Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
INCOME UNCERTAINTY AND
THE RELATIVE SIZE OF
GOVERNMENT
Abstract
Following the industrial revolution there has been a relentless upward trend in the size of the
government both in terms of public services and the redistribution of income. Focusing on
redistribution, I first review major political economy approaches to this subject, which combine utility
maximization principles, voter behaviour, and different political institutions. I then consider different
factors which are expected to influence the level of income redistribution in the economy, and model
how changes in these factors would affect the political equilibrium tax rate. Using the median voter
model in an environment subject to random economic shocks, I examine how changes in mean income,
income distribution, variance of income, probability of finding employment and consumer tastes affect
the politically optimal tax rate.
VASILIS WILLIAM TRIDIMAS
UNIVERSITY COLLEGE LONDON DEPARTMENT OF ECONOMICS
UNDERGRADUATE DISSERTATION 2014
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
Acknowledgements
I would like to thank my family for their support throughout my studies at UCL. My thanks are also
due to Dr Frank Witte for his guidance throughout writing this dissertation.
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
Table of Contents
1) Introduction
1
2) Literature Review
4
3) A General Form Model of Government Size under Employment Uncertainty
8
4) Government Size under Uncertainty and Logarithmic Utility
16
5) Government Size under Uncertainty and Quadratic Utility
20
6) A General Equilibrium Model of Government Size
23
7) Concluding Remarks
32
Appendices
33
References
36
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
1) Introduction
Following the industrial revolution there has been a relentless upward trend in the size of the
government both in terms of public services and the redistribution of income. A wealth of literature has
developed to explain this phenomenon, emphasizing factors such as demand and supply of government
services, and politics. Detailed surveys of the literature can be found in, amongst others, Mueller (2003)
and Hillman (2009).
The present dissertation focuses on the size of government redistribution expenditure driven by demand
for redistribution from voters, using an expected utility maximization framework. Under a government
behaving in accordance with the median voter theorem, it examines the effects of various economic
factors on the size of the government relative to total income. With the assumption of a balanced
government budget, where state expenditure equals tax revenue, different economic setups are used to
derive the tax rate optimally set by a government whose policy is determined entirely by the preferences
of the median voter. This tax rate represents the proportion of the economy taken by the state, and hence
the relative size of the government, and is referred to as the politically optimal equilibrium tax rate. The
effects of the following factors are analysed: median income/wage, mean income/wage, income
uncertainty as captured by the variance of income and uncertainty in finding a job, and the trade-off
between labour and leisure.
Total government expenditure is broken down into two broad categories. First, there are exhaustive
public expenditures which consist of claims on resources that can no longer be used by the private
sector. This includes defence, administration, law and order, education, health and infrastructure.
Second, there are transfer expenditures such as pensions, benefits, subsidies to producers, and interest
on public debt. This type of expenditure amounts to a transfer of the use of resources from some sectors
of the economy to others, or across generations of households when it comes to interest payments. Even
though both types of expenditure have a redistributive element, this dissertation deals with transfer
payments, and specifically benefits paid to improve the distribution of income.1
The Profile of Government Expenditure
To illustrate the growth of government, three figures are displayed. Figure 1 presents the time profile
of UK total public expenditure as a proportion of GDP from 1870 to 2012. Starting from 10% of GDP
in 1870, an unequivocal upward trend is observed, with two large spikes over WWI and WWII, reaching
48% in 2012. Figure 2 provides an international comparative picture. For all the countries listed, a clear
upward time profile of public spending is identified. It is nevertheless seen that different countries
display different patterns, with Sweden showing the largest rise in the relative size of government,
closely followed by France, while the United States and Japan are on the lower end.2 A detailed graph
of expenditure on social insurance as a percentage of GDP for the years 1960 to 2000 is plotted in Figure
1
It should be noted that the size of government in the economy is not fully captured by figures on expenditure
only, because these ignore the regulatory effect of government on the economy which may not involve public
spending, but still affect private economic activity. A simple example is the imposition of seatbelt laws which
require private spending on seatbelts, while also reducing public spending on medical care.
2
The data on which figure 2 is based is presented in the appendix, with additional countries.
1
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
3. This figure further attests the inexorable growth of government, even though for no country has the
path been smooth.
Figure 1: UK Total Public Expenditure as Proportion of GDP
Source: Lee (2012); The Growth of Public Expenditure in the United Kingdom from 1870 to 2005
Figure 2: International Trends in Total Public Expenditure as Percentage of GDP
70
60
50
Pct. GDP
40
30
20
10
0
1860
1880
United Kingdom
1900
1920
United States
1940
Germany
1960
1980
France
2000
Japan
2020
Sweden
Source: Tanzi and Schuknecht (2000); Public Spending in the 20 th Century; IMF; OECD
2
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
Figure 3: Recent International Trends in Social Insurance Expenditure as a Percentage of GDP
Source: Congleton and Bose (2010); The Rise of the Modern Welfare State, Ideology, Institutions and Income
Security: Analysis and Evidence
There are many channels through which demand for redistribution can take place. The dissertation
concentrates on demand driven by income uncertainty and income inequality. The work is structured as
follows. Section 2 selectively reviews existing literature on the size of government. Section 3 presents
a general model of demand for redistributive government spending through utility maximization, which
abstracts from labour disincentive effects and focuses purely on income uncertainty and inequality.
Section 4 removes variance of income and introduces heterogeneity in the probability of finding a job.
Section 5 explicitly derives indirect utility over income uncertainty upon assuming a quadratic utility
function. Section 6 introduces labour disincentive effects by assuming quasilinear preferences over
income and leisure, and proceeds to model uncertainty through a set of states of the world over which
wages vary. Section 7 concludes. An appendix compares the resulting tax rate of section 6 with those
derived from the government maximizing (a) social welfare, and (b) tax revenue, and so acting as a
Leviathan. In all cases examined, the equilibrium tax rate is derived as a function of the underlying
structural variables of the model, and comparative static properties are examined.
3
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
2) Literature Review
Three main theories exist to explain the determination of government policy and consequently the size
of the state. First, the government can be considered as a benevolent social planner which chooses tax
rates and public expenditure to maximize a social welfare function that is the weighted sum of individual
utilities (Mirrlees, 1971; Dixit and Sandmo, 1977). This government may place different emphasis on
the utility of different individuals, for example putting greater effort into maximizing the income of
lower income individuals, but ultimately takes into account everyone’s utility. The socially optimal tax
rate balances equity and efficiency trade-offs.
Second, the government may act to maximize its size (Brennan and Buchanan, 1980). This view starts
from the premise that competition between political parties cannot limit the size of government. The
reasons for this are (1) the rational ignorance of voters, which stipulates that the costs of casting an
informed vote outweigh the uncertain benefit of doing so, and (2) collusion between elected officials
who may pursue policies which bear little relevance to the preferences of the electorate. Further, the
government, as composed of elected officials and appointed bureaucrats, is assumed to have an inherent
desire to expand its budget. Under these conditions the public sector emerges as a gorging ‘Leviathan’
(as coined by Brennan and Buchanan), which expands up to the point where higher tax rates would
reduce spending as disincentive effects on output outweigh scaling effects of higher tax.
The third branch of literature considers the government as a political entrepreneur that wishes to
maximize votes so that it stays in office. An early progenitor of this approach to government growth is
the displacement hypothesis of Peacock and Wiseman (1967) which postulates that citizens enjoy the
benefits from public spending, but dislike paying taxes and there is some tolerable level of taxation,
which operates as a constraint on government spending. In normal times the constraint is binding and
only a gradual increase of public expenditures may be observed. However during periods of social
upheaval, like wars, famines or some large scale national disasters, taxpayers accept increased taxation
to finance urgent needs for larger public expenditure outlays (see Figure 1). After the crisis, public
expenditure does not return to its original path, but is displaced permanently upwards because voters
tolerate higher taxes in return for other civilian services which yield direct benefits.
The thesis will focus on the size of government that results from politically orientated behaviour, and
more specifically by the median voter theorem (Downs, 1957; Black, 1948; Bowen, 1943). It is assumed
that the following conditions hold: (1) politicians are elected/policies are decided with simple majority
voting – 50%+1 of the vote wins the election; (2) politicians are only interested in getting into power
and so want to maximize the number of votes they get such that ideological objectives are not pursued;
(3) each voter has one vote; (4) preferences are single peaked – the individual has an ideal policy choice,
and the further away from the ideal point the more unhappy he/she is; (5) there is only one policy choice
in question, so that voting is single dimensional as for example in a left – right or from small–to–big
choice; (6) voters vote honestly rather than strategically – they do not vote against what they don’t
want (voter sincerity); and (7) all policy choices can be compared with each other. Under these
assumptions the government will choose to maximize the utility of the individual with characteristics
at the median of the distribution of those characteristics.
Employing the median voter framework, Meltzer and Richards (1981) put forward seminal research by
developing a general equilibrium model in which the size of the government relative to income is
proxied by one tax rate in the economy, which exists solely to redistribute income. Individuals maximize
4
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
utility derived from consumption and leisure time subject to choosing working hours at their given
wage, in order to buy consumption. All individuals receive a uniform lump-sum benefit and are taxed
at a proportional income tax rate. Individuals are assumed to differ only in the hourly wage they can
earn (which depends on their own labour productivity), and so have identical preferences. The
government balances its budget such that taxes equal benefits, and chooses the tax rate through a
decisive voting rule.3 Either a dictator is the decisive individual, and chooses a tax rate based on their
own preferences, or the median voter is the decisive voter and the tax rate is chosen as the rate which
will maximize his/her utility. As individuals differ only in hourly wage, the median voter is the voter
with the median hourly wage. The theoretical model finds that the tax rate is an increasing function of
the mean-to-median wage ratio. When mean and median wages are equal, the tax rate is set at zero as
the median voter cannot gain from redistribution. Reducing median income from the mean raises the
tax rate continuously. The politically optimal tax rate however is always less than 100%, because at a
certain point labour disincentive effects which reduce output and hence tax revenue, exceed scaling
effects of additional taxation.
These findings are consistent with the empirical regularity known as Wagner’s Law – that a country’s
public sector becomes a larger proportion of its economy as per capita income rises, also interpreted as
viewing demand for public sector services as income elastic (a textbook exposition of Wagner’s law if
found in Hillman, 2009). The model is consistent with Wagner's law if economic growth raises the
income of skilled individuals relative to the income of unskilled (Kuznets, 1955). As the income of the
skilled rises faster than the unskilled, the income gap between the median and mean voter increases,
and hence the median voter opts for a higher tax rate to redistribute more income and the government
grows. Meltzer and Richards (1983) then go on to test the prediction of the model, and find that while
previous tests of Wagner’s Law itself gave mixed findings, their model supported it when relative
income across the economy was taken into account, as opposed to just absolute income.
More recently, Aidt and Jensen (2013) present and test historical hypotheses for the growth of
government – the historical ones they focus on are instrumental in explaining the growth of government
before the West had fully industrialised. However they also consider ‘secondary’ hypotheses which are
more applicable to developed economies today. The ‘primary’ hypotheses include (1) the franchise
extension hypothesis which suggests that the larger the electorate as a proportion of the population, the
greater the size of the government 4; (2) the retrenchment hypothesis which argues that initial elite
power sharing with emerging middle classes causes smaller government, while the government only
starts to expand once the emerging working classes are given suffrage; (3) the ballot hypothesis which
contends that the introduction of a secret ballot, for a given electorate, is what increases the size of the
government, as the ruling elite can no longer manipulate and suppress the voting intentions of the wider
electorate. Their econometric results rule out the ballot hypothesis, but find support for both the
franchise extension and retrenchment hypotheses.
The three secondary hypotheses are as follows: (1) the war-finance hypothesis, that is greater general
participation in war leads to a larger peace time government expenditures; (2) the modernization
hypothesis which stipulates income growth, urbanization and higher education drive growth in
government (as in Wagner's law); (3) the globalization hypothesis, where government growth is driven
by greater income uncertainty arising from increased trade integration. War finance is supported by the
data. Modernization receives some support for income and education standards as well. Globalization
3
This way the single dimensionality condition of the median voter theorem is satisfied, as a change in tax is
translated into a change in public spending.
4
As the right to vote is extended under a median voter setting, a poorer individual becomes the median voter.
5
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
is proxied by whether a country used the gold standard, and the level of trade protection in place. The
problem with this is that trade protection feeds into the size of government directly through tariffs;
countries which finance a significant portion of their spending through tariffs will also have a low level
of globalization, introducing a source of bias.
