The Public Provision of Private Goods
with Status Concerns
Tobias König, Tobias Lausen,∗ and Andreas Wagener
School of Economics and Management
University of Hannover
Koenigsworther Platz 1
30167 Hannover, Germany
Phone/Fax: +49 - 511 - 762 8214/4574
e-mail: [email protected],
[email protected], [email protected]
Abstract: Concerns for distinction, prestige and social status influence individual behaviour. We study how such concerns affect the political economy of the public provision
of private goods. Depending on their strength, status concerns give rise to three configurations of a majority voting equilibrium (MVE): first, the MVE can be of an endsagainst-the-middle type, where a coalition of rich and poor households is put against
the middle class. Second, an ‘ends-against-the-ends’ equilibrium may emerge where the
very poor form a coalition with upper middle class while the lower middle class votes in
coalition with the rich. Third, the most-preferred policy of the median-income individual
may also be a MVE. Status concerns affect the provision level of private goods: goods
may be provided which are not consumed by a majority of people, and societies where
status concerns prevail may have larger or lower levels of public provision than status-free
societies.
JEL-classification: D72, H42, D63.
Keywords:
Public Provision, Status, Majority Voting.
This version:
February 2012.
∗
Corresponding author.
1
Introduction
To a substantial degree, governments provide private goods (i.e., goods that rival in consumption) to their citizens. According to some estimates, the volume of public spending
on private goods like schooling, health care, public transportation, housing, daycare etc.
ranges from 20 to 30 percent of GDP in OECD countries (see Blomquist et al., 2010).
Typically, government-provided goods are made available to citizens free of charge or at
subsidized prices. However, they can only be consumed at a uniform level (of quantity
or quality) for everybody. As an alternative to consuming what is publicly provided,
individuals often can opt out of the public system and purchase similar goods or services
on private markets.1 In the latter case, they still have to pay the taxes that go to finance public provision; they do not benefit from public provision but rather subsidize the
consumption of those staying within the public system.
A substantial body of literature has been devoted to the question of how public provision
of private goods can be explained in terms of democratic majority voting (e.g., Epple and
Romano, 1996; Luelfesmann and Myers, 2011). A widely unchallenged assumption in this
literature is that the decision whether to consume on the public or private sector is solely
driven by quality concerns: individuals opt out of when they prefer another, often better
quality than the one provided by the public sector. For example, parents who consider
the quality of education provided by state-run schools or kindergartens too low may send
their children to private schools or childcare. Likewise, public transport may be regarded
as too slow or uncomfortable which makes people commute by car. However, there is
a still growing literature arguing that, in addition to quality aspects, consumer choices
are driven by desires for distinction, prestige and social status (see, e.g., Veblen, 1899;
Frank, 1985). These motives may affect the choice between the publicly provided option
and their private alternatives as well. E.g, taking one’s children out of public schools
and sending them to private institutions may satisfy needs of elitism and exclusiveness or
conspicuously signal higher status, greater wealth or refined tastes (see, e.g., Postlewaite,
1998; Foskett and Hemsley-Brown, 2002; Meyer and Smith, 1995). Likewise, commuting
in private cars conveys greater prestige and delivers larger ego-rents than using public
trains or busses (Steg, 2005). Consulting doctors in private practices and being treated
in private hospitals bestows superior prestige and higher status on patients than the
state-financed public alternatives (Simanis, 1970).
1
In several instances, citizens can also top up the publicly provided amount by additional purchases,
financed out of their pockets. We do not discuss this case here.
1
In this paper, we firstly study the political economy of the public provision of private goods
when – in addition to quality aspects – concerns for social status matter. Extending the
classical framework of Epple and Romano (1996), we consider a model where the level
of a publicly provided good, which is financed by an income tax, is determined by direct
majority voting. Individuals, who only differ in their exogenuous incomes, can opt out of
public provision and buy their desired level of the good on private markets. We introduce
status concerns by assuming that opting out of the public system and buying on the
private sector gives utility beyond the satisfaction derived from the private consumption
of the good. This additional utility, say status utility, is endogenous: it increases with
the number of individuals staying in the public system (status diminishes if opting out
becomes too common.) We analyze how allowing for endogenous status concerns change
the structure of a majority voting equilibrium (MVE) and the equilibrium level of the
publicly provided good compared to the non-status case.
We show that the existence and structure of a majority voting equilibrium (MVE) depend
on the distribution of voter types, specifically on the relationship between individuals’
marginal willingness to pay for the publicly provided good and their income. Income
determines whether one stays in or opts out of the public system, with the rich typically
opting out. For those staying in the public system, income typically affects the willingness
to pay for an expansion of the public system. In addition and different from models
without status concerns, those who opt out have a positive willingness to pay for the
publicly provided good for reasons of status and prestige. The reason is that an expansion
will lure some individuals (back) into the public system again, and the redistributive
subsidization of those in the public system by those who opt out keeps the public system
popular, thereby raising the prestige and social distinction of the declining group of those
who leave it. Although status concerns themselves are not directly income-dependent,
in the equilibrium of the economy the willingness to pay for public provision by private
consumers varies with their incomes. All this gives rise to interesting MVE patterns. Two
pure cases can be considered.
