tax evasion inequality and progressive taxes a … · 2012-11-08 · tax evasion, inequality, and...
TRANSCRIPT
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-
ii
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
iii
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.
iv
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
vi
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
xi
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: _________________________
xii
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.
Introduction Chapter 1
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.
Introduction Chapter 1
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.
Introduction Chapter 1
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
Introduction Chapter 1
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
Introduction Chapter 1
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.
Introduction Chapter 1
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.
Introduction Chapter 1
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.
Introduction Chapter 1
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).
Introduction Chapter 1
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.
Background and Motivation Chapter 2
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
Background and Motivation Chapter 2
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),
Background and Motivation Chapter 2
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
Background and Motivation Chapter 2
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).
Background and Motivation Chapter 2
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
Background and Motivation Chapter 2
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
Background and Motivation Chapter 2
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
Background and Motivation Chapter 2
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.
Background and Motivation Chapter 2
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
Background and Motivation Chapter 2
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.
Background and Motivation Chapter 2
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.
Background and Motivation Chapter 2
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
Background and Motivation Chapter 2
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
Background and Motivation Chapter 2
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
The Benchmark Model and Some Simple Extensions Chapter 3
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).
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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).
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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)
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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).
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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)
The Benchmark Model and Some Simple Extensions Chapter 3
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:
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
46
( )
(28)
Here variables are analogously defined with ‘ne’ representing ‘not-evading’.
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
3.3.2.1 Theoretical Analysis
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
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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).
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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).
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
64
Figure 3.9: Percentage of votes for various values of θ in the AS economy.
The Benchmark Model and Some Simple Extensions Chapter 3
65
Figure 3.10: Percentage of votes in favour of various values of θ in the ‘evade or not’
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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
3.3.3.1 Theoretical Analysis
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)
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
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.
The Benchmark Model and Some Simple Extensions Chapter 3
76
Figure 3.14: Percentage of votes for various values of θ in the AS economy.
Figure 3.15: Percentage of votes in favour of various values of θ in the ‘evade or not’
economy.
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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).
The Benchmark Model and Some Simple Extensions Chapter 3
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-
The Benchmark Model and Some Simple Extensions Chapter 3
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
The Benchmark Model and Some Simple Extensions Chapter 3
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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,
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
Here variables are analogously defined with ‘ne’ representing ‘not-evading’.
In this case the optimal consumption and bequest plans are given by
)25(,)()2(
1tt
ne
t WtWc
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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 ( ) .
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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).
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
105
Figure 4.5: Experiments with cost function parameter od .
Figure 4.6: Experiment with p, the probability of detection.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
108
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
109
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
111
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.
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
112
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
113
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
114
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
118
.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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
On Inequality, Tax Evasion and Progressive Taxes Chapter 4
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
Conclusion Chapter 5
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
Conclusion Chapter 5
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
Conclusion Chapter 5
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.
Conclusion Chapter 5
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
Conclusion Chapter 5
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.
Conclusion Chapter 5
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
BIBLIOGRAPHY
Akerlof, G. 1980. A Theory of Social Custom, of which Unemployment may be One
Consequence. Quarterly Journal of Economics. 94(4): 749-775.
Albanesi, S. 2000. Inflation and Inequality. Universita Bocconi. IGIER Working
Paper. No. 199.
Albanesi, S. 2007. Inflation and Inequality. Journal of Monetary Economics. 54(4):
1088-1114.
Alesina, A. and D. Rodrik. 1994. Distributive Politics and Economics Growth. The
Quarterly Journal of Economics. Vol 109(2): 465-490.
Allingham, M. G. and A. Sandmo. 1972. Income Tax Evasion: A Theoretical
Analysis. Journal of Public Economics. 1(3): 323-338.
Alm, J., R. Bahl and M. Matthew. 1991. Tax Base Erosion in Developing Countries.
Economic Development and Cultural Change. 39(4): 849-872.
Alm, J., R. Bahl and M. Murray. 1990. Tax Structure and Tax Compliance. The
Review of Economics and Statistics. 72(4): 603-613.
Alm, J. and J. Martinez-Vazquez. 2003. Institutions, Paradigms, and Tax Tvasion in
Developing Countries in Alm, J. and J. Martinez-Vazquez eds. Public Finance in
133
Developing and Transitional Countries: Essays in Honor of Richard Bird. Edward
Elgar, Cheltenham, UK.