Focusing on international trade, Rodrik (1998) explores the empirical regularity that the more open an
economy, the larger the relative size of its government, and provides a simple model to explain why
this is the case. Open economies are more likely to experience economic volatility, as they are prone to
external shocks through factors such as rapidly changing terms of trade, which affect a large portion of
the domestic economy. The government provides a safety net which balances out shocks by absorbing
surplus income from positive shocks, and spending more in the presence of negative shocks. Such
shocks can occur domestically, but their frequency increases in an open economy, and so does demand
for a government safety net. He finds a robust relation between openness and government spending. In
order to explore deeper the government's provision of social insurance to offset the risks of external
shocks, Rodrik sets out a model where the size of the government is defined by the proportion of labour
employed by the public sector, rather than the tax rate or level of expenditure. He shows that when
maximizing the utility of the representative household, higher variance in the terms of trade results in a
higher optimal proportion of labour employed in the public sector.
Taking a similar approach to Meltzer and Richards, Congleton and Bose (2010) set up a general
equilibrium model in which the individual chooses the allocation of their time endowment between
labour and leisure, and is subject to a random shock which reduces their time endowment. The shock
occurs with a given probability depending on the individual’s characteristics (Congleton and Bose use
age), and the individual maximizes utility across leisure and labour time in both states of the world of
normal hours available, and reduced hours available. In their model the median voter is assumed to be
decisive, and the state maximizes his/her utility subject to a balanced budget. They extend the model to
include the ideological preferences of the voter, so that the further away public provision is from the
ideal point of the median voter, the less happy he/she is. Other factors operating on the demand for
government include ideological shifts in the electorate; the effect of this would also depend on the
political institutions in place. Another factor includes pressure from interest groups, but the paper notes
that these groups appear no more powerful in the 1960s and 1970s (the period of greatest observed
growth of the welfare state), than in the 1930s or 1990s. Congleton and Bose also provide an outline of
institutional factors which are widely considered to be driving/have driven the size of government. In
democracies, the adoption or expansion of social welfare programmes takes place as perceived risks,
income, and suffrage increase for the decisive voters. Pent up demands for social insurance could not
be satisfied over the Second World War, leading to rapid expansions immediately after the war ended.
Peace and prosperity would reduce the rate of growth of social insurance in the 1960s and 1970s, by
reducing the perceived levels of risk across developed economies. However, the doubling and tripling
of the relative size of social welfare programmes in the 1960s and 1970s suggests that these programmes
are highly income elastic.
Contrary to explaining the growth of government by factors operating on the demand for government
services and redistribution programmes, a number of studies focus on the supply of government
services. Congleton and Bose (2010) also consider a supply side view, looking at whether growth in the
public sector can be explained through reductions in the costs of producing and administering such
services. They find that this cannot account for the magnitude of the increases.
Baumol (1967) argues that increases of public expenditures may be the result of increases in the cost of
producing public sector services and especially the price of their inputs. He divides the economy into
6
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
two sectors: a progressive sector and a non-progressive sector. The progressive sector is characterised
by cumulative increases in the productivity of labour, which arises because of technological progress
and economies of scale in production. On the contrary, labour productivity in the non-progressive sector
grows at a slower rate. The reason for such a productivity growth lag is that in the progressive sector,
e.g. manufacturing, capital can be substituted into the labour input without affecting the nature of the
product, while in the non-progressive sector it cannot. More specifically, in the non-progressive sector,
e.g. services, labour is an input into production and at the same time it is part of the output consumed,
so that capital substitution is very difficult if at all possible. The progressive sector can afford to pay
wages that grow in line with labour productivity while unit costs stay constant through time. In order to
prevent an exodus of labour from the non-progressive sector towards the progressive sector, where it
can earn a higher wage, the non-progressive sector must keep up with pay increases of the progressive
sector. However, with productivity increases in the non-progressive sector smaller than those in the
progressive sector, unit costs in the former will rise. With the demand for government services assumed
to be price inelastic, cost increases lead to larger government spending. We note however that the
productivity lag hypothesis applies to exhaustive public spending, and not to transfer expenditures.
Kau and Rubin (2002) emphasise the impact of technological advances which allow more efficient tax
collection through increased computerisation, less tax avoidance and less tax evasion. Further,
technological advances lead to a fall in the number of small-size farmers and an increase in the number
and size of manufacturing units, which are easier to tax. Similarly, the increased participation of women
in the labour force increases tax revenue. The upshot of those developments is that the size of tax
revenue rises, driving up the relative size of government.
7
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
3) A General Form Model of Government Size under Employment
Uncertainty
This section examines the equilibrium relative size of government under two dimensions of uncertainty,
namely (I) uncertainty in finding a job, and (II) uncertainty in the income earned from that job.
Individual 𝑖 is assumed to find a job with a probability 𝑃. If employed the individual receives an income
𝑌𝑖 which is randomly distributed with a mean 𝜇𝑖 depending on their own characteristics, and a variance
𝜎 2 . 𝑃 and 𝜎 2 are proxies for economic volatility, and are the same for all individuals. As a result
individuals differ only in income, and so we immediately identify the median voter as the individual
with the median 𝜇𝑖 . The individual spends all of their disposable income on consumption 𝑋𝑖𝑆 where 𝑆
denotes their state of employment; 𝑆 = 1 denotes in employment, and 𝑆 = 0 denotes in unemployment.
The utility function of the individual is written as:
𝑼𝒊 = 𝑼(𝑿𝒊𝑺 , 𝒗𝒂𝒓(𝒀𝒊𝑺 ))
(3.1)
All individuals have identical utility functions. The utility function is strictly concave with respect to
consumption, and strictly decreasing in the variance of income:
𝜕𝑈(𝑋𝑖𝑆 , 𝑣𝑎𝑟(𝑌𝑖𝑆 ))
𝜕 2 𝑈(𝑋𝑖𝑆 , 𝑣𝑎𝑟(𝑌𝑖𝑆 ))
𝜕𝑈(𝑋𝑖𝑆 , 𝑣𝑎𝑟(𝑌𝑖𝑆 ))
> 0,
< 0,
<0
2
𝜕𝑋𝑖𝑆
𝜕𝑣𝑎𝑟(𝑌𝑖𝑆 )
𝜕𝑋𝑖𝑆
Expected utility is therefore given by:
𝑬(𝑼𝒊 ) = 𝑷𝑼(𝑬(𝑿𝒊𝟏 ), 𝒗𝒂𝒓(𝒀𝒊𝟏 )) + (𝟏 − 𝑷)𝑼(𝑬(𝑿𝒊𝟎 ), 𝒗𝒂𝒓(𝒀𝒊𝟎 ))
(3.2)
Note that here an individual's income only has variance when in employment – when unemployed, he
receives a benefit which is not randomly distributed; by definition this income has no variance.
𝑋𝑖1 = (1 − 𝑡)𝑌𝑖 , 𝑋𝑖0 = 𝐺
𝐸(𝑋𝑖1 ) = 𝐸[(1 − 𝑡)𝑌𝑖 ] = (1 − 𝑡)𝜇𝑖 = 𝐶𝑖
𝑬(𝑼𝒊 ) = 𝑷𝑼(𝑪𝒊 , 𝝈𝟐 ) + (𝟏 − 𝑷)𝑼(𝑮, 𝟎)
(3.3)
The state can only spend in benefits its expected tax revenue, implying the following government budget
constraint:
𝑛
𝑛
∑(1 − 𝑃)𝐺 = 𝐸 [∑ 𝑡𝑃𝑌𝑖 ]
𝑖=1
𝑖=1
𝑛
𝑛(1 − 𝑃)𝐺 = 𝑡𝑃 ∑ 𝐸(𝑌𝑖 )
𝑖=1
𝑛
𝑛(1 − 𝑃)𝐺 = 𝑡𝑃 ∑ 𝜇𝑖
𝑖=1
𝑮=𝒕
𝑷
̅
𝝁
𝟏−𝑷
(3.4)
8
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
The state sets the tax rate in order to maximize the utility of the decisive voter, denoted by 𝑖 = 𝐷.
𝐸(𝑈𝐷 ) = 𝑃𝑈(𝐶𝐷 , 𝜎 2 ) + (1 − 𝑃)𝑈(𝐺, 0)
𝜕𝐸(𝑈𝑖 )
=0
𝜕𝑡
𝑃[
𝜕𝑈(𝐶𝐷 , 𝜎 2 ) 𝜕𝐶𝐷 𝜕𝑈(𝐶𝐷 , 𝜎 2 ) 𝜕𝜎 2
𝜕𝑈(𝐺, 0) 𝜕𝐺
+
] + (1 − 𝑃) [
+ 0] = 0
2
𝜕𝐶𝐷
𝜕𝑡
𝜕𝜎
𝜕𝑡
𝜕𝐺
𝜕𝑡
The variance of income is exogenous, so it does not vary with the tax rate:
𝜕𝜎 2
=0
𝜕𝑡
𝜕𝐶𝐷
= −𝜇𝐷 ,
𝜕𝑡
𝜕𝐺
𝑃
=
𝜇̅
𝜕𝑡 1 − 𝑃
𝜕𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝑈(𝐺, 0) 𝑃
(−𝜇𝐷 )] + (1 − 𝑃) [
𝑃[
𝜇̅ ] = 0
𝜕𝐶𝐷
𝜕𝐺 1 − 𝑃
𝝁𝑫
𝝏𝑼(𝑪𝑫 , 𝝈𝟐 )
𝝏𝑼(𝑮, 𝟎)
̅
=𝝁
𝝏𝑪𝑫
𝝏𝑮
(3.5)
In the present set up, the equilibrium tax rate implied by (3.5) shows the relative size of the of the
government; that is the ratio of tax revenue to total income, or equivalently the ratio of government
spending to total income. Setting up this problem with a general form does not provide an explicit
formula determining the politically optimal tax rate, but it does give the condition which the tax rate
must satisfy in order to be optimal. To check that expression (3.5) is a maximum, we check the second
derivative with respect to the tax rate as follows:
𝜕𝐸(𝑈𝐷 )
𝜕𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝑈(𝐺, 0)
= −𝜇𝐷 𝑃
+ 𝑃𝜇̅
𝜕𝑡
𝜕𝐶𝐷
𝜕𝐺
𝜕𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝑈(𝐺, 0)
𝜕(
) 𝜕𝐶
𝜕(
) 𝜕𝜎 2
𝜕(
) 𝜕𝐺
𝜕 2 𝐸(𝑈𝐷 )
𝜕𝐶𝐷
𝜕𝐶𝐷
𝜕𝐺
𝐷
=
−𝜇
𝑃
+
+
𝑃𝜇̅
+ 0]
]
[
𝐷 [
𝜕𝑡 2
𝜕𝐶𝐷
𝜕𝑡
𝜕𝜎 2
𝜕𝑡
𝜕𝐺
𝜕𝑡
𝜕 2 𝐸(𝑈𝐷 )
𝜕 2 𝑈(𝐶𝐷 , 𝜎 2 )
𝜕 2 𝑈(𝐺, 0) 𝑃
(−𝜇
)
=
−𝜇
𝑃
+
𝑃𝜇̅
𝜇̅
𝐷
𝐷
𝜕𝑡 2
𝜕𝐺 2 1 − 𝑃
𝜕𝐶𝐷 2
𝟐
𝟐
𝝏𝟐 𝑬(𝑼𝑫 )
𝑷𝟐 𝟐 𝝏𝟐 𝑼(𝑮, 𝟎)
𝟐 𝝏 𝑼(𝑪𝑫 , 𝝈 )
̅
=
𝑷𝝁
+
𝝁
𝑫
𝝏𝒕𝟐
𝟏−𝑷
𝝏𝑮𝟐
𝝏𝑪𝑫 𝟐
By the definition of the utility function we know that:
𝜕 2 𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝐶𝐷 2
< 0,
𝜕 2 𝑈(𝐺, 0)
<0
𝜕𝐺 2
And so we confirm that the tax rate gives a maximum expected utility for the decisive voter:
9
(3.6)
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
𝜕 2 𝐸(𝑈𝐷 )
<0
𝜕𝑡 2
We can now check the comparative static properties of the equilibrium tax rate, when 𝑃, 𝜎 2 , 𝜇̅ and
𝜇𝐷 vary.
Differentiating both sides of equation (3.5) with respect to 𝑃 gives (that is working by applying the
implicit function theorem):
𝜕𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝑈(𝐺, 0)
𝜕(
) 𝜕𝐶
𝜕(
) 𝜕𝜎 2
𝜕(
) 𝜕𝐺
𝜕𝐶𝐷
𝜕𝐶𝐷
𝜕𝐺
𝐷
𝜇𝐷 [
+
=
𝜇̅
+ 0]
]
[
𝜕𝐶𝐷
𝜕𝑃
𝜕𝜎 2
𝜕𝑃
𝜕𝐺
𝜕𝑃
𝜕𝜎 2
= 0, since 𝜎 2 and the probability of finding a job are independent of each other by assumption.
