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Federal Reserve Bank of Chicago
Public Investment and Budget Rules
for State vs. Local Governments
Marco Bassetto
WP 2008-21
Public Investment and Budget Rules for State vs. Local
Governments
Marco Bassetto∗
Federal Reserve Bank of Chicago
NBER
Abstract
Across different layers of the U.S. government there are surprisingly large differences
in institutional provisions that impose fiscal discipline, such as constitutionally mandated
deficit or debt limits, or specific tax bases. In this paper we develop a framework that can be
used to quantitatively assess their costs and benefits. The model features both endogenous
and exogenous mobility across jurisdictions, so we can evaluate whether the different degree
of mobility at the local vs. national level can justify different institutional restrictions. In
preliminary results, we show that pure land taxes have very beneficial incentive effects, but
can only raise limited amounts of revenues. In contrast, under exogenous mobility, income
taxes lead unambiguously to insufficient incentives to invest in public capital, unless the
fiscal constraints explicitly favor such investment. This conclusion seems to hold even with
the introduction of endogenous mobility, since adverse congestion effects from inefficient
migration offset the beneficial impact of (partial) capitalization of future taxes into land
prices.
∗
Very preliminary. I am indebted to Gadi Barlevy, Jeff Campbell, Daniele Coen–Pirani, Mariacristina De
Nardi, Leslie McGranahan, and seminar participants at the Federal Reserve Bank of Chicago and the University
of Chicago for helpful comments and discussions. I also thank Andrew Butters and Aspen Gorry for their valuable
research assistance, and NSF for financial support. The views expressed herein are my own and do not necessarily
reflect those of the Federal Reserve Bank of Chicago or the Federal Reserve System.
1
1
Introduction
The Unites States offers a rich variety of different institutions for limiting government indebtedness at different levels of decentralization. While almost all states are constitutionally restricted
to balance their budget every budget cycle, except for public investment, the federal government
is only constrained by a legislated debt cap that is routinely lifted whenever needed. The practice
of states does not extend to lower-level jurisdictions (such as counties and cities), where there is
a wide heterogeneity in the forms of restrictions to indebtedness.1
In previous work (Bassetto with Sargent [5]), we quantified the role of mobility and demographics in generating departures from Ricardian equivalence at the state level and the federal
level. Because of the higher degree of (gross) mobility, we found that states reap much bigger
efficiency gains than the federal government from adopting a “golden rule” that forbids borrowing for recurring expenses but allows it for capital projects. In that paper we assumed mobility
to be driven entirely by exogenous events, so that public investment has no effect on population
size and/or property prices. This assumption seems a reasonable approximation for large entities
such as states or countries, since mobility in that case is likely to be driven predominantly by
job opportunities, family decisions, and other factors that are only weakly linked to government
policy.2 However, the same assumption is no longer appropriate in considering mobility at the
local level, where households face a choice even when constrained by the job location and/or
family ties.
This paper develops a framework to quantitatively assess the role of different institutional
restrictions on government policy in delivering efficient public investment when mobility can be
driven both by exogenous events and by an optimal location choice.
In developing our analysis, we build upon two sets of existing papers. First, there is a very
large literature in local public finance that analyzes the interplay between mobility and the
efficient provision of public goods. This literature starts from Tiebout [21], and highlights the
role of the source of taxes (property vs. income or head taxes), the elasticity of the demand and
supply of land (with particular emphasis on zoning restrictions for the latter), and the role of
congestion externalities.3 This literature has mostly dealt with static environments, where the
location and policy decisions are taken at one point in time. A few papers have attempted to
bring dynamics into the picture to better analyze the role of public investment and debt,4 deriving
some of the qualitative results that we will discuss below. However, these dynamic models rely
on generations that live for two periods only, and are thus not well suited for quantitative work.
More recently, there has been research into computable models where agents face dynamic
choices both privately and as participants in the political process.5 These papers have mostly
1
See e.g. Miranda, Picur, and Straley [18].
For examples of the low estimate of endogenous mobility across states, see Meyer [16] and Gelbach [9].
3
For a literature survey, see e.g. Mieszkowski and Zodrow [17]; Stiglitz [20] contains a thorough discussion of
some of the central aspects in this debate.
4
See e.g. Kotlikoff and Rosenthal [15], Wildasin and Wilson [24], Schulz and Sj¨str¨m [19], and Conley and
o o
Rangel [7].
5
See e.g. Hassler et al. [11], Hassler et al. [10], Doepke and Zilibotti [8], Azzimonti [1], and Azzimonti,
Battaglini, and Coate [2].
2
2
worked with what Klein, Krusell, and R´
ıos-Rull [14] defined a “generalized Euler equation.” In
this paper, we show how to apply similar tools to local public finance, and we provide some
preliminary examples of the conclusions that can be drawn.