Andreoni, J., B. Erard and J. Feinstein. 1988. Tax Compliance. Journal of Economic
Literature. 36(2): 818-860.
Atkinson, A. B. and F. Bourguignon. 2000. Introduction: Income Distribution and
Economics. Handbook of Income Distribution. Vol 1: 1-58.
Atolia, M. 2009. Tax Evasion in an Overlapping Generations Model with Public
Investment. Florida State University, Working Paper Series.
Avi-Yonah, R. and Y. Margalioth. 2007. Taxation in Developing Countries: Some
Recent Support and Challenges to the Conventional View. TaxRev Vol27(1): (2007-
2008).
Barro, R. J. 2004. Inequality and Growth in a Panel of Countries. Journal of
Economic Growth. 5(1): 5-32.
Barro, R. 2000. Inequality and Growth in a Panel of Countries. Journal of Economic
Growth. 5(1): 5-32.
Bearse, P, G. Glomm, and E. Janeba. 2000. Why Poor Countries Rely Mostly on
Redistribution In-kind. Journal of Public Economics, Vol74(3): 463-481.
Benabou, R. 1996. Inequality and Growth. NBER Macroeconomics Annual.
134
Bhattacharya, J., H. Bunzel and J. Haslag. 2005. The Non-monotonic Relationship
between Seigniorage and Inequality. Canadian Journal of Economics. 38(2): 500-
519.
Black, D. 1948. On the Rationale of Group Decision Making. Journal of Political
Economy. 56(1): 23034.
Blomquist, N. S. 1985. Labour Supply in a Two-Period Model: The Effect of a
Nonlinear Progressive Income Tax. Review of Economics and Statistics. 52(3): 515-
524.
Bloomquist, K. M. 2003. Income Inequality and Tax Evasion: A Synthesis. OECD
Papers. Pisa.oecd.org.
Blundell, R., A. Duncan and C. Meghir. 1998. Estimating Labour Supply Responses
Using Tax Reforms. Econometrica. 66(4): 827-861.
Borck, R. 2009. Voting on Redistribution with Tax Evasion. Social Choice and
Welfare. Vol 32(3):439-454.
Burg, D. F. 2004. A World History of Tax Rebellions: An Encyclopedia of Tax
Rebels, Revolts, and Riots from Antiquity to the Present. New York: Routledge.
Caballe, J. and J. Panades. 2004. Inflation, Tax Evasion and the Distribution of
Consumption. Journal of Macroeconomics. 26(4): 567-595.
135
Chen, B. 2003. Tax Evasion in a Model of Endogenous Growth. Review of Economic
Dynamics. 6(2): 381-403.
Clotfelter, C. T. 1983. Tax Evasion and Tax Rates: An Analysis of Individual
Returns. The Review of Economics and Statistics. 65(3): 363-373.
Cooley, T and E, Prescott. 1995. Economic Growth and Business Cycles, in Cooley
and Prescott eds., Frontiers of Business Cycle Research, Princeton University Press.
Cowell, F. A. 1981. Taxation and Labour Supply with Risky Activities. Economica.
48(192): 365-379.
Cowell, F. A. and J. P. P. Gordon. 1989. On becoming a ghost: indirect tax evasion
and government audit policy. Discussion Paper 127, London School of Economics.
Crane, S. E. and F. Nourzad. 1986. Inflation and Tax Evasion: An Empirical
Analysis. The Review of Economics and Statistics. 68(2): 217-223.
Crowe, C. and E. Meade. 2008. Central Bank Independence and Transparency:
Evolution and Effectiveness. European Journal of Political Economy. Vol 24(4):
763-777.
Cukierman, A. 1992. Central Bank Strategy, Credibility, and Independence: Theory
and Evidence. Cambridge: MIT Press.
136
Cukierman, A., S. B. Web and B. Neyapti. 1992. Measuring the Independence of
Central Banks and Its Effects on Policy Outcomes. The World Bank Economic
Review. 6(3): 353-398.
Dearden, L., S. Machin and H. Reed. 1997. Intergenerational Mobility in Britain.
Economic Journal. 107(440): 47-66.
Deaton, A. and S. Zaidi. 2002. Guidelines for Constructing Consumption Aggregates
for Welfare Analysis. World Bank Publications.
Dixit, A. and S. Skeath. 2000. Games of Strategy. Second Edition. New York:
Norton.
Dolmas, J., G. W. Huffman and M. A. Wynne. 2000. Inequality, Inflation and
Central Bank Independence. Canadian Journal of Economics. 271-287.
