clear that the economy will converge to a unique

invariant long-run distribution with [ ] as support. The inequality in the

economy will fall and converge to the inequality level characterised by the ergodic

distribution. This means that in either of the two cases, tax evasion will not

contribute to persistence in inequality. The dynamics in both cases, however, are

reminiscent of the ‘catching points’ or poverty trap type of situation.

4.7 Concluding Remarks

The aim of this chapter was to build on the models of the previous chapter

with a view to providing an exploratory study of tax-evasion and inequality in a

Wt+11

11

Wt

W

W

450

bc

bnc


122

macroeconomic framework with heterogeneous agents. We propose a simple

variation of the Allingham and Sandmo (1972) model and integrate it to a dynamic

general equilibrium framework with overlapping generation of agents with wealth

and income heterogeneity. In contrast to the micro-theoretic literature on tax evasion,

and similar in spirit to asset-pricing models in macroeconomics, we model the

agent’s decision as a state-contingent plan. That is, the agents optimal plans of

consumption, saving and bequests are contingent on whether evasion is detected or

not.

A key finding of our results is that with the introduction of a non-linear tax

structure and a re-modelling of the penalty structure, the models produce slightly

more realistic results in relation the number of evaders in the economy. In this

model, we do not get a scenario in which 100% of the population is evading taxes

across the range of tax rates considered. Recall that this was not the case for certain

levels of the tax rate in the models of the previous chapter. In addition, we find a

non-monotonic relationship between the number of evaders and inequality. The

results also show, that in this instance, the effect of inequality seems to be non-

monotonic with respect to the number of evaders in the economy. This is similar to

the results of the ‘evade or not’ model with a cost-of-evasion function in the earlier

chapter. Most of the empirical evidence in the literature suggests a positive link

between inequality and tax evasion, whether tax evasion is measured directly or

indirectly (see Bloomquist 2003 and Gupta et al. 2001). Our model, however,

suggests that this evidence must be interpreted with caution Also, for a given level

of inequality, as was the case in the models of the previous chapter; we find that the

number of evaders is increasing in the tax rate. The income profile of non-evaders


123

remain unchanged: agents at the lower end of the distribution are the ones that do not

evade from the payment of taxes.

We also find that, in both the AS and ‘evade or not’ model, the proportion of

unreported income for agents who evade is increasing with the agent’s wealth. This

is a similar result to the model with a cost-of-evasion function analysed in the

Chapter 3. Several of the results in this chapter, while analogous to the ones

presented in chapter 3 are more realistic in relation to some dimensions. However,

we do find that the extent of evasion is much higher than the models presented in the

previous chapter.

In relation to the non-linear tax structure that we have introduced in the

models of this chapter, we find that a higher degree of tax progressivity increases the

number of evaders in the economy. For a given level of inequality, however, we find

that the relationship between the number of evaders and tax progressivity is non-

monotonic. This is similar in comparison to the previous models of non-

monotonicity between the tax rate and the number of evaders in the economy. In

addition, the non-monotonicity between the level of inequality and the number of

evaders renders the effect of former on the latter inconclusive.

The political economy outcomes of the models produce the most interesting

results in this chapter. We find that the voting outcome in the ‘evade or not’ model is

in favour of the lowest possible tax rate available to the agents. In addition, the

agents also vote for the lowest degree of tax progressivity presented to them. This is

in contrast to the ‘evade or not’ models in Chapter 3 where, apart from a very low

level of inequality, agents in the economy vote for the highest tax rate presented to

them. It seems likely that the agents in this economy choose low levels taxes as they


124

are associated with low levels of tax evasion and therefore achieve redistribution that

may be greater relative to high taxes. The results of the AS model in this chapter are,

however, similar to the previous chapter. In these models, the agents vote for the

highest tax rate and degree of progressivity presented to them. This is due to the fact

that since all agents are evading a proportion of their taxes in this economy, the

transfers received by the agents are maximised when the tax structure is at its most

progressive.