2
The Model
2.1
Preferences and Technology
We consider an economy populated by a continuum of households i ∈ I; each of them must
choose to live in one of a continuum of towns j ∈ J . We normalize to one the measure of
households and towns. From the aggregate perspective, all towns are identical; in particular,
¯
each of them encompasses L units of land.
Each household consumes a private good and enjoys the services from land and a durable
local public good. In addition, in each period the household has a set Nit of N towns where it can
potentially live, and for each of those an idiosyncratic, location-specific utility shock ψijt that is
independent across the N towns and distributed according to a cumulative distribution function
ˆ
F . Over time, the location-specific shocks evolve as follows: with probability θ, both the set of
towns and the shock remain the same as in the previous period. With probability 1 − θ, the
household draws a new set of towns and a new location-specific utility shock, both independent
of the previous realizations. This exogenous shock can be interpreted both as a major life event
that affects location preferences and as the death of a household, which is then replaced by a
newly born household to preserve a constant population. In the notation that follows, I will
assume the first interpretation, whereby a household is infinitely lived (as an altruistically linked
dynasty). However, the only change that is needed for the second interpretation to apply is to
assume that annuities markets for wealth are present. For the purpose of calibration, we will
interpret θ as stemming from both forces, and we will assume that the proper annuity markets
are open.
Summarizing, household i’s preferences are given by
∞
β t [cit + w (Lit ) + v
E0
t=0
Gj t t
µj t t
+ ψijt t ],
where cit is consumption by household i in period t, Lit is the amount of land whose services are
consumed by household i in period t, jt is the town where person i chooses to live in period t, Gjt t
is the amount of a durable public good (“public capital”) provided in town jt in period t, µjt t is
the density of population in town jt in period t, and w and v are continuously differentiable and
strictly concave functions. The assumption of linear preferences in private consumption greatly
simplifies the analysis by eliminating wealth effects from the household preferences over land
and public goods.6 For tractability, we also need to assume that all households benefit in the
6
Bassetto with Sargent [5] discusses robustness of the results with respect to this assumption in the context of
a model with exogenous mobility. In the present context, endogenous sorting by wealth would however potentially
generate further complications.
3
same way from the public good. Finally, note that we assume that public capital is subject to
congestion externalities, so a household only benefits from its level per capita.7 We assume v
and w to be strictly increasing and concave. Preferences are only defined over location decisions
that belong to the set Nit at each time t.
We assume that a law of large numbers applies across households,8 so that in each period
each town j has a density N of potential inhabitants, and the c.d.f. of the utility that each
potential inhabitant receives compared to living in the next best city is given by the c.d.f. of
ˆ
ˆ
X − Y , with X ∼ F and Y ∼ F N −1 . We denote by F this c.d.f., and we assume that it is
continuously differentiable.
In each period, households receive an exogenous endowment y.9 There is a constant-returnsto-scale technology that turns one unit of endowment into one unit of private consumption.
Likewise, another constant-returns-to-scale technology turns one unit of endowment from a unit
density of households into one unit of public investment in any given location.
Public capital depreciates at a rate δ.
2.2
Government
The provision of the public good is decided by majority voting by the residents of each town.
Public financing comes from either income or land taxes. We denote income taxes per capita as
Tt and land taxes per unit of land τt . Since land is homogeneous and there are no structures, a
land tax is equivalent to a property tax in this setup.
We assume that a fraction x of investment can be financed with debt, and that the town
government is committed to repay interest and a fraction α of outstanding debt in each period.
3
Efficient Allocations
Because of quasilinear preferences, the allocation of people, land, and public capital remains
the same along the Pareto frontier; only the allocation of private consumption is affected.10 To
avoid well-known measurability issues, it is convenient to index each household by the sequence
of realizations of idiosyncratic shocks.11 A Pareto-optimal allocation will thus be described by
a sequence of aggregate consumption {Ct }∞ , a sequence of government spending by each town
t=0
{{Gjt}j∈J }∞ ,12 and stochastic sequences of land holdings and locations {Lt , jt }∞ , adapted to
t=0
t=0
7
We could allow for a weaker congestion effect, as long as w is sufficiently increasing to ensure a uniform
distribution of households across all towns. The analysis would be considerably more complicated if clustering of
households emerged endogenously from the increasing returns intrinsic in the public good.
8
See Judd [13] and Uhlig [22] for a discussion of the law of large numbers in the context of models with a
continuum of households.
9
Introducing private capital and labor would have no effect on the results; see Bassetto with Sargent [4, 5].
10
We assume here that the nonnegativity constraint on consumption does not apply, or we restrict our attention
to the region where it is not binding.
11
See Jovanovic and Rosenthal [12] for related discussion.