First, suppose that the marginal willingness to pay for an increase in public provision rises
with individual incomes, both for those who stay in the public system and for those who
opt out. For government-provided goods like education this seems to be an appropriate
assumption for those who actually consume the good (see Epple and Romano, 1996). An
increasing willingness to pay for status-concerned people who opt out means, roughly,
that the very rich are more ready to sacrifice private consumption for gaining in prestige
than the not-so-rich. Without status concerns, a MVE when marginal willingnesses to
2
pay for public provision increase in income for those who consume the publicly-provided
good necessarily has an end-against-the-middle structure (Epple and Romano, 1996):
the decisive voter is a below-median income voter, who is in a coalition with rich nonconsumers and poorer households, put against the middle class. The public provision
level is lower than what is preferred by the median income earner. With status concerns,
the palette of possible equilibrium configurations is considerably richer, ranging from
ends-against-the-middle type equilibria (if status concerns are relatively unimportant)
to single-crossing type equilibria with the median income earner being pivotal (if status
concerns are sufficiently strong at the margin). In between, an interesting and novel ‘endsagainst-the-ends’ configuration may arise as an equilibrium: the rich side with the lower
middle-class in preferring high levels of public good provision (the former for reasons of
prestige, the latter because they value the publicly provided good highly), whereas the
upper middle class is in coalition with the very poor who both prefer a low level of public
provision.
We show that these types of political equilibria may also occur in our second scenario
where we assume that the marginal willingness to pay for the publicly provided good
is decreasing for individuals both staying in the public system and those who opt out.
Economically, this means that richer consumers of the publicly provided good are more reluctant to finance its expansion than poorer consumers and that status concerns of people
outside the system are marginally weak and decreasing with income. Using this scenario
we exemplify the consequences of social concerns for the level of public provision, i.e. for
the size of the public sector. Interestingly, a MVE may entail positive public provision
even if the median income earner opts out: tax-financing increased public provision buys
him valuable prestige by driving a larger crowd into the public system. Without status
concerns a decisive voter who stayed out of the public system would never be willing to
finance it. Thus, status concerns might explain public provision of private goods even in
cases where the middle classes consume privately (e.g., for public transport). For situations where the median income earner consumes in the public system, intuition suggests
that status concerns call for higher provision levels: status-seeking pushes more individuals to opt out of public provision such that, for a given tax rate, it is now easier to endow
the rest of the population with a higher level of the government-provided good. Yet, we
show that status concerns may also go along with the median individual choosing a lower
public provision level. The intution is that with stronger status concerns the individual
indifferent between public and private consumption is necessarily poorer, thus affecting
his/her sensitivity of jumping into the public system when the publicly provided level is
3
marginally increased. When the distribution function is locally concave, this sensitivity
gets higher, thereby increasing the marginal costs of public funds. If this counter effect is
sufficiently strong, status-driven societies will end up with less public spending.
The rest of this paper is organized as follows: Section 2 presents the model and provides
a necessary condition for a majority voting equilibrium. Sections 3 and 4 deal with the
cases that, respectively, the willingness to pay for public provision increases or decreases
in income. Section 5 provides comparative statics. Section 6 concludes.
2
The model
2.1
Framework
We consider an economy populated by a continuum of individuals with measure one. Individuals only differ in their exogenous incomes y. Income in the population is distributed
according to a twice continuously differentiable cumulative distribution function F (·) with
support on Y ⊆ [0, ∞).
There are two private goods, called x and c. Good c, which is the numéraire, is a standard
consumption good traded on a private market. The government provides good x in equal
per-capita levels x̄ to all individuals at a zero price. As an alternative to consuming x̄,
individuals can opt out of public provision and instead buy their desired level on a private
market at unit price px . For notational simplicity, we normalize units of good x such that
its market price equals one. Both alternatives are mutually exclusive, i.e. individuals
cannot choose the publicly provided quantity x̄ and supplement it via private market
purchases. To finance public provision a proportional income tax with rate t is levied.
Individuals have to pay taxes irrespective of whether they consume x̄ or opt out.
The production technology of good x is linear: one unit of the numéraire can be transformed into one unit of x; this technology is identical for the public and the private
sector.
Individuals have identical preferences over x and c, represented by a smooth, strictly
increasing and strictly quasi-concave utility function u(x, c). In addition, individuals care
about distinction and social status. In the context of a dual provision scheme as described
above, one way to achieve distinction from the masses and to enhance ones’s social status
is to opt out of public provision. We account for such motives in the following way: when
opting out, an individual receives an extra utility gain S which is beyond the benefit
derived from the pure consumption of x and c. This extra utility is endogenous and
4
depends on the expected number of individuals N e who stay in the public system: the
more individuals are expected to stick with public provision, the higher are the status
gains attainable through opting out. Specifically, status concerns are captured by some
function S = S(N e ) with
S(N e ) > 0,
S ′ (N e ) ≥ 0 for all N e ∈ (0, 1).
(1)
We will specify the formation of expectations later. Total individual utility is
u(x, c) + 1market · β · S(N e ),
(2)
where 1 is the indicator function that takes values 1 if the individual opts out and zero
otherwise. The parameter β ≥ 0 is used to measure the intensity of status concerns. We
require that status is not too important, relative to consuming good x: for all x > 0 and
N e ∈ [0, 1], β · S(N e ) is such that
β · S(N e ) < u(x, 0) − u(0, 0).