Conclusion Chapter 5

125

CHAPTER 5

Concluding Remarks

This chapter contains a brief summary of the main outcomes of the study

presented in previous chapters of this thesis, and provides some directions for future

research. The aim of this thesis was to explore tax evasion in the framework of a

macroeconomic model, and study its implications for the link between inequality and

tax progressivity from a political economy perspective. Our starting point was the

seminal work of Allingham and Sandmo (1972), who model the behavior of agents

as a decision involving choice of the extent of their income to report to tax

authorities, given a certain institutional environment, represented by parameters such

as the probability of detection and penalties in the event the agent is caught. The

approach followed in this thesis involves a step-by-step extension of this elegant

construct with the eventual aim of constructing a macroeconomic model of tax

evasion capable of examining the political economy implications of tax evasion for

the progressivity in tax structure of an economy.

A key motivation for this exercise, emphasized in Chapters 1 and 2 of this

thesis pertains to a well-known limitation of the Allingham and Sandmo model;

specifically, it indicates a level of compliance that is significantly below what is

observed in the data. Addressing this issue in a macroeconomic model is inspired by

the idea that some features of macroeconomic models could have the effect of

alleviating the extent of evasion observed in the basic model. Typically, agents in a


126

macroeconomic model face several trade-offs that are inextricably linked to the tax

evasion decision, such as their consumption plans over time and across different

goods, and these decisions could also have a bearing on their tax evasion decision.

Furthermore the issue of the political economy determination of the tax

structure is also essentially a macroeconomic one. It has long been recognized that

policies and institutions are endogenous, and there is a large body of literature,

reviewed in the second chapter of the thesis, involving political economy,

macroeconomic models of policy. There is, however a lack of such models

analyzing the implications for policy parameters in the presence of tax evasion.

Likewise, while there are several microeconomic models of tax evasion that examine

political economy outcomes for redistribution, but as emphasized earlier, are

inadequate in the sense they lack the macroeconomic realism that is relevant to such

issues. Empirical analyses of the political economy determination of tax policy are

also scant; with the exception of Lupu and Pontusson (2011) there are no studies that

directly examine the link between inequality and tax progressivity. The inconclusive

nature of the empirical correlations in the data then further motivate the need for

theoretical research in macroeconomics for the purpose of identifying the structural

relationships underpinning the link between tax evasion and the political

determination of tax structure. The models presented in the third and fourth chapter

take some steps in that direction, and involve the key contributions of this thesis.

One of the variations, considered in Chapter 3 of the thesis, involves

incorporating the Allingham and Sandmo construct into a two-period

macroeconomic model of a small open economy of the type originally attributed to

Fisher (1930), and studied, for example in various macroeconomics textbooks such


127

as Obstfeld and Rogoff (1996). A further variation of this simple construct involves

allowing agents to initially decide whether to evade taxes or not. In the event they

decide to evade, they then have to decide the extent of income or wealth they wish to

under-report. The results of our analysis lead to some interesting insights. The

introduction of the ‘Evade or Not’ feature of the model is a key contribution to the

literature because it reduces the extent of evasion even in the context of a very

simple macroeconomic model of tax evasion. We find that the ‘evade or not’

assumption has strikingly different and more realistic implications for the extent of

evasion, and demonstrate that it is a more appropriate modeling strategy in the

context of macroeconomic models, which are essentially dynamic in nature and

involve consumption smoothing across time and across various states of nature.

Specifically, since deciding to undertake tax evasion impacts on the consumption

smoothing ability of the agent by creating two states of nature in which the agent is

‘caught’ or ‘not-caught’, there is a possibility that their utility under certainty, when

they choose not to evade, is higher than the expected utility obtained when they

choose to evade.