Donder, P. and J. Hindriks. 2004. Majority Support for Progressive Income Taxation
with Corner Preferences. Public Choice. 118(3-4): 437-449.
Duncan, D. and K. P. Sabirianova. 2008. Tax Progressivity and Income Inequality.
Andrew Young School of Policy Studies. Research Paper Series No. 08-26.
Dzhumashev, R. and E. Gahramanov. 2010. A Growth Model with Income Tax
Evasion: Some Implications for Australia. Economic Record. 85(275): 620-636.
Easterly, W. and S. Rebelo. 1993. Fiscal Policy and economic Growth: An Empirical
Investigation. Journal of Monetary Economics. 32(3): 417-458.
137
Edwards, S. and G. Tabellini. Explaining Fiscal Policies and Inflation in Developing
Countries. NBER Working Paper 3493.
Eissa, N. and J. B. Liebman. 1996. Labor Supply Responses to the Earned Income
Tax Credit. Quarterly Journal of economics. 111(2): 606-637.
Elster, J. 1989. Social Norms and Economic Theory. Journal of Economic
Perspectives. 3(4): 99-117.
Epple, D. and E. Romano. 1996a. Ends against the middle: Determining public
service provision when there are private alternatives. Journal of Public Economics.
Vol 62(3): 297-325.
Feinstein, J. 1991. An Econometric Analysis of Income Tax Evasion and Its
Detection. The Rand Journal of Economics. 22(1): 14-35.
Feldstein, M. 1995. Behavioral Responses to Tax Rates: Evidence from the Tax
Reform Act of 1986. American Economic Review. 85(2): 170-174.
Fiorio, C. V and F. d'Amuri. 2005. Workers' Tax Evasion in Italy. Giornale degli
Economisti e Annali di Economia. 64(2/3): 247-270.
Fischer, S. 1995. Central Bank Independence Revisited. American Economic Review.
85(2): 201-206.
138
Fishburn, G. 1981. Tax Evasion and Inflation. Australian Economic Papers. 20{37):
325-332.
Fisman, R. 2001. Estimating the Value of Political Connections. American Economic
Review. Vol 91(4): 1095-1102.
Fisman, R. and S. J. Wei. 2004. Tax Rates and Tax Evasion: Evidence from
“Missing Imports” in China. Journal of Political Economy. 112(2): 471-496.
Forbes, K. J. 2000. A Reassessment of the Relationship between Inequality and
Growth. American Economic Review. 90(4): 869-887.
Friedman, E., S. Johnson, D. Kaufmann and P. Zoido-Lobaton. 1998. Dodging the
Grabbing Hand: The Determinants of Unofficial Activity in 69 Countries. Journal of
Public Economics. 76(3): 459-493.
Galor, O. and D. Tsiddon. 1997. The Distribution of Human Capital and Economic
Growth. Journal of Economic Growth. 2(1): 93-124.
Gibbon, E. 1776. History of the Decline and Fall of the Roman Empire. London
Printed for W. Strahan and T Cadell.
Glaesar, E., G. Ponzetto and A. Shleifer. 2006. Why Does Democracy Need
Education. NBER Working Paper 12128.
139
Glomm, G. and B. Ravikumar. 1992. Public versus Private Investment in Human
Capital: Endogenous Growth and Income Inequality. Journal of Political Economy.
100(4): 1127-1170.
Gordon, J. P. P. 1989. Individual Morality and Reputation Costs as Deterrents to Tax
Evasion. European Economic Review. 33(4): 797-805.
Gradstein, M, B. Milanovic, and Y. Ying. 2001. Democracy and Income Inequality:
An Empirical Analysis. World Bank Policy Research Working Papers. No. 2651.
Graetz, M. J., J. Reinganum and L. Wilde. 1986. The Tax Compliance Game:
Toward an Interactive Theory of Law Enforcement. Journal of Law, Economics and
Organization. 2(1): 1-32.
Green, W. 2008. Econometric Analysis. Prentice Hall; 7th
edition.
Gupta, R. 2008. Tax Evasion and Financial Repression. Journal of Economics and
Business. 60(6): 517-535.
Gupta, S., H. Davoodi and R. Alonso-Terme. 2001. Does Corruption Affect Income
Inequality and Poverty? Economics of Governance. 3(1): 23-45.