The model proceeds in two stages with the following chronology of events: In the first
stage, a pair of expenditure level x̄ and tax rate t is determined by simple majority voting
(political equilibrium). In the second stage, each individual decides whether to opt out of
public provision or not, taking as given the tax rate t and the expenditure level x̄. Given
this decision, every individual spends his exogenous income in order to maximize utility
(individual consumption choices). We solve the model by backward induction.
2.2
Stage 2: Individual consumption choices
An individual with income y who opts out of public provision chooses a bundle (x, c) that
maximizes u(x, c) subject to the budget constraint c + x = y(1 − t). Let x∗ = x∗ (y(1 − t))
and c∗ = c∗ (y(1 − t)) = y(1 − t) − x∗ (·) be the resulting Marshallian demand functions
for goods x and c, respectively. By separability of (2), x∗ (·) and c∗ (·) are independent of
N e and β. Define indirect utility from consumption as
v(y(1 − t)) := u (x∗ , y(1 − t) − x∗ ) .
5
In combination with social status, total indirect utility amounts to
v(y(1 − t)) + β · S(N e ).
(3)
If the individual consumes the publicly provided amount x̄, she spends her entire net
income y(1 − t) on good c. In this case, utility is given by
u(x̄, y(1 − t)).
(4)
Preferences are assumed to exhibit decreasing willingnesses to pay for consumption goods:
Assumption (DMU): For all x̄, u(x̄, y(1 − t)) − v(y(1 − t)) is decreasing in y:
uc (x̄, y(1 − t)) − v ′ (y(1 − t)) < 0.
In (DMU), observe that v ′ (y(1 − t)) = uc (x∗ (·), y(1 − t) − x∗ (·)) by the envelope theorem.
As a consequence of (DMU), the marginal willingness to pay for good c decreases in the
amount of c.
Each individual chooses between opting out of public provision or staying in the public
system. Given a tax rate t, public provision level x̄ and an expected number of individuals
N e who stay in, an individual with income y opts out if and only if
v(y(1 − t)) + β · S(N e ) > u(x̄, y(1 − t)).
(5)
The following lemma establishes that for any given policy (t, x̄) and N e , there exists a
unique income level ŷ such that an individual with income ŷ is just indifferent between
staying in the public system and opting out. Moreover, all individuals with incomes higher
than ŷ opt out while those with smaller income consume the publicly provided amount.
Lemma 1 Assumme that (DMU) holds. Then, given x̄, β ≥ 0, t ∈ (0, 1) and N e ∈ (0, 1),
there exists a unique ŷ such that
v(y(1 − t)) + β · S(N e ) ≤ u(x̄, y(1 − t))
if and only if y ≤ ŷ.
6
Proof: If a critical income level ŷ exists, it satisfies
u(x̄, y(1 − t)) = v(y(1 − t)) + β · S(N e )
(6)
Since status is not too important, the individual with zero income strictly prefers the
public alternative, i.e.
u(x̄, 0) > v(0) + β · S(N e )
This inequality is reversed for y sufficiently large (by the strict quasi-concavity of u(·)).
Thus, by continuity, there exists at least one ŷ satisfying (6).
Since the difference
u(x̄, y(1 − t)) − v(y(1 − t)) − S(N ) decreases in y by (DMU), the critical income level is
unique.
•
The properties of the critical income level ŷ = ŷ(t, x̄, N e ; β) are given by:
∂ ŷ
∂t
∂ ŷ
∂ x̄
∂ ŷ
∂N e
∂ ŷ
∂β
ŷ
>0
(1 − t)
ūx
= −
>0
(1 − t)(ūc − u∗c )
β · S′
<0
=
(1 − t)(ūc − u∗c )
S
< 0,
=
(1 − t)(ūc − u∗c )
=
(7)
(8)
(9)
(10)
where ūx := ux (x̄, y(1 − t)), ūc := uc (x̄, y(1 − t)), respectively. Ceteris paribus, a rise
in the tax rate t makes it more attractive to stay in the public system, since it reduces
an individuals net income. In order to maintain indifference, the critical income level ŷ
increases. An analogous argument applies for an increased public provision level x̄. In
contrast, a higher number of individuals N e expected to stay in the public system or
an increase in the strength of status concerns β raises the attractiveness of opting out
because of higher status utility. Thus, the critical income level ŷ must fall.
2.3
Stage 1: Political equilibrium
In stage 1, a pair (t, x̄) is selected by majority vote. Tax rate and provision levels are, of
course, not independent of one another; rather they are linked by the requirement that
7
the government runs a balanced budget. Government expenditures, in turn, depend on
how many people actually consume the government-provided good. Given the existence
of a critical income level, their number N is given by:
N = F (ŷ(t, x̄, N e ; β)).
(11)
When deciding whether to choose public provision or to opt out, individuals do not yet
know the actual number of individuals who remain in the public system. They only form
expectations N e . We assume that these expectations are rational in the sense that actual
and expected number of individuals opting for public provision coincide:
Assumption (RE): Individuals form rational expectations, i.e. N e = N .