𝜕𝑃
𝜕𝐶𝐷
𝜕𝑡
= −𝜇𝐷
𝜕𝑃
𝜕𝑃
𝜕𝐺
𝜕
𝑃
=
(𝑡
𝜇̅ )
𝜕𝑃 𝜕𝑃 1 − 𝑃
𝜕𝐺
𝑃
𝜕𝑡
𝜕
𝑃
=
𝜇̅
+ 𝑡𝜇̅
(
)
𝜕𝑃 1 − 𝑃 𝜕𝑃
𝜕𝑃 1 − 𝑃
𝜕𝐺
𝑃
𝜕𝑡
(1 − 𝑃) − 𝑃(−1)
=
𝜇̅
+ 𝑡𝜇̅
(1 − 𝑃)2
𝜕𝑃 1 − 𝑃 𝜕𝑃
𝜕𝐺
𝑃
𝜕𝑡
1
=
𝜇̅
+ 𝑡𝜇̅
(1 − 𝑃)2
𝜕𝑃 1 − 𝑃 𝜕𝑃
𝜕 2 𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝑡
𝜕 2 𝑈(𝐺, 0)
𝑃
𝜕𝑡
1
𝜇𝐷 [
(−𝜇
)]
=
𝜇̅
[
(
𝜇̅
+ 𝑡𝜇̅
)]
𝐷
2
2
(1 − 𝑃)2
𝜕𝑃
𝜕𝐺
1 − 𝑃 𝜕𝑃
𝜕𝐶𝐷
−𝜇𝐷 2
−
𝜕 2 𝑈(𝐶𝐷 , 𝜎 2 ) 𝜕𝑡 𝜕 2 𝑈(𝐺, 0) 𝑃
𝜕𝑡 𝜕 2 𝑈(𝐺, 0) 2
1
2
=
𝜇̅
+
𝑡𝜇̅
2
2
2
(1
𝜕𝑃
𝜕𝐺
1 − 𝑃 𝜕𝑃
𝜕𝐺
− 𝑃)2
𝜕𝐶𝐷
𝜕 2 𝑈(𝐺, 0) 2
1
𝜕𝑡 𝜕 2 𝑈(𝐺, 0) 𝑃
𝜕 2 𝑈(𝐶𝐷 , 𝜎 2 )
2
2
𝑡𝜇̅
=
[
𝜇̅
+
𝜇
]
𝐷
(1 − 𝑃)2 𝜕𝑃
𝜕𝐺 2
𝜕𝐺 2 1 − 𝑃
𝜕𝐶𝐷 2
𝝏𝟐 𝑼(𝑮, 𝟎) 𝟐
𝟏
̅
−
𝒕𝝁
𝝏𝒕
(𝟏 − 𝑷)𝟐
𝝏𝑮𝟐
=
𝝏𝟐 𝑼(𝑪𝑫 , 𝝈𝟐 )
𝝏𝑷 𝝏𝟐 𝑼(𝑮, 𝟎) 𝑷
̅ 𝟐 + 𝝁𝑫 𝟐
𝝁
𝟐
𝟏
−
𝑷
𝝏𝑮
𝝏𝑪𝑫 𝟐
(3.7)
The second derivatives of the utility function with respect to consumption are all negative, meaning that
the numerator of equation (7) is positive and the denominator is negative. This means that by this model:
𝝏𝒕
<0
𝝏𝑷
10
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
The higher the probability of finding a job in the economy, the lower the tax rate the decisive voter will
choose. This would suggest higher tax rates are associated with higher levels of unemployment.
Analytically this is because lower taxes make utility higher when an individual has a job - and so it
makes sense to take advantage of this higher potential utility by lowering tax rates when more likely to
find a job.
Differentiating both sides of equation (5) with respect to 𝜎 2 gives:
𝜕𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝑈(𝐺, 0)
𝜕(
) 𝜕𝐶
𝜕(
)
𝜕(
) 𝜕𝐺
𝜕𝐶𝐷
𝜕𝐶𝐷
𝜕𝐺
𝐷
𝜇𝐷 [
+
=
𝜇̅
+ 0]
]
[
𝜕𝐶𝐷
𝜕𝜎 2
𝜕𝜎 2
𝜕𝐺
𝜕𝜎 2
Noting that the variance of income when individual 𝑖 is employed is independent of the benefit received
when unemployed, we have that:
𝜕𝐺
=0
𝜕𝜎 2
𝜕𝐶𝐷
𝜕𝑡
=
−𝜇
𝐷
𝜕𝜎 2
𝜕𝜎 2
𝜇𝐷
𝜕 2 𝑈(𝐶𝐷 , 𝜎 2 )
𝜕𝐶𝐷 2
(−𝜇𝐷
𝜕𝑡
𝜕 2 𝑈(𝐶𝐷 , 𝜎 2 )
)
+
𝜇
=0
𝐷
𝜕𝜎 2
𝜕𝐶𝐷 𝜎 2
𝝏𝟐 𝑼(𝑪𝑫 , 𝝈𝟐 )
𝝏𝒕
𝝏𝑪𝑫 𝝈𝟐
=
𝝏𝟐 𝑼(𝑪𝑫 , 𝝈𝟐 )
𝝏𝝈𝟐
𝝁𝑫
𝝏𝑪𝑫 𝟐
(3.8)
While the sign of (3.8) may seem ambiguous, we can come to a conclusion based on how we assume
the utility function changes with respect to variance of income and consumption. It is easiest to think
of
𝜕2 𝑈(𝐶𝐷 ,𝜎 2 )
𝜕𝐶𝐷 𝜎 2
𝜕𝑈(𝐶𝐷 ,𝜎2 )
)
𝜕𝐶𝐷
2
𝜕𝜎
𝜕(
as
, which analytically is the first derivative of the marginal utility of
consumption with respect to variance of income. It seems plausible to accept that when expected income
falls to zero, the individual cannot be compensated with a lower variance of that income. In this case,
income variance has a scaling effect on utility. This means that every extra unit of consumption provides
more utility the lower the variance of income is. This is illustrated in Figure 4.
11
Undergraduate Dissertation
Vasilis William Tridimas
𝐸[𝑈(𝑋𝑖1 , 𝑣𝑎𝑟(𝑌𝑖1 ))]
Income Uncertainty and the
Relative Size of Government
Figure 4
𝜎 2𝐴
𝜎 2𝐴 < 𝜎 2 𝐵
𝑋𝑖𝑆
𝑀𝑈𝑋𝐴
𝜎2𝐵
𝑀𝑈𝑋𝐵
0
𝐸(𝑋𝑖1 )
On the vertical axis is expected utility of individual 𝑖 when employed, and on the horizontal is their
expected income. We graph the relation between these variables for a given level of income variance.
The lower the variance of income is, the further away from the horizontal axis the utility curve. This is
shown by two individuals 𝐴 and 𝐵, where 𝐴’s income has a lower than variance than 𝐵’s, and so for
every given expected income level, 𝐴 enjoys a higher level of expected utility than 𝐵. For two
individuals with the same income, the one with lower income variance will display a higher marginal
utility from income. Hence:
𝜕 2 𝑈(𝐶𝐷 , 𝜎 2 )
<0
𝜕𝐶𝐷 𝜎 2
Under these conditions we obtain that:
𝝏𝒕
>0
𝝏𝝈𝟐
The politically optimal tax rate increases with the variance of income. Intuitively, when the decisive
voter experiences higher variance of income, he responds by raising the tax rate, and consequently a
greater benefit, to compensate for this loss of utility when employed, through higher utility when
unemployed.
Total differentiation of (3.5) with respect to 𝜇̅ and 𝑡 gives
𝜕 2 𝐸(𝑈𝐷 )
𝑑𝑡
𝜕𝜇̅ 𝜕𝑡
=− 2
𝜕
𝐸(𝑈𝐷 )
𝑑𝜇̅
𝜕𝑡 2
𝑑𝑡
=−
𝑑𝜇̅
0−𝑃
𝜕𝑈(𝐺, 0)
𝜕 2 𝑈(𝐺, 0) 𝜕𝐺
− 𝜇̅ 𝑃
𝜕𝐺
𝜕𝐺 2 𝜕𝜇̅
2
𝜕 𝐸(𝑈𝐷 )
𝜕𝑡 2
𝜕𝐺
𝑡𝑃
=
𝜕𝜇̅ 1 − 𝑃
12
Undergraduate Dissertation
Vasilis William Tridimas
𝝏𝑼(𝑮, 𝟎)
𝝏𝟐 𝑼(𝑮, 𝟎) 𝒕𝑷
̅
+
𝝁
𝒅𝒕
𝝏𝑮
𝝏𝑮𝟐 𝟏 − 𝑷
=𝑷
𝟐
̅
𝝏 𝑬(𝑼𝑫 )
𝒅𝝁
𝝏𝒕𝟐
Income Uncertainty and the
Relative Size of Government
(3.9)
The first component of the numerator is positive, while the second is negative. The sign of this
derivative is therefore ambiguous and depends on the algebraic form of the utility function. There are
two forces in operation. Before the tax rate is optimally readjusted, an increase in mean income raises
utility in the unemployed state through a higher benefit, but leaves income in the employed state
untouched because the individual does not get a benefit when employed. (A) There is a positive effect
on 𝑡 as income in the unemployed state is now comparatively ‘cheaper’, as for every unit of income
taxed away in the employed state, a greater amount of benefit is received in the unemployed state. (B)
There is a negative effect on 𝑡 as marginal utility in the unemployed state has fallen due to diminishing
marginal utility, and so by the concavity of the utility function, utility can be raised by shifting income
from unemployed state to employed state. The direction of the change in the tax rate depends on which
of these two competing forces is stronger, and this is an issue which can be settled by empirical research.
Total differentiation of (3.5) with respect to 𝜇𝐷 and 𝑡 yields
𝜕 2 𝐸(𝑈𝐷 )
𝑑𝑡
𝜕𝜇 𝜕𝑡
=− 2 𝐷
𝜕 𝐸(𝑈𝐷 )
𝑑𝜇𝐷
𝜕𝑡 2
𝑑𝑡
= −𝑃
𝑑𝜇𝐷
𝜇𝐷
𝜕 2 𝑈(𝐶𝐷 , 𝜎 2 ) 𝜕𝐶𝐷 𝜕𝑈(𝐶𝐷 , 𝜎 2 )
+
−0
𝜕𝜇𝐷
𝜕𝐶𝐷
𝜕𝐶𝐷 2
𝜕 2 𝐸(𝑈𝐷 )
𝜕𝑡 2
𝜕𝐶𝐷
=1−𝑡
𝜕𝜇𝐷
𝒅𝒕
= −𝑷
𝒅𝝁𝑫
𝝁𝑫
𝝏𝟐 𝑼(𝑪𝑫, 𝝈𝟐 )
𝝏𝑼(𝑪𝑫 , 𝝈𝟐 )
(𝟏
−
𝒕)
+
𝝏𝑪𝑫
𝝏𝑪𝑫 𝟐
𝝏𝟐 𝑬(𝑼𝑫 )
𝝏𝒕𝟐
(3.10)
As was the case with (3.9), the sign of (3.10) is ambiguous since the first component of the numerator
is negative, and the second is positive. Qualitatively the reasoning is the same as before. Before the tax
rate is optimally readjusted, a mean preserving increase in median income raises utility in the employed
state, leaving the benefit unaffected, and so utility constant in the unemployed state. (A) There is a
negative effect on 𝑡 as income in the unemployed state becomes comparatively more ‘expensive’; that
is every unit of income gained in the unemployed state through taxation removes a larger amount of
earned income from the employed state. (B) There is a positive effect on 𝑡 as marginal utility in the
employed state rises, and so by the concavity of the utility function, utility can be raised by shifting
income from employed state to unemployed state.