Another realistic outcome that emerges is that the extent of evasion is

increasing in wealth. As mentioned earlier, tax evasion studies typically have to

resort to DRRA preferences to achieve levels of evasion that are increasing in

wealth, a feature that has some empirical support in the literature. In the context of

the model of this thesis, this is achieved while still maintaining CRRA preferences.

This is important in the sense that macroeconomic models require preferences to be

restricted to the CRRA class if they are to be consistent with some stylised facts

pertaining to business cycles and economic growth. In addition, the percentage of


128

evaders in the model economy is also reduced to numbers that are more consistent

with the empirical estimates in the literature.

Furthermore the simple-two period model incorporating an ‘evade or not’

choice can be used to demonstrate some strikingly different political economy

implications relative to its Allingham and Sandmo counterpart. In variations of the

two models that allow for voting on the tax parameter, we find that agents typically

choose to vote for a high degree of progressivity by choosing the highest available

tax rate from the menu of choices available to them. There is, however, a small

range of inequality levels for which agents in the ‘evade or not’ model vote for a

relatively low value of the tax rate.

The final steps in the model building procedure involve grafting the two-

period models with a political economy choice into a dynamic overlapping

generations setting with more general, non-linear tax schedules and a ‘cost-of

evasion’ function that is increasing in the extent of evasion. Results based on

numerical simulations of these models show further improvement in the model’s

ability to match empirically plausible levels of tax evasion. In addition, the

differences between the political economy implications of the ‘evade or not’ version

of the model and its Allingham and Sandmo counterpart are now very striking; there

is now a large range of values of the inequality parameter for which agents in the

‘evade or not’ model vote for a low degree of progressivity. This is because, in the

‘evade or not’ version of the model, low values of the tax rate encourages a large

number of agents to choose the ‘not-evade’ option, so that the redistributive

mechanism is more ‘efficient’ relative to the situations in which tax rates are high.


129

Some further implications of the models of this thesis relate to whether

variations in the level of inequality, and parameters such as the probability of

detection and penalties for tax evasion matter for the political economy results. We

find that (i) the political economy outcomes for the tax rate are quite insensitive to

changes in inequality, and (ii) the voting outcomes change in non-monotonic ways in

response to changes in the probability of detection and penalty rates. Specifically,

the model suggests that changes in inequality should not matter, although the

political outcome for the tax rate for a given level of inequality is conditional on

whether there is a large or small or large extent of evasion in the economy. This is in

contrast to the positive link between inequality and tax evasion suggested in, for

example, Bloomquist (2003), which implies that such a relationship needs to be

interpreted with caution. Similarly, the impact of institutional variables relating to

the tax evasion was also indeterminate. This would be expected if one were

attempting to estimate a linear, reduced-form relationship when the true underlying

structural relationship is a non-monotonic one, as implied, for example, by the

numerical analysis of the models of this thesis.

Of course, further development of the simple models presented in this thesis

is needed to shed further light on the inequality-progressivity link in the presence of

tax evasion. In what follows, therefore, we provide some directions for future

research. In the models of this thesis we looked at the equity related aspects and the

focus was entirely on distributional issues. However, the model in the form

developed in the last chapter is amenable to extensions along several directions. For

example, to make it suitable for analysing efficiency related issues one would need

to model a production economy with a labour-leisure choice for the agent. This

would also make it compatible with business cycle analysis – as modelling of a


130

labour leisure choice is essential for any study of the business cycle (see Cooley and

Prescott 1995, referenced earlier in this thesis). Furthermore it is of interest to

understand within a macroeconomic context how the tax evasion decision impacts on

the labour-leisure choice. Another interesting and arguably more realistic extension

would be in the direction of alternative political structures instead of the voting

model considered here, such as lobbies and power groups that are a characteristic of

economies with weak institutional settings. Models with lobbies or other complex

voting structures, and those which model an equity-efficiency trade-off by

incorporating work-effort, for example, could produce a diverse set of outcomes.

This would be an interesting direction unexplored in a macro-theoretic context.