Under (RE), condition (11) reads as:
N − F (ŷ(t, x̄, N ; β)) = 0.
(12)
The next lemma shows that a fixed point of (12) always exists and is unique:
Lemma 2 Under (DMU), given β ≥ 0, (12) possesses, for every (t, x̄), a unique solution
N = N (t, x̄; β).
Proof: Observe that, for any y ∈ Y, N − F (ŷ) is negative for N = 0 and positive for
N = 1. By continuity, there exists at least one zero. Since dF/dN = F ′ · ∂ ŷ/∂N < 0
under (DMU), it is unique.
•
For later use, note that the comparative statics of N = N (t, x̄; β) with respect to t, x̄ and
β are:
∂N
Nt :=
∂t
=
∂N
∂ x̄
=
Nx̄ :=
∂N
Nβ :=
∂β
=
F ′ (ŷ) ·
∂ ŷ
∂t
F ′ (ŷ) ·
∂ ŷ
∂ x̄
F ′ (ŷ) ·
∂ ŷ
∂β
1 − F ′ (ŷ) ·
1 − F ′ (ŷ) ·
1 − F ′ (ŷ) ·
8
∂ ŷ
∂N
∂ ŷ
∂N
∂ ŷ
∂N
>0
(13)
> 0,
(14)
< 0.
(15)
Now define indirect utilities when opting out and staying in, respectively, as
V out (t, x̄; y, β) := v(y(1 − t)) + β · S(N (t, x̄; β))
V in (t, x̄; y) := u(x̄, y(1 − t)).
Then, the induced utility funtion of an individual with income y is
V (t, x̄; y, β) = max V out (t, x̄; y, β), V in (t, x̄; y) .
(16)
The function V is continuous in (t, x̄; y, β) and continuously differentiable almost everywhere (specifically, except when V out (t, x̄; y, β) = V in (t, x̄; y), which is a set of measure
zero, however).
We restrict the analysis to pairs that balance the government budget:
Definition (Government budget constraint): A pair (t, x̄) is feasible if
t · Y = x̄ · N (t, x̄; β),
where Y =
R
Y
(17)
ydF (y) denotes average income in the economy.
Condition (17) describes the government budget constraint (GBC). By the Implicit Function Theorem, (17) defines the expenditure level x̄ as a function of t, i.e. x̄ = x̄(t).
Consequently, the policy space is one-dimensional.
A majority voting equilibrium (MVE) is defined as follows:
Definition: A pair (t∗ , x̄∗ ) is a majority voting equilibrium (MVE) if
(i) it satisfies the GBC: t · Y = x̄ · N (t, x̄; β),
(ii) it is compatible with (RE): N − F (ŷ(t, x̄, N ; β)) = 0,
(iii) and if at least half of the population prefers (t∗ , x̄∗ ) to any other (t, x̄) that satisfies
(i) and (ii).
Existence and properties of majority voting equilibria depend on the distribution of preferences over the policy space. These can be represented by the marginal willingness to
pay (MWTP) for an increase in x̄ (provided that conditions (GBC) and (RE) hold).
9
For an individual with income y who stays in the public system, indifference curves in
(t, x̄)-space are defined through
V in (t, x̄; y) = const.
(18)
By implicit differentiation of (18) we obtain the MWTP for increased x̄ when choosing
public provision
M W T P in (t, x̄; y) :=
∂V in (t, x̄; y)/∂ x̄
ūx
dt
=−
=
> 0,
in
dx̄
∂V (t, x̄; y)/∂t
y · ūc
(19)
which is unambiguously positive.
For individuals who opt out of public provision, indifference curves in (t, x̄)-space,
V out (t, x̄; y, β) = const.,
(20)
give rise to a MWTP for an increase in x̄ of
M W T P out (t, x̄; y, β) :=
∂V out (t, x̄; y, β)/∂ x̄
β · S ′ · Nx̄
dt
=−
=
.
dx̄
∂V out (t, x̄; y, β)/∂t
y · u∗c − β · S ′ · Nt
(21)
When status concerns are absent (S ′ ≡ 0), this marginal MWTP is zero. Since the
individual does not consume the publicly provided amount, a marginally higher x̄ would be
costly to him without offering any benefit. With status concerns, however, the individual
still has a positive MWTP for an expansion of x̄: By (14), extending the public provision
level x̄ drives some individuals back into the public system, which raises the degree of
distinction and exclusiveness of those who keep opting out.2
In a dual provision system where opting out is possible, we cannot directly apply the
Median-Voter-Theorem since preferences over (t, x̄) along the GBC may not be singlepeaked even without status concerns (see, e.g. Stiglitz, 1974; Epple and Romano, 1996;
Luelfesmann and Myers (2010)). However, we can provide a necessary condition for a
2
Note that (21) might be negative, which means that the individual has to be compensated for higher x̄
with a reduced tax rate t. To see this, imagine that x̄ is increased. As a first consequence, more people
will opt into public provision, see (14). Thus, if t were held constant, indirect utility V out (t, x̄; y, β)
would increase because of higher status. To maintain indifference we c ould raise the tax rate t such
that v(y(1 − t)) decreases. But this tax rise drives even more people to choose public provision, i.e.