The ambiguity of the results can be illustrated by graphing the budget constraint faced by the
government, with the preferences of the decisive voter over consumption in employment and
consumption in unemployment. This is shown below in Figures 5 and 6. Recalling that the decisive
13
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
voter’s expected income when employed and unemployed are given by 𝐸(𝑋𝐷1 ) = (1 − 𝑡)𝜇𝐷 and 𝑋𝐷0 ≡
𝑃
𝐺 = 𝑡 1−𝑃 𝜇̅ respectively, we rearrange:
𝑡 = 𝑋𝐷0
1−𝑃
𝑃𝜇̅
Which in turn yields:
𝐸(𝑋𝐷1 ) = (1 − 𝑋𝐷0
1−𝑃
) 𝜇𝐷
𝑃𝜇̅
𝟏−𝑷
𝑬(𝑿𝑫𝟏 ) = 𝝁𝑫 − (
𝝁 )𝑿
̅ 𝑫 𝑫𝟎
𝑷𝝁
(3.11)
We graph the above budget constraint. On the vertical axis we plot expected income in employed state,
and on the horizontal we plot income in the unemployed state, equal to the benefit which is certain.
Below the horizontal axis we also draw a line representing the income tax rate corresponding to each
point above on the budget constraint vertically above. By choosing a 100% tax rate the decisive voter
transfers all consumption into the unemployed state, giving them 𝐸(𝑋𝐷1 ) = 0 and 𝑋𝐷0 =
̅
𝑃𝜇
,
1−𝑃
and by
choosing a 0% tax rate he keeps all consumption in the employed state, and there is no redistribution of
income at all, giving him 𝐸(𝑋𝐷1 ) = 𝜇𝐷 and 𝐸(𝑋𝐷0 ) = 0. Under the assumptions of a positive
probability of being unemployed, and utility which is concave with respect to consumption, the median
voter’s optimal choice is graphed as below where the green indifference curve is tangent to the
redistributive budget constraint.
𝐸(𝑋𝐷1 )
Figure 5
𝜇𝐷
𝐸
𝑃𝜇̅
1−𝑃
𝑡=0
𝑡∗
𝑡=1
𝑋𝐷0 ≡ 𝐺
Now raising median income from 𝜇𝐷 0 to 𝜇𝐷 1 , holding mean instant constant, gives the following
possible results on Figure 6:
14
Undergraduate Dissertation
Vasilis William Tridimas
𝐸(𝑋𝐷1 )
Income Uncertainty and the
Relative Size of Government
Figure 6
𝜇𝐷 1
𝐸𝐴
𝜇𝐷 0
𝐸𝐵
𝐸
𝑃𝜇̅
1−𝑃
𝑡=0
𝑡∗𝐵
𝑡∗
𝑡=1
𝑡 ∗𝐴
𝑋𝐷0 ≡ 𝐺
None of the assumptions about the utility function are violated as long as the indifference curves do not
cross (so the new equilibrium indifference curve is above and to the right of the original green one).
When the optimal allocation moves to 𝐸𝐴 , the negative effect on the tax rate dominates; and the tax rate
falls. When the allocation moves to 𝐸𝐵 the positive effect on the tax rate dominates. We cannot know
the actual change without specifying a utility function and getting an idea of its parameters, as well as
the actual changes in relative income levels.
15
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
4) Government Size under Uncertainty and Logarithmic Utility
In this section we explore a simpler model of the determination of government size where we introduce
an explicit functional form of the utility function of the individual, with heterogeneous probability of
employment and certain income when employed. Individual 𝑖 is assumed to find a job with a probability
𝑃𝑖 . If employed the individual receives a certain income 𝑌𝑖 which depends on his own characteristics.
The individual spends all of their disposable income on consumption 𝑋𝑖𝑆 where 𝑆 denotes their state
of employment; 𝑆 = 1 denotes in employment, and 𝑆 = 0 denotes in unemployment. The utility
function of the individual is written as:
𝑼𝒊 = 𝐥𝐨𝐠(𝑿𝒊𝑺 )
(4.1A)
𝑋𝑖1 = (1 − 𝑡)𝑌𝑖
𝑋𝑖0 = 𝐺
The expected utility of individual 𝑖 is hence given by:
𝑬(𝑼𝒊 ) = 𝑷𝒊 𝐥𝐨𝐠((𝟏 − 𝒕)𝒀𝒊 ) + (𝟏 − 𝑷𝒊 ) 𝐥𝐨𝐠(𝑮)
(4.2A)
The state can only spend in benefits its expected tax revenue, implying the following government budget
constraint:
𝑛
𝑛
∑(1 − 𝑃𝑖 )𝐺 = ∑ 𝑡𝑃𝑖 𝑌𝑖
𝑖=1
𝑖=1
Using the covariance formula, the latter becomes:
𝑛
∑ 𝑃𝑖 𝑌𝑖 = 𝑛cov(𝑃, 𝑌) + 𝑛𝑃̅𝑌̅
𝑖=1
We would expect that an individual’s income and the chance of them finding a job depend on many of
the same factors such as health, education, training, and the like. Therefore we take cov(𝑃, 𝑌) to be
positive.
𝑛
𝐺 ∑(1 − 𝑃𝑖 ) = 𝑡[𝑛cov(𝑃, 𝑌) + 𝑛𝑃̅𝑌̅]
𝑖=1
𝐺[𝑛 − 𝑛𝑃̅] = 𝑡[𝑛cov(𝑃, 𝑌) + 𝑛𝑃̅ 𝑌̅]
𝐺=
𝑡[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
1 − 𝑃̅
Substituting this value of 𝐺 in, the expected utility of the decisive voter is now given by:
𝐸(𝑈𝑖 ) = 𝑃𝑖 log((1 − 𝑡)𝑌𝑖 ) + (1 − 𝑃𝑖 )log (
𝑡[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
)
1 − 𝑃̅
Maximizing the decisive voter's expected utility gives:
16
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
𝜕𝐸(𝑈𝐷 )
−1
1
= 𝑃𝐷
+ (1 − 𝑃𝐷 ) = 0
𝜕𝑡
1−t
t
𝑃𝐷
1
1
= (1 − 𝑃𝐷 )
1−t
t
1−t
𝑃𝐷
=
t
1 − 𝑃𝐷
𝐭 = 𝟏 − 𝑷𝑫
(4.3A)
Analytically, the equilibrium tax rate equals the probability of the median voter being unemployed. The
log form of the utility function means that other interesting parameters drop out of the equilibrium
solution. In order to gain a better understanding of what is involved, a taste parameter is added into the
log function, in a manner similar to the Stone-Geary utility function. Expected utility is now written as:
𝑼𝒊 = 𝐥𝐨𝐠(𝑿𝒊𝑺 + 𝜸)
(4.1B)
𝑬(𝑼𝑫 ) = 𝑷𝑫 𝐥𝐨𝐠(𝜸 + 𝑿𝑫𝟏 ) + (𝟏 − 𝑷𝑫 ) 𝐥𝐨𝐠(𝜸 + 𝑿𝑫𝟎 )
(4.2B)
In (4.1B), the parameter 𝛾 captures factors affecting the utility of the individual which are independent
of income. These cannot be affected by the government, so are constant for when employed and when
unemployed. Substituting the government budget constraint into (4.2B):
𝐸(𝑈𝐷 ) = 𝑃𝐷 log(𝛾 + (1 − 𝑡)𝑌𝐷 ) + (1 − 𝑃𝐷 )log (𝛾 +
𝜕𝐸(𝑈𝐷 )
−𝑌𝐷
= 𝑃𝐷 [
] + (1 − 𝑃𝐷 ) [
𝜕𝑡
𝛾 + (1 − 𝑡)𝑌𝐷
𝑌𝐷
𝑃 = (1 − 𝑃𝐷 )
𝛾 + (1 − 𝑡)𝑌𝐷 𝐷
(𝛾 +
𝑡[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
)
1 − 𝑃̅
[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
1 − 𝑃̅
]=0
𝑡[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
𝛾+
1 − 𝑃̅
[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
1 − 𝑃̅
𝑡[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
𝛾+
1 − 𝑃̅
[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
𝑡[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
𝑌𝐷
) 𝑃𝐷 = (1 − 𝑃𝐷 )
)(
𝛾 + (1 − 𝑡)𝑌𝐷
1 − 𝑃̅
1 − 𝑃̅
𝑌𝐷
1 − 𝑃𝐷
[𝛾(1 − 𝑃̅) + [cov(𝑃, 𝑌) + 𝑃̅𝑌̅]𝑡] (
[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
)=
𝛾 + (1 − 𝑡)𝑌𝐷
𝑃𝐷
𝛾(1 − 𝑃̅)
𝑌𝐷
1 − 𝑃𝐷
[
+ 𝑡] (
)=
[cov(𝑃, 𝑌) + 𝑃̅ 𝑌̅]
𝛾 + (1 − 𝑡)𝑌𝐷
𝑃𝐷
𝛾(1 − 𝑃̅)
1 − 𝑃𝐷
[𝛾 + (1 − 𝑡)𝑌𝐷 ]
[
+ 𝑡] 𝑌𝐷 =
̅
̅
[cov(𝑃, 𝑌) + 𝑃 𝑌]
𝑃𝐷
𝛾(1 − 𝑃̅)𝑌𝐷
1 − 𝑃𝐷
[𝛾 + 𝑌𝐷 − 𝑌𝐷 𝑡] − 𝑌𝐷 𝑡
=
[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
𝑃𝐷
17
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
𝛾(1 − 𝑃̅)𝑌𝐷
1 − 𝑃𝐷
1 − 𝑃𝐷
−
(𝛾 + 𝑌𝐷 ) =
(−𝑌𝐷 𝑡) − 𝑌𝐷 𝑡
[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
𝑃𝐷
𝑃𝐷
1 − 𝑃𝐷
𝛾(1 − 𝑃̅)𝑌𝐷
1 − 𝑃𝐷
𝑃𝐷
(𝛾 + 𝑌𝐷 ) −
= 𝑌𝐷 (
𝑡 + 𝑡)
[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
𝑃𝐷
𝑃𝐷
𝑃𝐷
1 − 𝑃𝐷
𝛾(1 − 𝑃̅)𝑌𝐷
1
(𝛾 + 𝑌𝐷 ) −
= 𝑌𝐷 𝑡
[cov(𝑃, 𝑌) + 𝑃̅𝑌̅]
𝑃𝐷
𝑃𝐷
̅ )𝑷𝑫
𝜸
𝜸(𝟏 − 𝑷
𝒕 = (𝟏 − 𝑷𝑫 ) ( + 𝟏) −
̅𝒀
̅]
[𝐜𝐨𝐯(𝑷, 𝒀) + 𝑷
𝒀𝑫
(4.3B)
The equilibrium tax rate now depends on all parameters specified in the model, that is, decisive and
mean probabilities of employment, decisive and mean income, covariance between income and
probability of employment, and the taste parameter. This specific case model provides an intuitive
framework which allows for heterogeneity in both income and probability of finding employment. More
specifically, checking the comparative statics we derive:
̅)
𝝏𝒕
𝜸
𝜸(𝟏 − 𝑷
= − ( + 𝟏) −
<0
̅𝒀
̅]
[𝐜𝐨𝐯(𝑷, 𝒀) + 𝑷
𝝏𝑷𝑫
𝒀𝑫
(4.4)
As the probability of the decisive voter getting a job rises, he chooses a lower income tax, since tax
only benefits him in the unemployed state, and he is now less likely to be unemployed.
𝝏𝒕
𝜸
= −(𝟏 − 𝑷𝑫 ) 𝟐 < 0
𝝏𝒀𝑫
𝒀𝑫
(4.5)
As the income of the decisive voter rises, he chooses a lower tax rate, as every unit of benefit in the
unemployed state reduces income in the employed state by a greater amount.
̅ )𝑷𝑫 𝑷
̅
𝝏𝒕
𝜸(𝟏 − 𝑷
=
>0
̅ [𝐜𝐨𝐯(𝑷, 𝒀) + 𝑷
̅𝒀
̅ ]𝟐
𝝏𝒀
(4.6)
The decisive voter chooses a higher tax rate as mean income rises and his income stays the same, as
every unit of income taxed away in the employed state now confers a greater amount of benefit in the
unemployed state.
̅𝒀
̅ ] + 𝜸𝒀
̅ (𝟏 − 𝑷
̅ )𝑷𝑫
𝝏𝒕 𝜸𝑷𝑫 [𝐜𝐨𝐯(𝑷, 𝒀) + 𝑷
=
>0
̅
̅𝒀
̅ ]𝟐
[𝐜𝐨𝐯(𝑷, 𝒀) + 𝑷
𝝏𝑷
(4.7)
Similarly to an increase in mean income, an increase in the mean probability of finding a job also
increases taxable income in the economy, increasing the benefit of a higher tax rate.
̅ )𝑷𝑫
𝝏𝒕
𝜸(𝟏 − 𝑷
=
>0
̅𝒀
̅ ]𝟐
𝝏𝐜𝐨𝐯(𝑷, 𝒀) [𝐜𝐨𝐯(𝑷, 𝒀) + 𝑷
(4.8)
An increase in the covariance raises the income tax for the same reason as for a rise in mean income
and mean probability of finding a job.