12
Measurability is not an issue for towns, since they will all be identical in the optimal allocation and in the
equilibria that we will analyze. Still, we use L∞ as the relevant space for {Gjt }j∈J , so public capital is only
4
the history of location options {Ns }t and the history of shocks {{ψjs }j∈Ns }t that solve the
s=0
s=0
following problem:
∞
Gj t t
β t [Ct + w (Lt ) + v
max E0
+ ψjt t ],
µj t t
t=0
subject to the output endowment constraints,
(Gjt − (1 − δ)Gj t−1 )dj ≤ y
Ct +
∀t
j∈J
land constraints,
t
N
ˆ
¯
dF (ψqs ) ≤ L ∀t, a.s. in j ∈ J .
Lt
({Ns }t ,{{ψjs }j∈Ns }t ):jt =j
s=0
s=0
s=0 q∈Ns
and the definition of µjt
t
ˆ
dF (ψqs ) ∀t, a.s. in j ∈ J .
µjt := N
({Ns }t ,{{ψjs }j∈Ns }t ):jt =j s=0 q∈N
s=0
s=0
s
By the symmetry of the shocks, it is straightforward to verify that a Pareto-optimal allocation assigns each household to its most preferred location according to the idiosyncratic shock;
¯
furthermore, all households consume L units of land (so µjt t ≡ 1).
Finally, the efficient condition for the provision of public capital requires the same level of
public capital in (almost) all towns:
v ′ (Geff ) = 1 − β(1 − δ)
(1)
The intuition behind equation (1) is straightforward: producing one additional unit of the public
good in each town costs one unit of aggregate consumption, but has an immediate benefit captured by v ′ (Geff ) and a future benefit in that it allows to cut investment and increase aggregate
consumption in the next period by 1 − δ units, which are discounted at the rate β.13
4
Markets and Competitive Equilibrium
We assume that there is no rental market for land;14 in order to live in each town, a household
must own the land whose services it consumes. There are no transaction costs from selling or
buying the land, nor are there moving costs, apart from the possible loss in location-specific
utility.
determined up to sets of towns of measure 0.
13
Because of symmetry, the sentence above applies to any set of positive measure of towns, so we are not
constraining the planner to deliver the same public good everywhere, but rather this is part of the optimal
solution.
14
The analysis will be extended to renters in future work.
5
Besides trading land, households can also trade both privately-issued and publicly-issued
bonds. To the extent that they face a probability of death, households can also trade with
financial intermediaries that annuitize the households’ wealth.
A symmetric steady state-competitive equilibrium is
• an allocation (c∗ , G∗ , L∗ );
• a price system (ρ∗ , p∗ ), where ρ∗ is the interest rate on government debt and p∗ is the price
of land;
• taxes and debt (T ∗ , B ∗ ) or (τ ∗ , B ∗ )
such that:
• Households maximize their utility taking prices and government policy as given, subject to
the budget constraint
Bit = Bi t−1 (1 + ρ∗ ) + y − Cit − T ∗ − p∗ (Lit − Li t−1 )
(2)
Bit = Bi t−1 (1 + ρ∗ ) + y − Cit − (p∗ + τ ∗ )Lit + p∗ Li t−1
(3)
or
and a lower bound on debt holdings −B whose only role is to rule out Ponzi schemes.
• the towns’ budget constraints are satisfied, i.e.,
ρ∗ B ∗ + G∗ (1 − δ) = T ∗
or
¯
ρ∗ B ∗ + G∗ (1 − δ) = τ ∗ L;
furthermore, the towns respect their borrowing limits and repayment schedules, i.e.,15
αB ∗ = δxG∗ .
(4)
¯
• The market for land clears in each town, i.e., L∗ = L. Since all towns have the same land
prices, households will always choose to locate where their idiosyncratic realization of ψijt
is highest.
• The market for debt clears, i.e., B ∗ is an optimal choice of assets for each household. Note
that linearity implies that the actual distribution of assets in the population is indeterminate (but it is also irrelevant for what follows).
15
We assume that the deficit limit is always binding for towns. This will be the case in all of the numerical
examples that we will consider, except for some cases in which Ricardian equivalence holds. When Ricardian
equivalence holds, (4) is a weakly optimal choice, and (without loss of generality) we assume that this is how
town debt is set.
6
Characterizing a symmetric steady-state competitive equilibrium given an exogenous level of
public capital G∗ is straightforward. From the household problem, the equilibrium interest rate
must be ρ∗ = (1 − β)/β. Given this interest rate, it is convenient to define the implicit land rent
as rt := pt − βpt+1 , where pt is the price of land in period t. In the steady state, this is yields
r ∗ = p∗ (1 − β). The demand for land by each household as a function of implicit rent is given by
ˆ
L(r) := arg max (w (L) − rL) ,
L
¯
ˆ
In the case of income taxes, a symmetric steady-state equilibrium requires L = L(r ∗ ): this
¯
is a (nonlinear) equation that can be solved for r ∗ .16 Similarly, for land taxes, we obtain L =
∗
∗ 17
ˆ
L(r + τ ).
Given G∗ , the budget constraint of the governments and their debt limits determine T ∗ and
∗
B . Finally, consumption is given by C ∗ = y − δG∗ .