N and, via social status, indirect utility increases. If this latter effect, which is captured by the term
β · S ′ · Nt , is larger than v ′ (y(1 − t)), indifference curves are (at least locally) downward sloping in
(t, x̄)-space. In what follows, we consider the case where the marginal WTP is positive.
10
policy (t, x̄) to be a MVE which relates the decisive voter to the individual with median
marginal willingness to pay (who is not necessarily the individual with median income):
Proposition 1 If (t∗ , x̄∗ ) is a majority voting equilibrium, there exists an individual such
that:
(i) (t∗ , x̄∗ ) is the individual’s most preferred policy,
(ii) the individual has the median MWTP at (t∗ , x̄∗ ).
Proof: Item (ii) requires that, for any policy (t, x̄) to be an equilibrium, the individual
with preferred policy (t, x̄) must have the median MWTP. To proof (ii), assume the contrary. Then, more than 50% of the population have either a larger or a smaller MWTP at
(t, x̄). Consequently, a majority of individuals would prefer a marginally higher or lower
level of x̄. Thus, the policy (t, x̄) cannot be a MVE. In fact, only at (t∗ , x̄∗ ) a majority
of the population opposes a local deviation in either direction. To proof item (i), assume
that the individual has the median MWTP at a certain policy (t, x̄), but that the latter
is not it’s preferred policy. Then, the individual prefers a policy on the GBC that either
entails a (marginal) larger or a smaller level of x̄. Since it has the median MWTP at
(t, x̄), the individual can find a majority that favors a local deviation from (t, x̄), such
that the latter f ails to be a MVE. Therefore, for (t∗ , x̄∗ ) as defined by item (ii) to be an
equilibrium, it has to be the individual’s most preferred policy.
•
By Proposition 1 the decisive voter (which in the following will be denoted by ys ) is
determined by the distribution of MWTPs over the full income range. This in turn
depends on preferences over x and c, i.e. u(x, c), the income distribution function, and –
in our case – by the specific status function.
Without status concerns, the MWTP is zero for individuals opting out. For those staying
in the public system, a change in one’s income may alter the marginal MWTP through an
income and a substitution effect which go into opposite directions regarding the demand
level for public provision. Epple and Romano (1996) analyze two polar cases: either
that the MWTP increases in income for all individuals staying in, or that it decreases.
However, in our case, the MWTP for those who opt out is positive, and by equation (21),
income-dependent as well. Thus, we also need assumptions about how the MWTPs vary
with income when opting out. Following Epple and Romano (1996), we consider two
scenarios: in the first, the MWTP is increasing for those who stay in as well as for those
11
opting out (section 3), while in the second the MWTP is increasing for both parts of the
population (section 4).
3
Willingness to pay increases in income
For some goods (like education) evidence suggests that for those who actually consume
the good, the MWTP for increased public provision rises with individual income. In
the absence of status concerns, Epple and Romano (1996) show that the only possible
MVE (if it exists) is of an ends-against-the middle type, with a coalition of rich and poor
households put against the middle class. The intuition for this result is as follows. If
no private sector exists, the MWTP for public provision is monotonic in income. Then,
the policy preferred by the median individual is the MVE. When, in contrast, public and
private provision of good x coexist and some individuals indeed opt out, these individuals
prefer a zero tax rate and expenditure level. Consequently, the policy preferred by the
median would loose against a coalition of rich and poor individuals who prefer a marginally
lower tax rate and expenditure level. In a MVE, an individual with income lower than
the median is decisive. At this individual’s preferred policy, 50 percent of the population
favor marginal increases in x̄, while a coalition of rich and poor individuals would prefer
a marginally lower public provision level.
Now suppose that status concerns are present, and that the MWTP for increased x̄ rises
with income for those oping out. This means that the very rich are more eager to spend
money for being in an elitist consumer circle than the not-so-rich. Specifically, assume
that if y ′ > y, then3
M W T P in (t, x̄; y ′ ) > M W T P in (t, x̄; y)
if
M W T P out (t, x̄; y ′ , β) ≥ M W T P out (t, x̄; y, β)
y ′ ≤ ŷ;
if
y ≥ ŷ.
(22)
In the following we illustrate that in the presence of status concerns, the MVE need have
the end-against-the-middle structure described above. Instead, depending on the strentgh
of M W T P out (t, x̄; y, β), two other interesting MVE patterns may emerge. To show this,
we calibrate the model such that we have a single parameter β that can be interpreted in a
meaningful way as the strength of status concerns, and at the same time, determines the
MTWP for individuals outside the public system monotonically (∂M W T P (y)/∂β > 0
3
For M W T P in , Epple and Romano refer to assumption (22) as SRI.
12
for all y > ŷ). Then, which property the political equilibrium has depends solely on the
status parameter.
Specifically, we assume that preference are given by
u(x, c) =
1
(α · x1−γ + (1 − α) · c1−γ ),
1−γ
with α = 0.02 and γ = 1.9; this ensures that (22) holds. Incomes are distributed according
to
F (y) =


0




 2y−2a
y≤a
a<y≤b
3(b−a)


1+





1
y−c
3(c−b)
b<y≤c
otherwise
where a = 0, b = 5, c = 100. The mean income is Y =
115
,
6
and median income is
ym = 3.75.