̅ )𝑷𝑫
(𝟏 − 𝑷
𝝏𝒕 𝟏 − 𝑷𝑫
=
−
=?
̅𝒀
̅]
[𝐜𝐨𝐯(𝑷, 𝒀) + 𝑷
𝝏𝜸
𝒀𝑫
(4.9)
18
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
The latter is ambiguous, since there are two opposing forces in operation. Increasing 𝛾 reduces the
marginal utility of consumption in both states 5, leading the individual to try to transfer income across
to the other state.
In sum, these results show that as mean income and probability of finding a job increase, the median
voter opts to increase the tax rate, a prediction consistent with Wagner’s law, while selfishly reducing
the tax rate when his own income and probability of finding a job increase relative to the mean.
5
Differentiating the marginal utility of consumption with respect to 𝛾 yields
19
𝜕2 𝑈
𝜕𝑋𝜕𝛾
1
= − (𝛾+𝑋)2.
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
5) Government Size under Uncertainty and Quadratic Utility
Another approach to modelling preferences over uncertainty is to employ a utility function which will
implicitly contain the variance of income, without explicitly specifying its position in the function as in
section 3. Using a quadratic utility function we can achieve this, and obtain the same general results as
in section 3.
Individual 𝑖 has a probability 𝑃 of finding a job. The income he receives 𝑌𝑖 is randomly distributed with
a mean 𝜇𝑖 and a variance 𝜎 2 . The levels of 𝑃 and 𝜎 2 are proxies for economic volatility, and are identical
for all individuals. Individuals spend all of their disposable income on consumption 𝑋𝑖𝑆 where 𝑆
denotes their state of employment; 𝑆 = 1 denotes in employment, and 𝑆 = 0 denotes in unemployment.
Utility is written in quadratic form as:
𝟏
𝟏
𝑼𝒊 = 𝑷 [𝑿𝒊𝟏 − 𝜸𝑿𝒊𝟏 𝟐 ] + (𝟏 − 𝑷) [𝑿𝒊𝟎 − 𝜸𝑿𝒊𝟎 𝟐 ]
𝟐
𝟐
(5.1)
Individuals are taxed a proportion 𝑡 of their income and the unemployed are given a benefit 𝐺. All
disposable income is spent on consumption (i.e. there is no saving).
𝑋𝑖1 = (1 − 𝑡)𝑌𝑖
and
𝑋𝑖0 = 𝐺
𝟏
𝟏
𝟐
𝑼𝒊 = 𝑷 [(𝟏 − 𝒕)𝒀𝒊 − 𝜸((𝟏 − 𝒕)𝒀𝒊 ) ] + (𝟏 − 𝑷) [𝑮 − 𝜸𝑮𝟐 ]
𝟐
𝟐
(5.2)
The state maximizes the expected utility of the decisive voter 𝐷. The state can only spend in benefits
its expected tax revenue:
𝒏
𝒏
∑(𝟏 − 𝑷)𝑮 = 𝑬 [∑ 𝒕𝑷𝒀𝒊 ]
𝒊=𝟏
(5.3)
𝒊=𝟏
Solving for 𝐺:
𝑛
𝑛(1 − 𝑃)𝐺 = 𝑡𝑃 ∑ 𝜇𝑖
𝑖=1
𝑮=𝒕
𝑷
̅
𝝁
𝟏−𝑷
(5.4)
Substituting this into the utility function of the decisive voter:
𝟏
𝑷
𝟏
𝑷𝟐
̅𝒕 − 𝜸
̅ 𝟐 𝒕𝟐 ]
𝑼𝑫 = 𝑷 [𝒀𝑫 − 𝒕𝒀𝑫 − 𝜸(𝟏 − 𝟐𝒕 + 𝒕𝟐 )𝒀𝑫 𝟐 ] + (𝟏 − 𝑷) [
𝝁
𝝁
𝟐
𝟏−𝑷
𝟐 (𝟏 − 𝑷)𝟐
In expected terms, the state maximizes this function with respect to 𝑡:
1
1
𝑃2
2
2
𝐸(𝑈𝐷 ) = 𝑃 [𝐸(𝑌𝐷 ) − 𝑡𝐸(𝑌𝐷 ) − 𝛾(1 − 2𝑡 + 𝑡 )𝐸(𝑌𝐷 )] + [𝑃𝜇̅ 𝑡 − 𝛾
𝜇̅ 2 𝑡 2 ]
2
2 (1 − 𝑃)
Recalling that 𝐸(𝑌𝑖 2 ) = 𝜎 2 + 𝜇𝑖 2 , 𝐸(𝑌𝑖 ) = 𝜇𝑖 , and substituting:
20
(5.5)
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
1
1
𝑃2
𝐸(𝑈𝐷 ) = 𝑃 [𝜇𝐷 − 𝑡𝜇𝐷 − 𝛾(1 − 2𝑡 + 𝑡 2 )(𝜎 2 + 𝜇𝐷 2 )] + [𝑃𝜇̅ 𝑡 − 𝛾
𝜇̅ 2 𝑡 2 ]
2
2 (1 − 𝑃)
𝜕𝐸(𝑈𝐷 )
1
1
𝑃2
= 𝑃 [−𝜇𝐷 − 𝛾(−2 + 2𝑡)(𝜎 2 + 𝜇𝐷 2 )] + [𝑃𝜇̅ − (2)𝛾
𝜇̅ 2 𝑡] = 0
(1 − 𝑃)
𝜕𝑡
2
2
[−𝜇𝐷 − 𝛾(−1 + 𝑡)(𝜎 2 + 𝜇𝐷 2 )] + [𝜇̅ − 𝛾
𝑃
𝜇̅ 2 𝑡] = 0
(1 − 𝑃)
−𝜇𝐷 + 𝛾(𝜎 2 + 𝜇𝐷 2 ) − 𝛾𝑡(𝜎 2 + 𝜇𝐷 2 ) + 𝜇̅ − 𝛾
𝜇̅ − 𝜇𝐷 + 𝛾(𝜎 2 + 𝜇𝐷 2 ) = 𝛾𝑡 [𝜎 2 + 𝜇𝐷 2 +
𝑃
𝜇̅ 2 𝑡 = 0
(1 − 𝑃)
𝑃
𝜇̅ 2 ] = 0
(1 − 𝑃)
̅ − 𝝁𝑫
𝝁
𝟐
𝟐
𝜸 + 𝝁𝑫 + 𝝈
𝒕=
𝑷
𝟐
𝟐
̅𝟐
𝟏 − 𝑷 𝝁 + 𝝁𝑫 + 𝝈
(5.6)
The determinants of the equilibrium tax rate are mean income, decisive voter’s income, income
variance, probability of finding a job, and the preference parameter 𝛾.
As 0 < 𝑃 < 1, the denominator is always positive. Under the median voter theorem where 𝜇𝐷 = 𝜇𝑀 ,
a positively skewed distribution of income means that 𝜇̅ − 𝜇𝐷 > 0 guaranteeing that the numerator is
also positive.
For the above equation to yield 𝑡 < 1, it must be that:
𝜇̅ − 𝜇𝐷
𝑃
<
𝜇̅ 2
(1 − 𝑃)
𝛾
𝜇̅ 𝜇𝐷
𝑃
𝜇̅ − 𝜇̅
<
𝜇̅
(1 − 𝑃)
𝛾
1−𝛿
𝑃
𝜇𝐷
<
𝜇̅ , where 𝛿 ≡
(1 − 𝑃)
𝛾
𝜇̅
𝛿 <1−
𝑃
𝛾𝜇̅
(1 − 𝑃)
If we let
̅ −𝜇𝐷
𝜇
𝛾
𝑃
+ 𝜇𝐷 2 ≡ 𝐴 and (1−𝑃) 𝜇̅ 2 + 𝜇𝐷 2 ≡ 𝐵, 𝑡 is rewritten as 𝑡 =
𝐴+𝜎 2
𝐵+𝜎 2
. Turning to the
comparative statics, we find:
𝝏𝒕
𝑩−𝑨
=
>0
𝟐
(𝑩 + 𝝈𝟐 )𝟐
𝝏𝝈
(5.7)
The more volatile the income is, the higher the tax rate, and hence the larger the size of the government.
21
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
𝑃
𝜇̅ − 𝜇𝐷
2 𝜕
2
2
𝜕𝑡 −𝜇̅ 𝜕𝑃 (1 − 𝑃) [ 𝛾 + 𝜇𝐷 + 𝜎 ]
=
2
𝜕𝑃
𝑃
𝜇̅ 2 + 𝜇𝐷 2 + 𝜎 2 ]
[
(1 − 𝑃)
𝜕
𝑃
2
𝜕𝑡 −𝜇̅ 𝐴 𝜕𝑃 (1 − 𝑃)
=
𝜕𝑃
𝐵2
̅𝟐 𝑨
𝝏𝒕
−𝝁
=
<0
𝝏𝑷 (𝟏 − 𝑷)𝟐 𝑩𝟐
(5.8)
The higher the probability of finding a job, the lower the equilibrium tax rate.
̅ − 𝝁𝑫
𝑷
𝟏
𝝁
𝑷
𝟐
𝟐
̅ 𝟐 + 𝝁𝑫 𝟐 + 𝝈𝟐 ) ( ) − (
̅
𝝏𝒕 (𝟏 − 𝑷 𝝁
𝜸
𝜸 + 𝝁𝑫 + 𝝈 ) (𝟐 𝟏 − 𝑷 𝝁)
=
𝟐
̅
𝝏𝝁
𝑷
̅ 𝟐 + 𝝁𝑫 𝟐 + 𝝈𝟐 )
(𝟏 − 𝑷 𝝁
̅ − 𝝁𝑫
𝑷
𝟏
𝝁
̅ 𝟐 + 𝝁𝑫 𝟐 + 𝝈𝟐 ) (𝟐𝝁𝑫 − ) − (
(
𝝁
+ 𝝁𝑫 𝟐 + 𝝈𝟐 ) (𝟐𝝁𝑫 )
𝝏𝒕
𝟏−𝑷
𝜸
𝜸
=
𝟐
𝝏𝝁𝑫
𝑷
̅ 𝟐 + 𝝁𝑫 𝟐 + 𝝈𝟐 )
(
𝝁
𝟏−𝑷
(5.9)
(5.10)
The signs of equations (5.9) and (5.10) are ambiguous and depend on the actual parameter values. The
ambiguity of these results mirrors that found in the general form model of section 3, such that the same
graphical analysis applies.
22
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
6) A General Equilibrium Model of Government Size
6.A
Choice of Government Size under Certainty
Now I set up a general equilibrium model under which individuals choose their labour/leisure hours
based on their hourly wage and the taxes/benefits set by the government. Individuals 𝑖 all earn a wage
𝑤𝑖 which depends upon their own characteristics such as health and education, but they have identical
preferences. They enjoy consumption 𝑋𝑖 and leisure time 𝐻𝑖 with the following quasilinear utility
function:
𝑼𝒊 = 𝜶 𝐥𝐨𝐠(𝑿𝒊 ) + 𝑯𝒊
(6.1A)
Individuals all have a total available time 𝑇, and choose to allocate 𝐿𝑖 into work which earns them
consumption, such that 𝐿𝑖 + 𝐻𝑖 = 𝑇. Individuals are taxed a proportion 𝑡 of their income, and are all
given a benefit in the amount of 𝐺 which is equal for everyone. Their consumption is equal to their
disposable income plus the benefit they receive:
𝑿𝒊 = (𝟏 − 𝒕)𝑳𝒊 𝒘𝒊 + 𝑮
(6.2A)
Utility is maximized subject to this constraint. Rewriting in terms of 𝐻𝑖 :
𝑋𝑖 = (1 − 𝑡)(𝑇 − 𝐻𝑖 )𝑤𝑖 + 𝐺
Substituting this constraint into the utility function, we maximize with respect to leisure time:
𝑈𝑖 = 𝛼 log[(1 − 𝑡)(𝑇 − 𝐻𝑖 )𝑤𝑖 + 𝐺] + 𝐻𝑖
𝜕𝑈𝑖
−𝛼(1 − 𝑡)𝑤𝑖
=
+1=0
𝜕𝐻𝑖 (1 − 𝑡)(𝑇 − 𝐻𝑖 )𝑤𝑖 + 𝐺
𝛼(1 − 𝑡)𝑤𝑖 = (1 − 𝑡)(𝑇 − 𝐻𝑖 )𝑤𝑖 + 𝐺
𝛼 = 𝑇 − 𝐻𝑖 +
𝐺
(1 − 𝑡)𝑤𝑖
Optimal leisure time choice: 𝑯𝒊 = 𝑻 − 𝜶 +
Optimal labour supply choice: 𝑳𝒊 = 𝜶 −
𝑮
(𝟏 − 𝒕)𝒘𝒊
𝑮
(𝟏 − 𝒕)𝒘𝒊
Optimal consumption choice: 𝑿𝒊 = 𝜶(𝟏 − 𝒕)𝒘𝒊
(6.3A)
(6.4A)
(6.5A)
While this model allows for disincentive effects to influence the policy decision, it is strictly applicable
to the case where all individuals choose a positive level of labour supply.6 For concreteness I assume
that all individuals work, i.e. government policy only affects the intensive decision of how much labour
In order for individual 𝑖 to participate in the labour market, it must be the case that 𝐿𝑖 ≥ 0, which using equation
𝐺
(6.4A), requires the individual to be offered a wage greater than or equal to the level 𝑊𝑅 =
. Optimal
6
𝛼(1−𝑡)
consumption then becomes equal to 𝐺.