It is straightforward to verify that a symmetric steady-state competitive equilibrium is Pareto
efficient, subject to the exogenously given amount of public spending.
To describe a political-economic equilibrium, we need to analyze the problem faced by voters
of a town that contemplates setting a policy different from that of all other towns. Accordingly,
we now define a competitive equilibrium with one town deviating from the symmetric steady
state. This is given by the same objects as the symmetric steady-state equilibrium (with the
same equilibrium conditions), complemented by what follows.
• An (exogenous) sequence {Gj0t }∞ , where Gj0 t is public capital in town j0 in period t;
t=0
• A location decision as a function of the history of idiosyncratic shocks for all households
for whom j0 ∈ Nit at some time t. It is straightforward to prove that the household
location decision is only affected by the contemporaneous realization of the idiosyncratic
shocks, since there are no wealth effects nor moving costs. We denote this choice as
J(ψij0 t , ψij1 t , ...ψijN−1 t ), where (j1 , . . . , jN −1 ) are the other elements of Nit . Since all other
towns behave identically and there is a continuum of them, the effect of the deviating
town on their population is negligible. We thus only use the J function to characterize the
population in town j0 . Accordingly, we set J(ψij0 t , ψij1 t , ...ψijN−1 t ) = j0 for a household that
optimally chooses to live in j0 , and normalize J(ψij0 t , ψij1 t , ...ψijN−1 t ) = 0 if the optimal
choice is to live anywhere else.
• Consumption, land, and asset holdings as a function of the history of idiosyncratic shocks
for all households for whom j0 ∈ Nit at some time t. Thanks again to quasilinearity
of utility, all households that choose to live in the same town will choose the same land
holdings, independently of the history of their idiosyncratic shocks. It thus follows that
land holdings for households living in town j0 can be summarized by a sequence {Lj0 t }∞ .
t=0
• a land price sequence for town j0 , {pj0 t }∞ ;18
t=0
16
Our assumptions about preferences ensure that a solution exists and is unique.
Existence of a solution is now only guaranteed if τ ∗ is not too high, while uniqueness is unaffected.
18
The interest rate is an economy-wide variable and remains at ρ∗ .
17
7
• a sequence of taxes and debt in town j0 , (Tj0 t , Bj0 t ) or (τj0 t , Bj0 t )
such that:
• For all households for whom j0 ∈ Nit at any time t, the location decision, consumption,
asset and land holdings maximize their utility taking prices and government policy as given,
subject to the budget constraints
Bit = Bi t−1 (1 + ρ∗ ) + y − Cit − Tjt − pjt (Lit − Li t−1 )
or
Bit = Bi t−1 (1 + ρ∗ ) + y − Cit − (pjt + τjt )Lit − pjtLi t−1
and the lower bound −B, where j is the index of the town they choose to live in period t,
and pjt = p∗ , Tjt = T ∗ , τjt = τ ∗ for j = j0 .
• Towns j0 ’s budget constraint and borrowing limits are satisfied, i.e.,
(1 + ρ∗ )Bj0 t−1 + Gj0 t = Tj0 t + (1 − δ)Gj0 t−1 + Bj0 t
¯
(1 + ρ∗ )Bj0 t−1 + Gj0 t = τj0 t L + (1 − δ)Gj0 t−1 + Bj0 t
(6)
Bj0 t − Bj0 t−1 (1 − α) = x(Gj0 t − Gj0 t−1 (1 − δ)).
or
(5)
(7)
and
• The market for land clears in town j0 , i.e.,
µj 0 t ≡
¯
L
ˆ
ˆ
ˆ
dF (ψij0 t )dF (ψ1 ) · · · dF (ψN −1 ) =
Lt
(ψij0 t ,ψ1 ,...,ψN−1 ):J(ψij0 t ,ψ1 ,...,ψN−1 )=j0
(8)
We next characterize such an equilibrium. Consider first the case of income taxes. Given the
sequence of public capital and initial debt Bj0 0 , equation (7) describes the evolution of debt, and
equation (5) determines the sequence of taxes. Define
w(r) := max (w (L) − rL) .
ˆ
L
For any household for whom j0 ∈ Nit , simple algebra shows that the net benefit of living in town
j0 rather in the best alternative town is given by
w(rj0 t ) + v
ˆ
Gj 0 t
µj 0 t
−
Tj0 t
− u∗ + ψij0 t − max ψijt ,
j∈Nit \j0
µj 0 t
where u∗ is the utility level attained in a town at the symmetric steady state, net of the private
endowment and the idiosyncratic preference shock, i.e.,
u∗ = w(r ∗ ) + v(G∗ ) − δG∗ 1 +
ˆ
8
x(1 − β)
.