To find a policy (t∗ , x̄∗ ) that can be a MVE, the necessary conditions of proposition 1
must hold. Moreover, this policy has to satisfy the GBC and rational expectations (RE).
Figure 1: Ends Against The Middle with Status Concerns
MWTP
0.30
Vote for Hx* ,t* L
1
0.25
0.75
0.20
0.15
0.5
0.10
0.25
0.05
0.00
0
20
40
60
80
100
y
0
0
1
2
3
4
5
6
7
x
(b)
(a)
Example 1: First, we set the status parameter β at a moderate value (β = 0.5) such that
the MWTP for all y > ŷ is below that of the decisive voter ys . Panel (a) in Figure 1 depicts
the resulting equilibrium MWTP as a function of y. Income ys and the corresponding
13
MWTP are represented by dashed lines, respectively. Between ys and the indifferent
individual ŷ lay 50% of the population, as in the Epple-Romano case. To check that this
situation is indeed a MVE, we let the policy prefered by ys compete against alternative
policy pairs on the GBC. The share of individuals voting for ys ’ s most prefered policy
alternative is depicted against a dense grid of feasible expenditures levels x̄ in panel (b). As
can be seen, ys ’s most preferred policy always obtains more than 50% of the votes. Thus,
the possibility of a political equilibrium of an end-against-the-ends type is established in
the presence of status concerns.
Example 2: Now we set the status parameter β at a higher value (β = 1). Then, the
MWTP of some individuals opting out is larger than that of ys (see figure 2). This is
the case for incomes higher than that represented by the second dashed vertical line in
panel (a). These individuals form a coalition with individuals earning incomes between
ys (first vertical dashed line) and the indifferent individual ŷ. Since ys has the median
MWTP, this coalition is balanced by a coalition of the lower end of those who stay in and
those who opt out, respectively. We again let this policy pair run against alternative pairs
on the GBC and show that it gains a majority in every case. Thus, we have a political
equilibrium with an, say, ends-against-the-ends property.
Figure 2: Ends Against The Ends with Status Concerns
MWTP
0.30
Vote for Hx* ,t* L
1
0.25
0.75
0.20
0.15
0.5
0.10
0.25
0.05
0.00
0
20
40
60
80
100
y
0
0
1
2
3
4
5
6
7
x
(b)
(a)
Example 3: Finally, there exists a critical value of β above which the MWTP structure
is such that all individuals above ys have a higher MWTP. This is the case for β = 6.5.
Then, the median MWTP coincides with that of the individual with median income (see
panel (a) of Figure 3). Panel (b) demonstrates that this median-income earner program
is not beaten by any other feasible alternative.
14
Figure 3: Median MVE with Status Concerns
MWTP
0.30
Vote for Hx* ,t* L
1
0.25
0.75
0.20
0.15
0.5
0.10
0.25
0.05
0.00
0
20
40
60
80
100
y
0
x
0
1
2
3
4
5
6
7
(b)
(a)
If status concerns are sufficiently strong, it is even possible that the MWTP is monotonically increasing over the full income range. Then, the single crossing condition by Gans
and Smart (1996) is satisfied, and a MVE is guaranteed to exist with the median income
individual decisive.
Thus, under increasing MWTP, we have shown that a MVE must have one of the following
configurations, depending on the strength of status concerns: it is either of an endsagainst-the-middle, an ends-against-the-ends or of an median income type.
4
Willingness to pay decreases in income
Now consider the case where the MWTP is decreasing for individuals both staying in the
public system and those who opt out. Specifically, assume that if y ′ > y, then
M W T P in (t, x̄; y ′ ) < M W T P in (t, x̄; y)
if
M W T P out (t, x̄; y ′ , β) < M W T P out (t, x̄; y, β)
y ′ ≤ ŷ;
if
y ≥ ŷ.
(23)
Economically, this means that richer consumers of the publicly provided good are more
reluctant to finance its expansion than poorer consumers. Moreover, in the group of
individuals opting out, the willingness to bear increases in the tax rate in order to entice
some people back into the public system is lower for the ”‘super-rich”’ than for those
income levels close to critical income level ŷ. Under this assumption, a MVE (if it exists)
must again be of one of the configurations described in section 3. In analogy to the
15
previous section, it can be shown that which property the equilibrium has depends solely
on the strength of status concerns – just the ’order’ reversed. For sufficiently low status
concerns, the Epple-Romano case can be replicated where the MWTPs are decreasing over
the full income range (such that the median income voter is decisive). This monotonicity
breaks for intermediate levels of status concerns. Then, the upper segment of those who
opt out goes together with the upper segment of those who stay in, but now this coalition
votes for marginal tax-expenditures decreases. For sufficiently strong status concerns, the
‘ends-against-the-middle’ equilibrium emerges where the rich and the poor end will vote
for lower levels of public provision, i.e. as they would under increasing MWTP without
status concerns.
5
Effects on public provision level
So far we have shown that status concerns can lead to rich MVE patterns. In addition to
political structure, it is worthwhile to characterize the MVE from an economic perspective.