23
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
time to allocate, not the extensive decision of whether to work or not. Individuals who would be
considered unemployed can be viewed as choosing 𝐿𝑖 very close to or equal to zero.
We now have the indirect utility function:
𝑽𝒊 = 𝜶 𝐥𝐨𝐠[𝜶(𝟏 − 𝒕)𝒘𝒊 ] + 𝑻 − 𝜶 +
𝑮
(𝟏 − 𝒕)𝒘𝒊
(6.6A)
The government must balance its budget such that taxes equal benefits:
𝑛
𝑛
∑ 𝑡 𝑤𝑖 𝐿𝑖 = ∑ 𝐺
𝑖=1
𝑖=1
Substituting in the individual’s labour supply choice:
𝑛
𝑛
𝑖=1
𝑖=1
𝐺
∑ 𝑡 𝑤𝑖 (𝛼 −
) = ∑𝐺
(1 − 𝑡)𝑤𝑖
𝑛
𝑛
𝑛
𝑖=1
𝑖=1
𝑖=1
𝑡𝐺
𝛼𝑡 ∑ 𝑤𝑖 −
∑1 = ∑𝐺
(1 − 𝑡)
𝛼𝑡𝑛𝑤
̅−
𝐺 (1 +
𝑡𝐺
𝑛 = 𝑛𝐺
(1 − 𝑡)
𝑡
) = 𝛼𝑡𝑤
̅
1−𝑡
1−𝑡+𝑡
𝐺(
) = 𝛼𝑡𝑤
̅
1−𝑡
7
̅ 𝒕(𝟏 − 𝒕)
𝑮 = 𝜶𝒘
(6.7A)
Now the government acts to maximize the utility of the decisive voter:
𝑉𝐷 = 𝛼 log[𝛼(1 − 𝑡)𝑤𝐷 ] + 𝑇 − 𝛼 +
𝐺
(1 − 𝑡)𝑤𝐷
Substituting in 𝐺, we maximize with respect to 𝑡:
𝑉𝐷 = 𝛼 log[𝛼(1 − 𝑡)𝑤𝐷 ] + 𝑇 − 𝛼 +
𝛼𝑡(1 − 𝑡)𝑤
̅
(1 − 𝑡)𝑤𝐷
𝑽𝑫 = 𝜶 𝐥𝐨𝐠[𝜶(𝟏 − 𝒕)𝒘𝑫] + 𝑻 − 𝜶 +
̅
𝜶𝒕𝒘
𝒘𝑫
(6.8A)
𝜕𝑉𝐷
−1
𝛼𝑤
̅
= 𝛼(
)+
=0
𝜕𝑡
1−𝑡
𝑤𝐷
Note that the quasilinear utility function results in a significant reduction in algebraic complexity; as 𝑡 cancels
from the numerator on the left hand side, the resulting expression for 𝐺 is only a second order polynomial in terms
of 𝑡.
7
24
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
𝑤
̅
1
=
𝑤𝐷 1 − 𝑡
1−𝑡 =
𝑤𝐷
𝑤
̅
𝒕=𝟏−
𝒘𝑫
̅
𝒘
Defining 𝛿 =
𝑤𝐷
̅
𝑤
as the skewness of the wage distribution, we have:
𝒕=𝟏−𝜹
(6.9A)
𝝏𝒕
= −𝟏
𝝏𝜹
(6.10A)
𝝏𝒕
<0
𝝏𝜹
This model mirrors the findings of Meltzer and Richards (1981), where as mean income rises relative
to median income, the median individual votes to redistribute more income towards themselves; as
median income rises relative to mean, the median individual reduces the tax rate as redistribution is
now more costly. It however goes further by finding an explicit solution for the tax rate, which is not
possible with a general form or more complicated set up. When mean income equals (or is higher than)
median income, the median voter has nothing to gain from redistribution and so votes for a zero level
of taxation.
6.B
Choice of Government Size under Wage Uncertainty
While the above model provides a simple and intuitive framework capturing disincentive effects of
taxation, and the decisive individual’s desire to redistribute income towards themselves, it does not
allow for income uncertainty. We can extend the initial assumptions to consider the individual now
choosing their optimal labour supply and leisure time in a given state of the world, where the wage they
can earn differs across states. States differ because of unexpected idiosyncratic shocks affecting
individual wage, as well as synchronised shocks to the entire economy, to which the wages of different
individuals may respond differently.
In this model we assume that voting is not state contingent. This means that even though income varies
across different states, all individuals have perfect foresight as to how their wage varies across states.
They do not know what states will occur when, but they know the probability of each state occurring.
As a result, a median voter is identified as the decisive voter across all given states.
The government is again assumed to provide a benefit 𝐺 to all individuals and to levy a universal income
tax at rate 𝑡. These levels are mandated to be constant across different states of the world, reflecting the
electorate’s demand for redistribution under uncertainty. The utility function and budget constraint of
the individual are as before, but are now state contingent as shown below.
𝑼𝒊𝑺 = 𝜶 𝐥𝐨𝐠(𝑿𝒊𝑺 ) + 𝑯𝒊𝑺
(6.1B)
𝑿𝒊𝑺 = (𝟏 − 𝒕)𝑳𝒊𝑺 𝒘𝒊𝑺 + 𝑮
(6.2B)
25
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
This is solved as essentially the same optimization problem as in 6.A, yielding state dependent optimal
choices:
Optimal leisure time choice: 𝑯𝒊𝑺 = 𝑻 − 𝜶 +
Optimal labour supply choice: 𝑳𝒊𝑺 = 𝜶 −
𝑮
(𝟏 − 𝒕)𝒘𝒊𝑺
𝑮
(𝟏 − 𝒕)𝒘𝒊𝑺
Optimal consumption choice: 𝑿𝒊𝑺 = 𝜶(𝟏 − 𝒕)𝒘𝒊𝑺
(6.3B)
(6.4B)
(6.5B)
This yields the indirect utility function:
𝑽𝒊𝑺 = 𝜶 𝐥𝐨𝐠[𝜶(𝟏 − 𝒕)𝒘𝒊𝑺 ] + 𝑻 − 𝜶 +
𝑮
(𝟏 − 𝒕)𝒘𝒊𝑺
(6.6B)
The tax revenue collected per individual 𝑖 in state 𝑠, 𝑐𝑖𝑆 , is also state dependent:
𝑐𝑖𝑆 = 𝑡𝐿𝑖𝑆 𝑤𝑖𝑆
𝑐𝑖𝑆 = 𝑡 [𝛼 −
𝐺
]𝑤
(1 − 𝑡)𝑤𝑖𝑆 𝑖𝑆
𝑐𝑖𝑆 = 𝛼𝑤𝑖𝑆 𝑡 −
𝑡
𝐺
(1 − 𝑡)
In order to determine the expected tax revenue per individual, we must further specify the nature of
uncertainty in this economy. We assume 𝑚 states of the world over which the wage varies, each state
occurring with probability 𝜆𝑆 where ∑𝑚
𝑆=1 𝜆𝑆 = 1.
The expected tax revenue collected from individual 𝑖 is therefore:
𝑚
𝐸(𝑐𝑖𝑆 |𝑆) ≡ 𝐶𝑖 = ∑ 𝜆𝑆 𝑐𝑖𝑆
𝑆=1
𝑚
𝐶𝑖 = ∑ 𝜆𝑆 [𝛼𝑤𝑖𝑆 𝑡 −
𝑆=1
𝑚
𝐶𝑖 = ∑ 𝛼𝑡𝜆𝑆 𝑤𝑖𝑆 −
𝑆=1
𝑡
𝐺]
(1 − 𝑡)
𝑡
𝐺𝜆
(1 − 𝑡) 𝑆
𝑚
𝑚
𝑆=1
𝑆=1
𝑡
𝐶𝑖 = 𝛼𝑡 ∑ 𝜆𝑆 𝑤𝑖𝑆 −
𝐺 ∑ 𝜆𝑆
(1 − 𝑡)
𝑚
∑ 𝜆𝑆 𝑤𝑖𝑆 = 𝑚cov(𝜆𝑆 , 𝑤𝑖𝑆 ) + 𝑚𝜆̅̅̅̅
𝑤i
𝑆=1
𝑚
,
∑ 𝜆𝑆 = 𝑚𝜆̅ = 1
𝑆=1
Where cov(𝜆𝑆 , 𝑤𝑖𝑆 ) ≡ 𝐾𝑖 is the covariance between the probability of a state 𝑆 occurring, and the wage
individual 𝑖 earns in that state, and may vary across individuals. ̅̅̅
𝑤i is the expected wage earned by
individual 𝑖 across all states. If a state of high wage carries a low probability of occurrence, then
26
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
cov(𝜆𝑆 , 𝑤𝑖𝑆 ) < 0. Conversely, if a state of high wage carries a high probability of occurrence, then
cov(𝜆𝑆 , 𝑤𝑖𝑆 ) > 0. If the probability of any given wage is the same across states, cov(𝜆𝑆 , 𝑤𝑖𝑆 ) = 0.
𝐶𝑖 = 𝛼𝑡[𝑚𝐾𝑖 + ̅̅̅]
𝑤i −
𝑡
𝐺
(1 − 𝑡)
Total tax revenue is then the sum of these expected revenues across all individuals, and is equated to
total government expenditure on benefits in order to satisfy the government budget constraint:
𝑛
𝑛
∑ 𝐶𝑖 = ∑ 𝐺
𝑖=1
𝑖=1
𝑛
∑ (𝛼𝑡[𝑚𝐾𝑖 + ̅̅̅]
𝑤i −
𝑖=1
̅ + 𝑛𝑊
̅]−
𝛼𝑡[𝑚𝑛𝐾
𝑡
𝐺) = 𝑛𝐺
(1 − 𝑡)
𝑡
𝐺𝑛 = 𝑛𝐺
(1 − 𝑡)
̅ is the mean wage of all individuals across all states, and 𝐾
̅ is the mean of all individuals’
Where 𝑊
covariances between probability of a state occurring and the wage they earn in that state.