αβ
(9)
Population density in j0 is thus
µj 0 t = N
1 − F u∗ − w(rj0 t ) − v
ˆ
Gj 0 t
µj 0 t
+
Tj0 t
µj 0 t
(10)
¯ ˆ
From equation (8), µj0 t = L/L(rj0 t ), which we substitute into (10) to obtain an implicit function
to be solved for rj0 t . We cannot prove in general that a solution to this problem exists, nor that it
is unique. However, in any computed example, both existence and uniqueness are straightforward
to check numerically, given G∗ . Given the implicit rent, the land price is given by
∞
β s−trj0 t .
pj 0 t =
s=t
Finally, when ρ∗ = (1 − β)/β households are indifferent among all consumption and asset plans
that satisfy their budget constraint, so the description is completed by assuming that households
for whom j0 ∈ Nit at any time t follow one of these paths.19
Similar equations apply in the case of land taxes. In this case, the implicit equation to be
solved for rj0 t is
¯
L
ˆ
L(rj0 t + τj0 t )
= N 1 − F u∗ − w(rj0 t + τj0 t ) − v
ˆ
ˆ
Gj0 t L(rj0 t + τj0 t )
¯
L
.
(11)
A non-symmetric competitive equilibrium will not be Pareto efficient in general, even conditional on the exogenous levels of public goods.20 This is because, by moving, a household does
not take into account the additional congestion that it causes in its destination nor the relief in
the town of origin. In general, these effects cancel only in a symmetric equilibrium.
5
Political Process and Political-Economic Equilibrium
We assume the following timing of events within each period t:
1. Households realize their idiosyncratic shocks, decide where to live for the period, and trade
land accordingly;
19
Existence is guaranteed, since we do not impose a lower bound on consumption. Also, note that the set of
households in question is of measure 0, so it is immaterial for market clearing conditions.
20
Strictly speaking, a single town deviating has no efficiency consequences, since the allocation is defined up to
sets of towns of measure 0. Nonetheless, taking limits from deviations of sets of positive measure, the efficiency
condition would require
Gj0 t Gj0 t
− v ′ (G∗ )G∗ .
µj0 t = 1 − F v ′
µj0 t µ20 t
j
This equation balances the direct benefit/cost accruing to the marginal mover with the congestion benefits/costs
at the town j0 and at any of the symmetrically-behaving towns.
9
2. Residents of each town vote over the provision of local public capital, subject to the town’s
budget constraint and borrowing limit;
3. The public good is produced and taxes are levied according to the outcome of the vote.
Consumption takes place.21
For the case of income taxes, we define a Markov political-economic equilibrium (MPEE) as a
vector (c∗ , G∗ , L∗ , ρ∗ , p∗ , T ∗ , B ∗ ), and functions (Γ, Λ, Π) that depend on public capital and debt
inherited from the previous period, such that:
(i) Starting from G−1 = G, B−1 = B, define recursively Gt = Γ(Gt−1 , Bt−1 ), Lt = Λ(Gt−1 , Bt−1 ),
pt = Π(Gt−1 , Bt−1 ). Set Tt and Bt to the values implied by (5) and (7). Given any value of
(G, B), (c∗ , G∗ , L∗ , ρ∗ , p∗ , T ∗ , B ∗ ) and (Gt , Lt , pt , Tt , Bt )∞ , together with a suitable choice
t=0
of household consumption, asset, and location decisions, must form a symmetric competitive equilibrium with one town deviating from the symmetric steady state.
¯
(ii) Γ(G∗ , B ∗ ) = G∗ , Λ(G∗ , B ∗ ) = L, and Π(G∗ , B ∗ ) = p∗ . In words, a town that starts at the
symmetric steady state remains there.
(iii) Start from any initial condition (G, B). Let the distribution of idiosyncratic preferences
for living in the town among the residents be the one that would prevail in the competitive
equilibrium described in point (i). Specifically, this means that, among residents, the
cumulative distribution function of the preference z for living in the current town relative
to the next best alternative is given by
F (z) − F (˜)
z
,
1 − F (˜)
z
where z is defined as the solution to
˜
¯
L
= N(1 − F (˜)).
z
Λ(G, B)
Let the current residents vote over choices of G′ . Let they assume that the resulting
allocation will be a competitive equilibrium with one town deviating from the steady state,
characterized by current-period taxes and debt issuance from (5) and (7), and a future
allocation that satisfies Gt = Γ(Gt−1 , Bt−1 ), Lt = Λ(Gt−1 , Bt−1 ), pt = Π(Gt−1 , Bt−1 ) at all
future times t. Then, Γ(G, B) is a Condorcet winner.
In words, what we require is that a majority of the residents choose the policy Γ(G, B) when
they expect the future response of the economy to be described by the same functions that
represent the current choices. While in principle this vote could take place starting from an
arbitrary population density and distribution of tastes, we only need to check optimality
for the density and the distribution of tastes that stem from an optimal location choice by
households that (correctly) anticipate that the policy Γ(G, B) will be implemented.
21
Debt can be traded both at stages 1 and 3.