Hayes, K. J., P. J. Lambert and D. J. Slottje. 1995. Evaluating Effective Income Tax
Progression. Journal of Public Economics. 56(3): 461-474.
140
Hines, J. and L. Summers. 2009. How Globalization Affects Tax Design. NBER
Working Papers 14664.
Hsiao, C. 2003. Analysis of Panel Data. Econometric Society Monographs.
Cambridge University Press.
Hume, D. 1752. Political Discourses. Indianapolis, IN.
Johnson, S., D. Kaufmann and P. Zoido-Lobaton. 1998. Regulatory Discretion and
the Unofficial Economy. American Economic Review. 88(2): 387-392.
Kakwani, N. C. 1977. Measurement of Tax Progressivity: An International
Comparison. The Economic Journal. 87(345): 71-80.
Kaufmann, D., A. Kraay and M. Mastruzzi. 2009. Governance Matters VIII:
Governance Indicators for 1996-2008. World Bank Policy Research. June 2008.
Koreshkova, T. A. 2006. A Quantitative Analysis of Inflation as a Tax on the
Underground Economy. Journal of Monetary Economics. 53(4): 773-796.
Kuznets, S. 1955. Economic Growth and Income Inequality. American Economic
Review. 45(1): 1-28.
Kuznets, S. 1963. Quantitative Aspects of the Economic Growth of Nations: VIII.
Distribution of Income by Size. Economic Development and Cultural Change. 11(2):
1-80.
141
Lahiri, R. and E, Magnani. 2007. On Skill Heterogeneity, Human Capital and
Inflation. International Advances in Economic Research. 13(3): 393-435.
Lahiri, R. and S. Ratnasiri. 2010. A political economy perspective on persistent
inequality, inflation, and redistribution. Economic Modelling. 27(5): 1199-1210.
Lin, W. Z. and C. C. Yang. 2001. A dynamic portfolio choice model of tax evasion:
Comparative statics of tax rates and its implication for economic growth. Journal of
Economic Dynamics and Control. 25(11): 1827-1840.
Ljunggvist, L. and T. Sargent. 2004. Recursive Macroeconomic Theory, 2nd Edition.
Cambridge: MIT Press.
Lupu, N and J. Pontusson. 2011. The Structure of Inequality and he Politics of
Redistribution. American Political Science Review. Vol 105(2): 316-336.
Marshall, M and K. Jaggers. 2010. Polity IV Project: Political Regime
Characteristics and Transitions, 1800-2010.
Meltzer, A. and S. Richard. 1981. A Rational Theory of the Size of Government.
Journal of Political Economy. 89(51): 914-927.
Mendoza, E., A. Razin and L. Tesar. 1994. Effective Tax Rates in Macroeconomics:
Cross-Country Estimates of Tax Rates on Factor Incomes and Consumption. Journal
of Monetary Economics. 34(3): 297-323.
142
Mirrlees, J. 1971. An Exploration in the Theory of Optimum Taxation. Review of
Economic Studies. 38(2): 175-208.
Musgrave, R. A. and T. Thin. 1948. Income Tax Progression, 1929-48. Journal of
Political Economy. 56(6): 498-514.
Myles, G. D. and R. A. Naylor. 1996. A Model of Tax Evasion with Group
Conformity and Social Customs. European Journal of Political Economy. 12(1): 49-
66.
Naylor, R. A. 1989. Strikes, Free Riders, and Social Customs. Quarterly Journal of
Economics. 104(4):771-785.
Neumann, R. J. Holman and J. Alm. 2003. Globalization and Tax Policy. The North
American Journal of Economics and Finance. Vol 20(2): 193-211.
Peltzman, S. 1980. The Growth of Government. Journal of Law and Economics.
23(2): 209-287.
Pencavel, J. 1979. A note on income tax evasion, labor supply, and nonlinear tax
schedules. Journal of Public Economics. 12(1): 115-124.
Perotti, R. 1992. Income Distribution, Politics, and Growth. American Economic
Review. 82(2): 311-316.
Persson, T and G. Tabellini. 1994. Is Inequality Harmful for Growth. American
Economic Review. Vol 84(3): 600-621.
143
Persson, T. and G. Tabellini. 1999. Political Economics and Public Finance. NBER
Working Paper. No. 7097.
Petrocik, J. R. and D. Shaw. 1991. Nonvoting in America. In: Crotty, W., Editor.
Political participation and American democracy. Greenwood Press, New York.