1−𝑡+𝑡
̅ + 𝑛𝑊
̅] = 𝐺(
𝛼𝑡[𝑚𝑛𝐾
)
(1 − 𝑡)
̅ +𝑾
̅̅̅]𝒕(𝟏 − 𝒕)
𝑮 = 𝜶[𝒎𝑲
(6.7B)
Given this constraint the utility of individual 𝑖 in state 𝑆 is:
𝑉𝑖𝑆 = 𝛼 log[𝛼(1 − 𝑡)𝑤𝑖𝑆 ] + 𝑇 − 𝛼 +
̅+𝑊
̅ ]𝑡(1 − 𝑡)
𝛼[𝑚𝐾
(1 − 𝑡)𝑤𝑖𝑆
𝑉𝑖𝑆 = 𝛼 log[𝛼(1 − 𝑡)𝑤𝑖𝑆 ] + 𝑇 − 𝛼 +
̅+𝑊
̅ ]𝑡
𝛼[𝑚𝐾
𝑤𝑖𝑆
The expected utility of individual 𝑖 is the weighted sum of state contingent utilities across all states:
𝑚
𝐸(𝑉𝑖𝑆 |𝑆) = ∑ 𝜆𝑆 𝑉𝑖𝑆
𝑆=1
𝑚
𝐸(𝑉𝑖𝑆 |𝑆) = ∑ (𝜆𝑆 𝛼 log[𝛼(1 − 𝑡)𝑤𝑖𝑆 ] + 𝜆𝑆 𝑇 − 𝜆𝑆 𝛼 +
𝑆=1
𝑚
𝑚
𝜆𝑆
̅+𝑊
̅ ]𝑡)
𝛼[𝑚𝐾
𝑤𝑖𝑆
𝑚
𝑚
̅+𝑊
̅ ]𝑡 ∑
𝐸(𝑉𝑖𝑆 |𝑆) = 𝛼 log(1 − 𝑡) ∑ 𝜆𝑆 + ∑ 𝜆𝑆 𝛼 log(𝛼𝑤𝑖𝑆 ) + ∑ 𝜆𝑆 (𝑇 − 𝛼) + 𝛼[𝑚𝐾
𝑆=1
𝑆=1
𝑆=1
And for the decisive voter 𝐷:
𝑬(𝑽𝑫𝑺 |𝑺) =
27
𝑆=1
𝜆𝑆
𝑤𝑖𝑆
Undergraduate Dissertation
Vasilis William Tridimas
𝒎
𝒎
𝒎
Income Uncertainty and the
Relative Size of Government
𝒎
̅ + ̅𝑾
̅̅]𝒕 ∑
𝜶 𝐥𝐨𝐠(𝟏 − 𝒕) ∑ 𝝀𝑺 + ∑ 𝝀𝑺 𝜶 𝐥𝐨𝐠(𝜶𝒘𝑫𝑺 ) + ∑ 𝝀𝑺 (𝑻 − 𝜶) + 𝜶[𝒎𝑲
𝑺=𝟏
𝑺=𝟏
𝑺=𝟏
𝑺=𝟏
𝝀𝑺
𝒘𝑫𝑺
(6.8B)
Maximization of the utility of the decisive voter gives:
𝑚
𝑚
𝑆=1
𝑆=1
𝜕𝐸(𝑉𝐷𝑆 |𝑆)
−1
𝜆
̅+𝑊
̅]∑ 𝑆 = 0
=𝛼
∑ 𝜆𝑆 + 𝛼[𝑚𝐾
𝜕𝑡
1−𝑡
𝑤𝐷𝑆
𝑚
𝛼
𝜆
̅+𝑊
̅]∑ 𝑆
= 𝛼[𝑚𝐾
1−𝑡
𝑤𝐷𝑆
𝑆=1
Defining 1⁄𝑤𝐷𝑆 as a measure of poverty equal to the inverse of the wage of the decisive voter, we
obtain:
𝑚
1
𝜆
𝑚𝜆̅
̅+𝑊
̅ ] ∑ 𝑆 = [𝑚𝐾
̅+𝑊
̅ ] [𝑚cov(𝜆𝑆 , 1⁄𝑤 ) +
= [𝑚𝐾
]
𝐷𝑆
1−𝑡
𝑤𝐷𝑆
𝑤𝐷𝑆
𝑆=1
1
̅̅̅̅̅̅̅̅
̅+𝑊
̅ ] [𝑚cov(𝜆𝑆 , 1⁄𝑤 ) + 1
[𝑚𝐾
⁄𝑤𝐷𝑆 ]
𝐷𝑆
𝒕=𝟏−
=1−𝑡
𝟏
̅̅̅̅̅̅̅̅̅
̅ +𝑾
̅̅̅] [𝒎𝐜𝐨𝐯(𝝀𝑺 , 𝟏⁄𝒘 ) + 𝟏
[𝒎𝑲
⁄𝒘𝑫𝑺 ]
𝑫𝑺
(6.9B)
Equation (6.9B) defines the median voter equilibrium income tax rate in the presence of wage
uncertainty under different states of the world. If there is no uncertainty, there is only one state and we
̅ = 0, cov(𝜆𝑆 , 1⁄𝑤 ) = 0. This simply reduces to the original equation (6.9A) in
have 𝑚 = 1 and 𝐾
𝐷𝑆
the simple model with no uncertainty, 𝑡 = 1 −
𝑤𝐷
.
̅
𝑤
In order to sign 6.9B we work as follows. We would expect that given an individual’s characteristics
such as education, age, health and the like, higher wages are associated with a lower probability state.
This is shown graphically in Figure 7. The vertical axis depicts the wage earned by individuals, and the
horizontal axis shows the probability of earning those wages. The downward sloping relation represents
the negative correlation we would expect to observe, that is, that higher wages carry a lower probability
of occurrence. 8 The graph shows the relations for two individuals 𝑎 and 𝑏, where 𝑎 has greater earning
abilities than 𝑏 because of, say, higher education.
8
It bears noting that with this discrete approach to observing wage vs. probability of wage, we do not need to
specify a formal probability density function over which an individual’s wage is distributed, but only need to
invoke a general assumption that the individual is less likely to be offered high wages.
28
Undergraduate Dissertation
Vasilis William Tridimas
𝑤𝑖𝑆
Income Uncertainty and the
Relative Size of Government
Figure 7
𝑤𝑎𝑆 1
Individual 𝑎
𝑤𝑏𝑆1
Individual 𝑏
𝜆𝑆 1
𝜆𝑆
Specifically, it is shown that with the same probability 𝜆𝑆 1 , individual 𝑎 gets a higher wage 𝑤𝑎𝑆 1 than
individual 𝑏 who earns 𝑤𝑏𝑆 1 . It is also important to note that these two wages do not necessarily
correspond to the same state of the world, only that the two states of the world occur with the same
probability. It could indeed be case that in a given state of the world, individual 𝑏 is offered a higher
wage than individual 𝑎.
̅ is negative,
Given the assumption that 𝜆𝑆 and 𝑤𝑖𝑆 are negatively correlated, the mean covariance 𝐾
while the decisive voter’s covariance between 𝜆𝑆 and the inverse of the wage 1⁄𝑤𝐷𝑆 , cov(𝜆𝑆 , 1⁄𝑤𝐷𝑆 ),
is positive. Henceforth, for equation (6.9B) to yield a positive tax rate which is also smaller than 100%,
̅̅̅̅̅̅̅̅
̅+𝑊
̅ ] [𝑚cov(𝜆𝑆 , 1⁄𝑤 ) + 1
it must be the case that [𝑚𝐾
⁄𝑤𝐷𝑆 ] > 1. When this condition is satisfied,
𝐷𝑆
̅ and cov(𝜆𝑆 , 1⁄𝑤 ) being negative and positive respectively
and our assumptions about the signs of 𝐾
𝐷𝑆
̅ > 𝑚𝐾
̅ , otherwise the tax rate would not make economic sense.
are met, it must be the case that 𝑊
Examining the comparative static properties of (6.9B), we have:
𝝏𝒕
=
̅̅̅ [𝒎𝑲
𝝏𝑾
̅
𝟏
̅̅̅̅̅̅̅̅̅
̅̅̅]𝟐 [𝒎𝐜𝐨𝐯(𝝀𝑺 , 𝟏⁄𝒘 ) + 𝟏
+𝑾
⁄𝒘𝑫𝑺 ]
𝑫𝑺
>0
𝝏𝒕
𝟏
=
>0
̅̅̅̅̅̅̅̅̅
̅̅̅̅̅̅̅̅̅ 𝟐 [𝒎𝑲
𝝏 (𝟏⁄𝒘𝑫𝑺 ) [𝒎𝐜𝐨𝐯(𝝀𝑺 , 𝟏⁄𝒘 ) + 𝟏
̅
̅
̅̅
+ 𝑾]
⁄𝒘𝑫𝑺 ]
𝑫𝑺
(6.10B)
(6.11B)
Equation (6.10B) shows that when the average wage in the economy across all states and individuals
increases, the politically optimal tax rate increases. Analytically, this is because the decisive voter takes
advantage of the higher income of everybody, by raising the transfer benefit he receives. On the other
hand, equation (6.11B) reveals that as the decisive voter’s average inverse wage increases, and
equivalently his average wage across all states decreases, he raises the tax rate levied on everybody to
compensate for his loss of earning power. These results reflect the earlier findings of the model without
uncertainty.
29
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
̅ , relating to the uncertainty of all
Uncertainty in the economy is captured by average covariance 𝐾
individuals, and cov(𝜆𝑆 , 1⁄𝑤𝐷𝑆 ), relating only to the uncertainty of the decisive individual. Partially
differentiating (6.9B) with respect to these two parameters, we obtain:
𝝏𝒕
𝒎
=
>0
̅ [𝒎𝑲
𝝏𝑲
𝟏⁄
̅ +𝑾
̅̅̅]𝟐 [𝒎𝐜𝐨𝐯(𝝀𝑺 , 𝟏⁄𝒘 ) + ̅̅̅̅̅̅̅̅̅
]
𝒘
𝑫𝑺
𝑫𝑺
(6.12B)
̅ raises the political equilibrium tax rate. As discussed, 𝐾
̅ is negative, so increasing its value
Increasing 𝐾
means reducing its magnitude, implying a reduction in average uncertainty. The upshot of this is that
as the economy becomes less uncertain, the decisive voter is better off with a higher tax rate. The reason
for this is that a more certain economy allows him to redistribute more income towards himself.
𝝏𝒕
=
𝝏𝐜𝐨𝐯(𝝀𝑺 , 𝟏⁄𝒘𝑫𝑺 ) [𝒎𝐜𝐨𝐯(𝝀 , 𝟏⁄
𝑺
𝒘
𝒎
̅̅̅̅̅̅̅̅̅ 𝟐 [𝒎𝑲
𝟏
̅ +𝑲
̅]
+
)
⁄𝒘𝑫𝑺 ]
𝑫𝑺
>0
(6.13B)
An increase in cov(𝜆𝑆 , 1⁄𝑤𝐷𝑆 ) translates into an increase in the magnitude of cov(𝜆𝑆 , 𝑤𝐷𝑆 ), and
therefore higher wage uncertainty for the decisive individual. As a result, he raises the equilibrium tax
rate to compensate himself by redistributing others’ income towards himself.
Given that the relation between probability 𝜆𝑆 and wage 𝑤i𝑆 depends on the underlying structure of the
economy, the covariance between these two variables can be decomposed in order to better illustrate
where wage uncertainty originates from, by using the correlation coefficient formula corr(𝑋, 𝑌) =
cov(𝑋,𝑌)
. This means that for every individual 𝑖:
√var(𝑋)√var(𝑌)
𝐾𝑖 ≡ cov(𝑤i𝑆 , 𝜆𝑆 ) = corr(𝑤i𝑆 , 𝜆𝑆 ) × √var(𝑤i𝑆 ) × √var(𝜆𝑆 )
The variances of wages and of states of the world capture the true essence of economic uncertainty.
Holding other factors constant, we see that raising var(𝑤i𝑆 ) across individuals, and raising var(𝜆𝑆 ) both
̅ . The same is also true for
drive up the magnitude of 𝐾𝑖 and hence raise the magnitude of 𝐾
cov(𝜆𝑆 , 1⁄𝑤𝐷𝑆 ) where:
cov(𝜆𝑆 , 1⁄𝑤𝐷𝑆 ) = corr(𝜆𝑆 , 1⁄𝑤𝐷𝑆 ) × √var(1⁄𝑤𝐷𝑆 ) × √var(𝜆𝑆 ) 9
Equations (6.10B) to (6.13B) show that the skewness of the wage distribution, that is the relative sizes
of the decisive voter’s wages and the mean wages, are pivotal in determining the effect of an income
shock or an uncertainty shock on equilibrium tax rate. For example, two economies with different wage
distributions, and/or distributions of covariances of individual wage and probability of a given state
materializing, subject to a shock of the same magnitude, will end up with different relative sizes of their
governments. It follows that it would be interesting to extend the existing empirical research on the
relative size of the government by explicitly considering wage uncertainty as an additional determinant.
A noteworthy aspect of the present model is that by averaging income across all individuals and across
all states of the world, the government balances its budget on average, but at any given point in time, it
It should be borne in mind that raising var(λS ) has an ambiguous effect on the equilibrium tax rate. Intuitively,
an increase in the above variance raises uncertainty on average, resulting in a downward effect on 𝑡, while
simultaneously it raises uncertainty for the decisive voter, resulting in an upward effect on 𝑡.
9
30
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
may run a deficit or a surplus. This is a significant departure from the previous models examined where
at all times tax revenue equals benefit expenditure. This departure offers promising avenues for future
research which would explore the costs incurred by the government from interest paid on debt
accumulated while in deficit, and conversely the interest earned on budget surpluses.
31
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
7) Concluding Remarks
The present work utilises the standard median voter model of the determination of the size of
government, and includes the effects of income uncertainty in the sense that there is a probability that
voters are unemployed, and in addition that their incomes vary when employed. Although cases of
uncertainty have been explored before in the literature, the combination of these two factors is novel.