In this section, we analyze how the introduction of status affects the equilibrium level (or
quality) of the publicly provided good. For convenience, we consider cases where the
MWTP is decreasing over the full range of the income distribution. Formally, we assume
that for any y ′ > y, we have
M W T P in (t, x̄; y ′ ) < M W T P in (t, x̄; y)
if
M W T P out (t, x̄; y ′ , β) < M W T P out (t, x̄; y, β)
y ′ ≤ ŷ,
if
M W T P out (t, x̄; y ′ , β) < M W T P in (t, x̄; y)
y ≥ ŷ,
if
and
(24)
y < ŷ < y ′ .
This assumption has the advantage that preferences are monotone in income such that
we do not need to endogenize the decisive voter: the decisive individual is always the
individual with median income (which simplifies a comparative static analysis considerably).4 Second, and opposed to the opposite case that the MWTPs are globally increasing
(which would also ensure a median income voter equilibrium), we can provide numerical
examples without imposing overly strong motives for social status.5 To analyze the effects of status on public good provision, we distinguish between two different scenarios,
4
We also do not need to care for existence since under (24) ordinal single-crossing is satisfied, and
consequently, a majority voting equilibrium with the median voter being decisive always exists (see
Gans and Smart, 1996).
5
As example 3 indicates, the median income voter equilibrium under increasing MTWP only emerges for
relative high levels of β.
16
depending on whether the decisive voter (here, median income earner) prefers to stay in
the public system or not. We finally analyze the implications of status for public versus
private sector quality.
5.1
Positive provision levels in spite of non-consumption
Consider a case where the decisive voter (=median income earner) opts out of public
provision. As a benchmark, imagine that status concerns are absent (β = 0). Then,
(t∗ , x̄∗ ) = (0, 0) is the unique MVE, since the median income earner’s MWTP for x̄ is
zero. When individuals care about social status, this result need not hold. The reason
is that the median income earner might be willing to finance a positive level of x̄ even if
he opts out of public provision: By providing a positive amount of x̄ the number N of
individuals choosing the public option rises, which enhances indirect utility via the status
function S(N ). It may happen that this effect exceeds the utility decrease associated with
the rise in the tax rate. Then we have:
v(ym (1 − t)) + β · S (N (t, x̄; β)) > v(ym ) + β · S (N (0, 0; β)) .
To see that this result may indeed emerge, consider
Example 4: Assume that direct preferences are represented by u(x, c) =
√
√
x + c. The
status function is linear, i.e. S(N ) = N . Then, indirect utility when opting out is
V out (t, N ; y, β) = 2 ·
p
0.5 · y(1 − t) + β · N.
Let income be uniformly distributed on an interval [y, y], such that median and mean
incomes are ym = Y = 0.5 · [y + y]. The median income individual solves
max
t,x̄
V out (t, x̄; ym , β) s.t. t · Y = x̄ · N ∗
where N ∗ is the solution to N − F (ŷ(t, x̄, N ; , β)) = 0. Suppose further that β = 5,
y = 0 and y = 100. Then, the solution to the above optimization problem is (t∗ , x̄∗ ) =
(0.17, 20.55). The corresponding indifferent income level and share of individuals staying
in the public system are given by ŷ = 41.88 and N ∗ = 0.42, respectively. It follows that
V out (0.17, 20.5; 50, 5) = 11.19 > 10 = V out (0, 0; 50, 5).
17
One can also verify that the maximum attainable utility, given the GBC, when staying
in the public system is V in (0.32, 28.66; 50) = 11.18 such that the median income earner
strictly prefers to opt out of public provision. Hence, (t∗ , x̄∗ ) ≫ 0 is the unique MVE.
To sum up, we have
Proposition 2 With status concerns, public provision (x̄ > 0) may occur even with the
decisive voter staying out of the public system.
Thus, the status-enhanced model (unlike the standard framework) might explain why we
observe public provision of private goods even in cases where middle classes opt out of
public provision, for example, transportation services or housing.
5.2
Provision levels are non-monotonic in status
We now study a situation where the median income earner strictly prefers to stay in
the public system. Here, status does not affect the median income individual’s MWTP
for public provision. However, as (10) indicates, introducing status concerns pushes the
critical income level ŷ down, which in turn changes the equilibrium number of individuals
choosing public provision, i.e. the function N = N (t, x̄; β). Consequently, concerns for
social status might alter the government budget constraint (17). We want to analyze how
this affects the equilibrium level of public provision x̄. Consider the median individual’s
preferred policy (t∗ , x̄∗ ) when β = 0. Since we are in an interior optimum, this policy
must satisfy:
ūx
N + x̄ · Nx
=
ym · ūc
Y − x̄ · Nt
(25)
The left hand side of (25) is the median individual’s MWTP, while the right hand side
shows the marginal cost of public provision, i.e., the rise in the tax rate necessary to
finance a marginal expansion of x̄. To analyze how status concerns affect these marginal
costs, differentiate the right hand side of (25) with respect to β. If marginal costs decrease,
the median individual can implement a higher public provision level, whereas increasing
marginal cost would force him to choose less public provision.
>0
∂
∂β
N + x̄ · Nx
Y − x̄ · Nt
>0
z
}|
{ z
}|
{
(Nβ + x̄ · Nx̄β ) · (Y − x̄ · Nt ) + (N + x̄ · Nx̄ ) ·x̄ · Ntβ
=
.