The results of our models suggest future directions for empirical research. A

stylised fact, for example, associated with the tax structure of developing economies

is their greater reliance on indirect as opposed to direct taxation. According to Avi-

Yonah and Margalioth (2006), the structure of taxation in developing countries is

radically different from that of developed countries. About two thirds of the tax

revenue in developed countries is obtained from direct taxes, mostly personal income

tax and social security contributions. The remaining one-third comes primarily from

domestic sales tax. The situation is exactly reversed in developing countries: about

two-thirds of the tax revenue comes from indirect taxes, mostly VAT, sales tax,

excises and taxes on trade. The latter characteristic is driven by the practical

implications of tax evasion for revenue collection by governments. Specifically, in

the presence of tax evasion, direct taxes are harder to collect and administer, leading

to a shift towards indirect taxation as a source of revenue (Avi-Yona and Margalioth

2006). This supports the idea that tax structures are determined differently in the

presence of corruption and tax evasion.


131

As there is substantial empirical evidence supporting the fact that such

countries tend to rely on indirect taxation as a source of revenue (see Avi-Yona and

Margalioth 2006) and given that tax evasion is very difficult to measure directly, one

could introduce an indirect, albeit unconventional, measure of tax evasion, namely

‘indirect tax as a percentage of total revenue’. The motivation for using indirect taxes

as a proxy for the extent of tax evasion is rationalised as follows: Governments that

have weak institutions are subject to a higher extent of tax evasion, and as such the

tax revenue collected from direct taxes (income taxes) is compromised. The

authorities would then have to raise revenue by other means, that is, through indirect

taxes. A higher degree of indirect taxes, therefore, could signify a higher level of tax

evasion. This innovative measurement of tax evasion would make for interesting

empirical analysis. These avenues are left for future research.

132

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Appendix for Chapter 3

147

APPENDIX


Appendix 3.1: Comparison of Indirect Utility IUFAS and IUFNE.

Comparing the indirect utilities of the AS model (labeled IUFAS) and the model

without evasion (labeled IUFNE), and assuming log utility, gives the following:

iff

( ) ( ) ( ( ))

( ) ( ) ( )

( ) ( ( ))

( ) ( )

( ) ( ( ))

( ) ( )

(A1)

Recall that ( ( )

).

For the indirect utility in the AS model to always be greater than the indirect utility

of the not-evade alternative, we would also require the second term on the LHS to

greater than the second term on the RHS.

iff


148

( ) ( ( ))

( ) ( )

( ) ( ) ( )

(A2)

If the conditions for an interior solution are satisfied, however, the second term on

the LHS is less than the second term on the RHS, making it difficult to compare the

expressions of both side of the inequality.


149

Appendix 3.2: Derivation of Conditions for an Interior Solution.

For an interior solution:

( )

( ) (A3)

This implies that for (upper bound condition):

(A4)

and for (lower bound condition):

( )

( ) (A5)

which gives:

( )

( )

(A6)


150

Appendix 3.3: Comparison of Indirect Utility IUFAS and IUFNE Two-Period

Model.

Comparing the indirect utilities of the AS two-period model (IUFAS) and the ‘evade

or not’ two-period model (IUFNE) gives the following:

iff

( ) [(

)( )

]

( ) [(

)

]

( ) ( )

Rearranging terms we get:

[

(

)( )

] ( ) [

(

)

]

( )

[(

)

] ( )

( )

(A7)

Comparing the utilities of the two models, it is again not possible to prove the

proposition that the utility with evasion (IUFAS) is higher than the certainty scenario

(IUFNE) given that the conditions for an interior solution are satisfied.