10
In the case of property taxes, the definition is identical, except that τ ∗ replaces T ∗ and (6)
replaces (5).
In deciding which policy they prefer, households have to anticipate whether the policy will
lead them to move to a different jurisdiction next period or to stay put (assuming that they are
not hit by the exogenous mobility shock). It is straightforward to see that two households i and
j will have the same ranking over the set of policies that lead both of them to remain, and also
over the set of policies that lead both of them to leave. Conflict will only emerge for policies
that lead one of them to stay and the other to leave, or between choosing a policy that would
lead both of them to stay and another one that would lead both of them to leave.
At the steady state, all households will remain put until hit by the exogenous shock. By
continuity, around the steady state, stayers will remain a majority of the population, and we
can characterize the equilibrium by looking at the first-order conditions for their preferred policy
(assuming a differentiable equilibrium). Once we have characterized the equilibrium in such a
neighborhood, we could then expand the search to regions where the pivotal voter might be
moving, and check robustness to policies that might lead more than 50% of the population to
decide to move out. Notice, however, that this case is relevant theoretically, but cannot be
relevant empirically. We will typically consider a period to be a budget cycle (a year), and such
a rapid population drop is implausible (except perhaps for tiny villages).
Let W (G, B) be the value that a household expects to attain in the future by living in a
community that has state variables described by G and B, net of the cost of acquiring the land
and of the idiosyncratic location shock. Consider a household that anticipates to stay in the
same location with probability θ and that lives in a town characterized by an initial population
µ,22 inherited capital (before depreciation) G, and debt B. Such a household will prefer a policy
that solves the following problem:
max v
′
G
G′
µ
+w
¯
L
µ
−
(α + (1 − β)/β)B + (1 − x)(G′ − (1 − δ)G)
+
µ
¯
L
β Π(G′ , B ′ ) + βθW (G′, B ′ ) + β(1 − θ)W (G∗ , B ∗ ),
µ
(12)
where
B ′ := (1 − α)B + x(G′ − (1 − δ)G).
(13)
Equation (12) illustrates the trade-offs faced by a household in choosing G′ . First, the choice has
a contemporaneous effect on current utility and taxes. Since all households face the same costs
and benefits along this dimension, this component does not involve any externalities. Looking
to the following period, the household is affected by the current choice in two ways:
• To the extent that higher G′ and/or lower B are reflected into future land prices, the
household realizes a capital gain on the land that it has purchased.
¯
We assume that the household owns L/µ units of land. We could further generalize the description to nodes
in which households picked different land holdings. However, this complication is irrelevant for the same reasons
that will lead us to drop µ as a separate state variable.
22
11
• The current choice of G affects future levels of spending and taxes; to the extent that the
household anticipates remaining in the town, it will take these considerations into account.
These will be reflected in the value W (G′ , B ′ ). Note that part of the change in W will
undo the benefits of the capital gains above: in order to remain in the town, the household
needs to forgo realizing part of those capital gains, to retain land where to live.
ˆ
The solution to (12) yields a function Γ(G, B, µ): it would thus appear that the population
size should appear as a state variable. However, in their location decision, households correctly
ˆ
anticipate the policy that will be undertaken. Substituting the resulting value Γ into (10) or (11),
we see that in equilibrium the resulting population level will only depend on G and B inherited
from the past.23 Furthermore, while it would be easy to compute ex post, characterizing an
equilibrium can be done without keeping track of the choices that would be made in a node in
which (10) or (11) do not hold.24
In the case of income taxes, the first-order conditions of the policy preferred by a household
that anticipates to remain in town with probability θ are thus given by the solution to the
following system of four functional equations:
1. Value function, along the equilibrium path:
Γ(G, B)Λ(G, B)
+ w (Λ(G, B)) −
¯
L
Λ(G, B) [(α + (1 − β)/β) B + (1 − x) (Γ(G, B) − (1 − δ)G)]
+
¯
L
βΠ (Γ(G, B), B ′ ) Λ(G, B) + βθW (Γ(G, B), B ′) + β(1 − θ)W (G∗ , B ∗ );
W (G, B) = − Π(G, B)Λ(G, B) + v
(14)
2. First-order condition for the policy preferred by a household that anticipates to stay:
0=
Λ(G, B) ′
v
¯
L
Γ(G, B)Λ(G, B)
¯
L
′
− (1 − x) +
(15)
′
′
′
′
′
β ΠG (Γ(G, B), B ) + xΠB (Γ(G, B), B ) Λ(G, B) + βθ WG (G , B ) + xWB (G , B ) ;
3. Market clearing for land:
¯
L
= N 1 − F u∗ − w L−1 (Λ(G, B)) − v
ˆ ˆ
Λ(G, B)
Γ(G, B)Λ(G, B)
+
¯
L
Λ(G, B)
[(α + (1 − β)/β)B + (1 − x)(Γ(G, B) − (1 − δ)G)]
;
¯
L
4. Land price:
ˆ
Π(G, B) = L−1 (Λ(G, B)) + βΠ (Γ(G, B), B ′) .