According to the median voter theorem, vote maximizing political parties offer the policy platform that
maximizes the utility of the median voter. With a positively skewed distribution of income, the median
income is smaller than the mean income. The median voter uses his decisive position to redistribute
away from the richer mean towards himself. In addition, recognising that income is subject to
uncertainty, the median voter redistributes the income risk to insure against income fluctuations.
This thesis has explicitly modelled the government’s behaviour under the median voter theorem. An
initial set up was assumed where voters gain utility from consumption and disutility from income
uncertainty, while facing uncertainty in whether they have a job or not, and in how much they will earn
if they find a job. In section 3, abstracting from disincentive effects of taxation on labour supply, we
have derived that greater uncertainty in finding a job and in income drive up tax rates. Changes in mean
and median voter income produce ambiguous changes in the politically optimal tax rate, as a ceteris
paribus increase in mean income raises availability of funds for redistribution, creating a negative effect
on taxation to balance income across employed and unemployed state, while also creating a positive
effect on taxation as it renders income in the unemployed state ‘cheaper’. The opposite effects apply to
changing median income. Section 4 removes income uncertainty but introduces heterogeneity in
individuals’ probability of finding a job, adding a dimension through which the median voter takes
advantage of others’ improved employment prospects through greater taxation. In order to gain further
traction, section 5 assumes utility is quadratic over income, and then derives indirect utility over income
uncertainty as measured by the variance of income. Solving for the political equilibrium tax rate gives
the same qualitative results as the general form, that is, an increase in income uncertainty and reduction
in chance of finding a job raise the tax rate. Once again, changes in mean and median income have
ambiguous effects.
Allowing for disincentive effects through a labour supply set up reduces the scope through which an
intuitive model can immediately identify median preferences over the tax rate. In section 6A, using
quasilinear preferences for algebraic tractability allows us to derive an explicit solution for the
equilibrium tax rate, closely mirroring the Meltzer and Richards (1981) original work in the area. The
tax rate rises with the (positive) skewness of the wage distribution. Extending this framework in section
6B, we introduce uncertainty in the form of a set of states of the world over which wages differ, but the
government tax rate and expenditure on benefits remains constant. This enables us to explore the effects
of uncertainty parameters on the size of government. Increasing uncertainty and reducing expected
wage for the median voter increase the tax rate, an intuitive result. Increasing uncertainty and reducing
expected wage for the mean of the economy reduces the tax rate, as taxation becomes costlier and so
there exists less scope for redistribution. A key conclusion from this analysis is that the effects of
uncertainty depend critically on the distribution of wages across the economy.
32
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
Appendices
Appendix 1
Table 1: International Trends in Total Public Expenditure as Percentage of GDP
Austria
Belgium
Britain
Canada
France
Germany
Italy
Japan
Netherlands
Spain
Sweden
Switzerland
United States
Average
1870
10.5
na
9.4
na
12.6
10.0
13.7
8.8
9.1
na
5.7
16.5
7.3
10.4
1913
17.0
13.8
12.7
na
17.0
14.8
17.1
8.3
9.0
11.0
10.4
14.0
7.5
12.7
1920
14.7
22.1
26.2
16.7
27.6
25.0
30.1
14.8
13.5
8.3
10.9
17.0
12.1
18.4
1937
20.6
21.8
30.0
25.0
29.0
34.1
31.1
25.4
19.0
13.2
16.5
24.1
19.7
23.8
1960
35.7
30.3
32.2
28.6
34.6
32.4
30.1
17.5
33.7
18.8
31.0
17.2
27.0
28.4
1980
48.1
58.6
43.0
38.8
46.1
47.9
42.1
32.0
55.8
32.2
60.1
32.8
31.4
43.8
1990
38.6
54.8
39.9
46.0
49.8
45.1
53.4
31.3
54.1
42.0
59.1
33.5
33.3
44.7
2000
52.1
49.1
36.6
40.6
51.6
45.1
46.2
37.3
44.2
39.1
52.7
33.7
32.8
43.2
2005
50.2
52.0
40.6
39.2
53.4
46.8
48.2
34.2
44.8
38.4
51.8
37.3
36.1
44.1
2009
52.3
54.0
47.2
43.8
56.0
47.6
51.9
39.7
50.0
45.8
52.7
36.7
42.2
47.7
Source: Tanzi and Schuknecht (2000); Public Spending in the 20 th Century; IMF; OECD
Appendix 2
Using the same general equilibrium set up as in section 6.A we can briefly compare the median voter
solution of government size to the solutions given by the other two theories of government mentioned
in the literature review; the government acting as (a) as a benevolent social planner aiming to maximize
social welfare, and (b) as a Leviathan aiming to maximize its own size.
With the same assumptions about the individual having quasilinear preferences over consumption and
leisure time, and the government having to balance its tax revenue with its benefit expenditure,
regardless of government policy determination we arrive at an individual 𝑖’s indirect utility function as
a expressed in terms of the tax rate as equation (6.8A) from section 6A:
𝑉𝑖 = 𝛼 log[𝛼(1 − 𝑡)𝑤𝑖 ] + 𝑇 − 𝛼 +
𝛼𝑡𝑤
̅
𝑤𝑖
(a) Social Welfare Optimization
The government’s objective function becomes:
𝒏
Maximize:
𝑺 = ∑ 𝒃𝒊 𝑽 𝒊
(𝐈)
𝒊=𝟏
Where 𝑏𝑖 = the weight on individual 𝑖’s utility by the government. As a weighting parameter the
absolute size of 𝑏𝑖 does not matter, only the sizes of 𝑏𝑖 relative to one another. For convenience we will
take ∑𝑛𝑖=1 𝑏𝑖 = 1. Substituting in indirect utility:
𝒏
𝑺 = ∑ 𝒃𝒊 [𝜶 𝐥𝐨𝐠[𝜶(𝟏 − 𝒕)𝒘𝒊 ] + 𝑻 − 𝜶 +
𝒊=𝟏
̅
𝜶𝒕𝒘
]
𝒘𝒊
(𝐈𝐈)
33
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
Maximization of the above yields:
𝑛
𝑛
𝑖=1
𝑖=1
𝜕𝑆
−𝛼
𝑏𝑖
=
∑ 𝑏𝑖 + 𝛼𝑤
̅∑ =0
𝜕𝑡 1 − 𝑡
𝑤𝑖
1
Defining 𝑤 as 𝑝𝑖 , as a measure of poverty:
𝑖
𝑛
𝑛
𝑖=1
𝑖=1
1
∑ 𝑏𝑖 = 𝑤
̅ ∑ 𝑏𝑖 𝑝𝑖 = 0
1−𝑡
𝑛
1
∑ 𝑏𝑖 = 𝑤
̅(𝑛cov(𝑏, 𝑝) + 𝑛𝑏̅𝑝̅ ) = 0
1−𝑡
𝑖=1
With the government pursuing a redistributive policy, cov(𝑏, 𝑝) is positive, meaning that the higher the
wage of an individual (and hence the lower their 𝑝𝑖 ), the lower the weight placed on their utility in the
social welfare function, and vice versa.
𝑛
Using ∑ 𝑏𝑖 = 𝑛𝑏̅ = 1,
𝑖=1
1
=𝑤
̅(𝑛cov(𝑏, 𝑝) + 𝑝̅ ) = 0
1−𝑡
𝒕𝑺 = 𝟏 −
𝟏
̅)
̅ (𝒏𝐜𝐨𝐯(𝒃, 𝒑) + 𝒑
𝒘
(𝐈𝐈𝐈)
𝝏𝒕𝑺
𝟏
= 𝟐
>0
̅)
̅ 𝒘
̅ (𝒏𝐜𝐨𝐯(𝒃, 𝒑) + 𝒑
𝝏𝒘
(𝐈𝐕)
𝝏𝒕𝑺
𝟏
=
>0
̅ (𝒏𝐜𝐨𝐯(𝒃, 𝒑) + 𝒑
̅ )𝟐 𝒘
̅
𝝏𝒑
(𝐕)
𝝏𝒕𝑺
𝒏
=
>0
̅ )𝟐
̅ (𝒏𝐜𝐨𝐯(𝒃, 𝒑) + 𝒑
𝝏𝐜𝐨𝐯(𝒃, 𝒑) 𝒘
(𝐈𝐕)
While it would seem contradictory that raising average wage raises the tax rate, and raising poverty also
raises the tax rate, these two factors are both negatively related as 𝑛𝑤
̅ is the sum of all 𝑤𝑖 , and 𝑛𝑝̅ is the
sum of the inverses of all 𝑤𝑖 . The effect on taxation of changing mean income depends on the
distribution of income, represented by 𝑤
̅𝑝̅ 10. The higher the covariance between social weight, and the
poverty of an individual, the higher the tax rate, ie a government which places greater emphasis on the
utility of the poor will levy a higher income tax rate to redistribute more income to these individuals.
In the event of a Benthamite utility function where all individuals have equal weighting, and
cov(𝑏, 𝑝) = 0, redistribution depends purely on the distribution of income captured by 𝑤
̅𝑝̅ .
Generally speaking, the more positively skewed the distribution of wages, the higher the value of 𝑤
̅𝑝̅ relative
to an even distribution of wages, and so greater inequality necessitates higher tax rates.
10
34
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
(b) Leviathan
The government’s objective function becomes the maximization of state expenditure, 𝐺.
̅ 𝒕(𝟏 − 𝒕)
𝑮 = 𝜶𝒘
(𝐕𝐈)
𝜕𝐺
= 𝛼𝑤
̅(1 − 2𝑡) = 0
𝜕𝑡
𝒕𝑳 =
𝟏
𝟐
(𝐕𝐈𝐈)
This result comes from the quasilinear nature of the individual’s utility function, and gives a maximum
tax revenue found as the point where 50% of income is taxed. The government takes no interest
whatsoever in the preferences of the electorate.
35
Undergraduate Dissertation
Vasilis William Tridimas
Income Uncertainty and the
Relative Size of Government
References
Aidt, T., Jensen, P. (2013). Democratization and the size of government: evidence from the long 19th
century. Public Choice 157, 511-542.
Baumol, W.J. (1967). The macroeconomics of unbalanced growth: The anatomy of urban crisis,
American Economic Review, 57, 415-426.
Brennan, G., Buchanan, J. (1980). The Power to Tax: Analytical Foundations of a Fiscal Constitution.
Cambridge University Press, Cambridge.
Black, D. (1948). The theory of committees and elections, Cambridge University Press.
Bowen, H.R. (1943). The interpretation of voting in the allocation of economic resources, Quarterly
Journal of Economics, 58, 27-48.
Congleton, R. Bose, F. (2010). The Rise of the Modern Welfare State, Ideology, Institutions and Income
Security: Analysis and Evidence. Public Choice 144.
Dixit, A.K., Sandmo, A. (1977). Some Simplified Formulae for Optimal Income Taxation.
Scandinavian Journal of Economics, Wiley Blackwell, vol. 79(4), pages 417-23.
Downs, A. (1957). An Economic theory of democracy. Harper and Row.
Hillman, A.L. (2009). Public Finance and Public Policy: Responsibilities and Limitations of
Government (2nd ed), Cambridge UP, Cambridge
Kau, J.B., Rubin, P.H. (2002). The growth of government: sources and limits. Public Choice 113, 389402.
Kuznets, S. (1955). Economic Growth and Income Inequality. A.E.R. 45.
Lee, C. (2012). The Growth of Public Expenditure in the United Kingdom from 1870 to 2005. Palgrave
Macmillian.
Meltzer, A.H., Richards, S.F. (1981). A rational theory of the size of government. Journal of Political
Economy 89, 914-927.
Meltzer, A.H., Richards, S.F. (1983). Tests of a rational theory of the size of government. Public Choice
41, 403-418.
Milanovic, B. (2000). The median-voter hypothesis, income inequality, and income redistribution: an
empirical test with the required data. European Journal of Political Economy 16, 367-410.
Mirrlees, J.A., 1971, An exploration in the theory of optimum income taxation, Review of Economic
Studies, 38, 175-208.
Mueller, D.C., 2003. Public Choice III. Cambridge: Cambridge University Press.
Peacock, A.T. and Wiseman, J. (1961). The growth of public expenditure in the United Kingdom,
George Allen and Unwin, London.
Rodrik, D. (1998). Why Do More Open Economics Have Bigger Governments? Journal of Political
Economy 106.
Tanzi, V. and Schuknecht, L. (2000). Public Spending in the 20th Century: A Global Perspective.
Cambridge University Press.
36