(Y − x̄ · Nt )2
{z
}
|
>0
18
(26)
Introducing status has two effects. First, for any given (t, x̄), the number of individuals in
the public sector falls which decreases marginal costs. This effect is captured by the term
Nβ in (26) and conforms to intuition: with some additional individuals opting out, it is
now easier to endow the rest of the population with a higher provision level x̄. Second,
since the individual with critical income ŷ changes with status concerns (i.e. we are in a
different position of the income distribution), the reactions of equilibrium N with respect
to an increase in t or x̄ are affected. In (26) these changes are reflected in Ntβ and Nx̄β ,
respectively. If Ntβ and Nx̄β are both negative in sign, such that less individuals opt into
public provision when t or x̄ are marginally increased, marginal costs decrease with status
concerns. However, Ntβ and Nx̄β can be positive as well, depending in a complex way on
the shape of the income distribution, the status function S(N ) and individual preferences
u(x, c). Thus, in general it is not clear which effect dominates, such that status concerns
may lead to higher or lower provision levels.
Proposition 3 Compared to a situation with no status concerns, provision levels of x̄ in
the new MVE may be higher or lower.
Interestingly, an important factor driving this result is the shape of the income distribution
function. Indeed, using two numerical examples, we show that less public provision occurs
if (all else being equal) the distribution function is sufficiently concave. To highlight the
importance of the income distribution, we consider two economies that only differ with
respect to the shape of their income distributions. Direct preferences and the status
√
√
function are identical in both examples. Specifically, assume that u(x, c) = x + c and
(for simplicity) that status motives are invariant, i.e., S(N ) = β.
Example 5: Assume that income in the economy is log-normally distributed, ln N (µ, σ 2 ),
with µ = 1.5 and σ = 1. With this specification, median and mean incomes are given
by ym = 4.48 and Y = 7.39, respectively. In the case where individuals are not concerned with their status, the indirect utility of the median income earner is maximized at
(t∗ , x̄∗ ) = (0.611097, 4.53031). With status concerns β = 0.25, the median income earner
prefers (t∗ , x̄∗ ) = (0.596879, 4.44575), which entails a lower tax rate and expenditure level.
19
Example 6: Suppose that income is piecewise uniformly distributed on the interval
[y, y] = [0, 108.4939] in the following way:



0






0.111565057 · y





0.4138625 + 0.028792557 · y
F (y) =


0.986789601 + 0.000146202 · y






0.008576463 + 0.00007905 · y





1
y<0
y≤y≤5
5 < y ≤ 20
20 < y ≤ 69
69 < y < 108.4939
otherwise
Then median and mean income are the same as in Example 5. Without status concerns,
the median income earner prefers the same combination of tax rate and expenditure
level as in Example 5, i.e., (t∗ , x̄∗ ) = (0.611097, 4.53031). With positive status concerns
β = 0.25, the median voter now prefers a higher provision level as well as a higher tax
rate: (t∗ , x̄∗ ) = (0.613594, 4.55838).
This confirms that optimal provision levels for government-supplied private goods are
generally non-monotonic in the intensity of status concerns.
5.3
Publicly versus privately provided quality
In the absence of status concerns, an individual opting out of public provision always
chooses a higher quantity of good x than offered by the public sector, since in that case,
consumption of good c is necessarily lower.6 However, if we allow for status concerns, it
is possible that some individuals who opt out privately choose a lower level of good x.
The intuition is that status utility from opting out may more than outweigh the utility
loss from the associated decrease in c such that one would accept a lower level of x when
consuming privately. To see that a lower quality in the private sector can indeed occur
as in equilibrium, we employ the same setup as in example 4 (section 5.1) and depict
individual’s chosen quality of good x as a function of income in Figure 4. As can be seen,
individuals with incomes above and in the vicinity of the indifferent income, ŷ), consume
less than the uniform quality in the public sector, x̄ = 20.55.
We sum up this in
6
Without status, for an individual opting out we have u(x∗ (·), y(1 − t) − x∗ (·)) > u(x̄, y(1 − t)). As
y(1 − t) − x∗ (·) < y(1 − t), it follows that x∗ (·) > x̄.
20
Figure 4: Equilibrium demand for good x as a function of income
x,x*
40
20
0
20
`
y
ym
60
80
100
y
Proposition 4 With status concerns, quality on the private market can be lower than
that provided by the public system.
In that sense, our model captures the circumstance that not every private school or
university seem to have a higher quality than its state-run counterpart – a stylized fact
which cannot be explained by the standard framework of public provision of private goods.
6
Conclusion
Motives of social positioning may explain why people would avoid publicly provided goods
even if they have to contribute to their tax financing and have to finance private alternatives out of their own pockets. Such status concerns add a motive why taxpayers would
support the public provision of private goods that they themselves would not consume in
the public system. This then affects the political economy of publicly providing private
goods – however, in unclear directions: societies where status concerns prevail may have
larger or lower government sectors than status-free societies. Status-laden democracies
might even provide goods that a majority of their citizens would not consume. Moreover,
prestige gives rise to a fascinating variety of political equilibria, seemingly unlikely political coalitions may form, and status concerns may override the common stratification of
political preferences along the income spectrum.
21
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22