Piketty, T. 2000. Theories of Persistent Inequality and Intergenerational Mobility in
Handbook of Income Distribution. Atkinson, A. B and F. Bourguignon (ed.). North
Holland. 1(8): 429-476.
Pissarides, C. and G. Weber. 1989. An Expenditure-Based Estimate of Britain's
Black Economy. Journal of Public Economics. 39(1): 17-32.
Pommerehne, W. W. and H. Weck-Hannemann. 1996. Tax Rates, Tax
Administration, and Income Tax Evasion in Switzerland. Public Choice. 88(1-2):
161-170.
Ramsey, F. 1927. A Contribution to the Theory of Taxation. Economic Journal.
37(145): 47-61.
Reiter, M. 2000. Relative Preferences and Public Goods. European Economic
Review. 44(3): 565-585.
Roubini, N. and X. Sala-i-Martin. 1995. A Growth Model of Inflation, Tax Evasion,
and Financial Repression. Journal of Monetary Economics. 35(2): 275-301.
144
Saint-Paul, G. and T. Verdier. 1993. Education, Democracy and Growth. Journal of
Development Economics. 42(2): 399-407.
Sandmo, A. 1981. Income Tax Evasion, Labour Supply, and the Equity-Efficiency
Tradeoff. Journal of Public Economics. 16(3): 265-288.
Sandmo, A. 2005. The Theory of Tax Evasion: A Retrospective View. National Tax
Journal. LVIII(4): 643-633.
Sargent, T. 1987. Dynamic Macroeconomic Theory. Harvard University Press.
Slemrod, J. 1985. An Empirical Test for Tax Evasion. Review of Economics and
Statistics. 67(2): 232-238.
Slemrod, J. 1994. Fixing the Leak in Okun's Bucket Optimal Tax Progressivity when
Avoidance can be Controlled. Journal of Public Economics. 55(1): 41-51.
Slemrod, J. 1996. High-Income Families and the Tax Changes of the 1980’s: The
Anatomy of Behavioral Response. NBER Chapters in: Empirical Foundations of
Household Taxation, 169-192.
Slemrod, J. 2007. Cheating Ourselves: The Economics of Tax Evasion. Journal of
Economic Perspectives. 21(1): 25-48.
Slemrod, J. and J. Bakija. 2000. Does Growing Inequality Reduce Tax
Progressivity? Should It? NBER Working Paper 7576.
145
Slemrod J. and S. Yitzhaki. 2000. Tax Avoidance, Evasion, and Administration.
NBER Working Paper 7473.
Solon, G. 1992. Intergenerational Income Mobility in the United States. American
Economic Review. 82(3): 393-408.
Suits, D. B. 1977. Measurement of Tax Progressivity. American Economic Review.
67(4): 747-752.
Torgler, B. 2003. To Evade Taxes or Not to Evade: That is the question. Journal of
Socio-Economics. 32(3): 283-302.
Vogel, J. 1974. Taxation and Public Opinion in Sweden: An Interpretation of Recent
Survey Data. National Tax Journal. Vol 27: 499-513.
Yamarik, S. 2001. Nonlinear Tax Structures and Endogenous Growth. Manchester
School, University of Manchester, vol. 69(1): 16-30.
Yitzhaki, S. 1974. A Note on `Income Tax Evasion: A Theoretical Analysis'. Journal
of Public Economics. 3(2): 201-202.
Zimmerman, D. J. 1992. Regression Toward Mediocrity in Economic Stature.
American Economic Review. 82(3): 409-429.
Zweimuller, J. 2000. Inequality, Redistribution, and Economic Growth. Empirica.
27(1): 1-20.
146
Zweimuller, J. 2000. Shumpeterian Entrepreneurs Meet Engle's Law: The Impact of
Inequality on Innovation-Driven Growth. Journal of Economic Growth. 5(2): 185-
206.
Appendix for Chapter 3
147
APPENDIX
Appendix for Chapter 3
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
Appendix for Chapter 3
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.
Appendix for Chapter 3
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)
Appendix for Chapter 3
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.
Appendix for Chapter 4
151
Appendix for Chapter 4
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
Appendix for Chapter 4
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
Appendix for Chapter 4
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
Appendix for Chapter 4
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
Appendix for Chapter 4
155
)( tt WtWydyne .
Given that the log transformation is monotonic, straightforward manipulation yields
the result of Proposition 1.
Appendix for Chapter 4
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).
Appendix for Chapter 4
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.
Appendix for Chapter 4
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).