23
(16)
(17)
Of course, if the resulting equation does not have a solution, then an equilibrium does not exist. Conversely,
if multiple solutions can be found, then we have multiple equilibria.
24
For a more comprehensive explanation of this point, see Chari and Kehoe [6]. However, notice that keeping
track of such nodes is essential when uniqueness of an equilibrium is an issue; see Bassetto [3].
12
In equations (14)-(17), we used for brevity the definitions of u∗ and B ′ of (9) and (13), with
G′ = Γ(G, B).
In the case of land taxes, equations (14) and (15) remain unchanged, but equations (16) and
(17) are replaced by
¯
L
= N 1 − F u∗ − w L−1 (Λ(G, B)) − v
ˆ ˆ
Λ(G, B)
Γ(G, B)Λ(G, B)
¯
L
(18)
and
1
ˆ
Π(G, B) = L−1 (Λ(G, B))− ¯
L
α+
1−β
β
B + (1 − x)(Γ(G, B) − (1 − δ)G) +βΠ(Γ(G, B), B ′).
(19)
6
The Power and Limits of Land Taxes
It is straightforward to verify that, in the case of land taxes, we obtain the following simple
solution:
Γ(G, B) = Geff
(20)
¯
Λ(G, B) = L,
Π(G, B) =
ˆ
¯
¯
B
L−1 (L) − δGeff /L (1 − δ)(G − Geff )
+
− ¯,
¯
1−β
L
βL
W (G, B) =
¯
v(Geff ) + w(L) − δGeff ¯
B∗
− LΠ(Geff , B ∗ ) −
,
1−β
β
and
where Geff is the efficient level of public capital defined in equation (1) and B ∗ = xδGeff /α. Land
taxes induce an efficient provision of the public good in all periods. This is because future land
prices fully reflect future taxes and debt: the current voter thus internalizes the effects of her
choices on future residents of the town. This result follows Schultz and Sj¨str¨m [19] and Conley
o o
25
and Rangel [7].
With capitalization of amenities and debt into future land prices, the degree of endogenous or
exogenous mobility becomes irrelevant: the solution is independent of F and θ. The solution is
also independent of x and α, since Ricardian equivalence holds: if the current voter leaves bigger
debt to future generations, she will fetch a correspondingly lower price upon selling her land.
Land taxes would thus appear a promising source of revenue to provide appropriate incentives
for public investment.
25
In Schultz and Sj¨str¨m land taxes are not sufficient to restore efficiency, since the determination of the public
o o
good occurs before moving decisions, and the current benefits of the public good are not fully priced into land.
Conley and Rangel distinguish between public goods that generate fiscal vs. direct intergenerational spillovers.
Our paper focuses only on goods that generate fiscal spillovers: future generations can make up for the lack of
past investment by increasing their own investment, with the only cost of additional taxes.
13
In order for land taxes to reap such efficiency gains, it is very important that they only hit
land, a factor on fixed net supply that enjoys pure rents. Hence, the tax does not resemble
property taxes as currently assessed by most localities in the United States: those taxes apply
both to land and to structures. It also is different from a tax on land values, which (in a richer
model with nonuniform land) would distort the incentives to aggregate into cities.26
For quantitative analysis, land taxes are limited by the value of the land. As an example, the
average value of farm real estate per acre in Illinois in 2007 was $4,330.27 Assuming a discount
factor of 4%, and even assuming that this was the value of the marginal plot of land, Illinois
would be able to raise less than 1.5% of Gross State Product in pure land taxes before driving
to 0 the value of the marginal plot. Any increases beyond that point would generate distortions,
since they would lead residents to abandon tracts of land.28
We conclude thus that, while appealing, land taxes alone are not likely to generate enough
revenues to sustain sufficient government investment.
7
Some Numerical Examples about Income Taxes
We will develop a full quantitative assessment of income taxes in future versions of this work.
We consider here two numerical examples that illustrate the consequences of endogenous and
exogenous mobility on the efficiency of the allocation chosen by voters.
In the first example, we focus on the role of endogenous mobility by assuming that the
ˆ
distribution F is degenerate on 0. In this case, a symmetric equilibrium requires households to
be indifferent on where to live among the N towns that are available to them. Substituting (16)
and (17) into (14), it follows that W (G, B) is independent of (G, B): since households have no
particular attachment to a specific town, they care about the provision of the public good and
taxes in that town only to the extent that they affect the price of land that they own. For this
reason, the degree of exogenous mobility becomes irrelevant.