151


Appendix 4.1: Derivation of Variables for Expression in terms of Wt and α

Assuming log utility, one can manipulate equations (4.2)-(4.9) and (4.23)-(4.28) in

order to express the variables nc

t

c

t

nc

t

c

t

nc

t

c

t

nc

t

c

t bbccsscc 1111 ,,,,,,, in terms of tW and α:

First-order conditions:

)20(,1

)1( 11 Abcrc c

t

c

tt

c

t

similarly,

)21(,1

)1( 11 Abcrc nc

t

nc

tt

nc

t

and

)22(,1

)1( 11 Abcrc ne

t

ne

tt

ne

t

Using equation (9) one gets,

)23(,)1( 11 Acsrc c

t

c

tt

c

t

)24(1

1 1 Ar

cs

t

c

tc

t


152

)25(1

)1)(1(A

r

crs

t

c

tc

t

)26()1( Acs c

t

c

t

Substituting back in equation (10) gives:

)27(,)()()2(

)1(AdWtWc tt

c

t

Likewise the same applies to the rest of the equations to give the following:

(A28),)()()2(

1

dWtWc d

t

nc

t

A29)(,)()()2(

)1)(1(1

dWtW

rc tt

c

t

(A30),)()()2(

)1(1

dWtW

rc d

t

nc

t

A31)(,)()()2(

)1)(1(1

dWtW

rb tt

c

t


153

A32)(,)()()2(

)1(1

dWtW

rb d

t

nc

t

(A33),)()()2(

)1)(1(

dWtWs tt

c

t

)A34(.)()()2(

)1(

dWtWs d

t

nc

t


154

Appendix 4.2: Proof of Proposition 1

For a comparison of indirect utility functions in the two situations, it is first

convenient to exploit the log utility form so that expected utility if you choose to

evade is written as (note that the variables which appear as arguments in the utility

function are optimal levels):

(A35).])([])(ln[ 1

1111

pnc

t

nc

t

nc

t

pc

t

c

t

c

t bccbcc

Likewise, if agents do not evade taxes, preferences may be written as:

(A36)].)(ln[ 11

ne

t

ne

t

ne

t bcc

We can simplify further by using the first order conditions and budget constraints to

express all variables in terms of first period consumption. In that case, (A35) can be

written as )2)(1()2( )()ln( pnc

t

pc

t cc , and (A36) can be written as 2)ln( ne

tc .

Substituting (11) and (12) into the former and (25) into the latter, we can then

compare indirect utilities in the two situations. We can then show that agents will

choose to evade if and only if

.2

ln22

ln

1

dynedyncdycpp

In the above,

,)()()1( dWtWydyc tt

,)()( dWtWydync d

t and


155

)( tt WtWydyne .

Given that the log transformation is monotonic, straightforward manipulation yields

the result of Proposition 1.


156

Appendix 4.3: Results of Experiments - Basic AS Model

This section presents some sensitivity analyses varying the parameters d0, p, and ϕ.

(a) Experiments with cost function parameter od :

From the diagram above, we can see that the proportion of unreported income falls

as the cost of evasion (d0) increases in the basic AS model. Qualitatively, this is

similar to the ‘evade or not’ model discussed in Section 4.5 of Chapter 4. In addition,

there is 100 percent evasion in the AS model (all 501 of the agents evade taxes).


157

(b) Experiments with probability of detection p:

With regards to the probability of detection, p, the results show that, for most part,

the higher the probability of detection the proportion of undeclared income. These

results are similar to the ‘evade or not’ model presented in Chapter 4. In this

instance, however, there is a range of wealth levels (from around Wt=120 to 160)

where the proportion of unreported income falls for detection rates of p=0.2 but not

for p=0.3 or p=0.5.


158

(c) Experiments with the penalty rate ϕ:

Changes in the penalty rate, ϕ, in the AS model does not seem to have an effect on

the proportion of unreported income (α). Once again, this result is similar to the

‘evade or not’ model presented in Chapter 4. Unlike the ‘evade or not’ model

though, the number of evaders in the AS model remains unchanged at 501 (all agents

in the economy evade some form of taxes).

tax evasion inequality and progressive taxes a … · 2012-11-08 · tax evasion, inequality, and...

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