¯
We set the following parameter values: β = 0.96, δ = 0.03, L = 1, v(G) ≡ G−2 /2, w(L) ≡
1−σ
L /(1 − σ). We consider the case in which the town is not allowed to borrow to pay for capital
expenses, i.e., x = 0, and look at the resulting equilibrium distortion (at the symmetric steady
state). As in Bassetto with Sargent [5], we measure distortions as a wedge between the marginal
utility of public capital under the efficient allocation and under the equilibrium one:
wedge =
v ′ (G∗ ) − v ′ (Geff )
v ′ (Geff )
We choose this metric because it is more likely to be robust to departures from the assumption of
quasilinear preferences.29 A positive (negative) wedge means that public capital is underprovided
26
However, Wallis and Weingast [23] point out that a tax on land value may actually be beneficial for incentives
when the benefit of the public good does not accrue uniformly within the jurisdiction: when capitalization reflects
the differences in benefits, owners of land that receives the most benefits will pay a larger share of the costs.
27
Source: USDA web site. I thank David Oppedahl for pointing me to this information.
28
It is also likely that this figure is close to the top of the Laffer curve for such a tax: once the tax led people
to abandon farmland, the amount of land in use and available for taxes would shrink dramatically.
29
See the appendix of Bassetto with Sargent [5] for further discussion.
14
(overprovided).
The degree to which the voting outcome is inefficient depends on two forces:30
• Congestion: when a town has higher public capital than others, it becomes a more desirable
location. This drives more households to live there, and the resulting congestion dilutes
the benefits of the additional investment undertaken in the past.
• Capitalization: the newly arriving households bid up the price of land, which rewards the
previous residents (as owners of land) for the investment undertaken.
250
200
150
100
50
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Elasticity of demand for land
Figure 1: Efficiency wedge (%) with degenerate F
Whether congestion or capitalization will be more prevalent depends on whether equilibrium is
achieved mostly by varying quantity or price. With a fixed supply, this is in turn driven by the
elasticity of the demand for land (1/σ), which is thus the key parameter of interest. As figure 1
shows, for the case of degenerate F inefficiency is likely to be very large under a pure balanced
budget, unless the demand for land is extremely inelastic. In the limit, as σ → +∞, efficiency is
restored, since only a fixed number of households can live in the town, and the benefits of public
investment are thus fully factored into land prices.
In our second example, we consider the opposite polar extreme: we now assume that N = 1,
so that households have no location choice. In this case, it is Π that is independent of (G, B):
the price of land will be determined by the demand of the households that are captive to each
town, and the demand is unrelated to the provision of public goods.31 In equation (15), all longterm benefits that are internalized by the voters must now come from W . The key parameter
is thus no longer σ, which is irrelevant, but θ, the probability that a household will remain in
30
31
For further discussion of these forces, see e.g. Stiglitz [20] and Schultz and Sj¨str¨m [19].
o o
Of course, this relies on the assumption that preferences are strongly separable in the public good and land.
15
300
250
200
150
100
50
0
0.8
0.85
0.9
θ
0.95
1
Figure 2: Efficiency wedge (%) with N = 1
the same town. Results are shown in figure 2, where we retain the same parameter values and
functional forms as above. When θ ≈ 1 the town looks like a closed economy with infinitelylived households, and Ricardian equivalence holds; as figure 2 shows, this leads to efficiency. In
contrast, the higher the probability that a household will move out of town (or die), the bigger
the distortion generated by the fact that it neglects benefits accruing to future immigrants.
8
Conclusion
This paper has developed a framework to analyze the relevance of endogenous and exogenous
mobility on the incentives that local communities face in providing efficient levels of public investment. The preliminary numerical examples presented here suggest the following conclusions:
• Land taxes (purely based on acreage owned by each taxpayer) deliver efficient incentives,
but land per se is not valuable enough to fully support the needs of government investment.
• When households move in response to public amenities and debt, the adverse effect of
congestion is likely to be as important as the beneficial incentive effect of capitalization.
When taxes are primarily levied on income, relying on capitalization alone does not seem
thus a good substitute for explicit rules that encourage investment at the local level, such
as the golden rule.
In future research, we will calibrate the model to fully assess the quantitative relevance of
the distortions introduced by different budget rules. We will also incorporate investment in
structures, to consider the effect of property taxes that hit both land and capital.
16
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18
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Torben G. Andersen and Luca Benzoni
WP-08-14
Revenue Bubbles and Structural Deficits: What’s a state to do?
Richard Mattoon and Leslie McGranahan
WP-08-15
The role of lenders in the home price boom
Richard J. Rosen
WP-08-16
Bank Crises and Investor Confidence
Una Okonkwo Osili and Anna Paulson
WP-08-17
Life Expectancy and Old Age Savings
Mariacristina De Nardi, Eric French, and John Bailey Jones
WP-08-18
Remittance Behavior among New U.S. Immigrants
Katherine Meckel
WP-08-19
Birth Cohort and the Black-White Achievement Gap: The Role of Health Soon After Birth
Kenneth Y. Chay, Jonathan Guryan, and Bhashkar Mazumder
WP-08-20
Public Investment and Budget Rules for State vs. Local Governments
Marco Bassetto
WP-08-21
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