week 2

About 250 words per activity with reference page

use class references

Week Two Discussion One

Provide and discuss an example of how the federal government uses funding to influence the

behavior and policies of local level public safety related programs.

Week Two Discussion Two

Considering the concept of federalism, who bears the greatest responsibility for public safety:

Federal state or local government? Or does the greatest responsibility for public safety lay

elsewhere. Explain your response.

. Federalism: U.S. v. The States, Topic Overview

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Topic Overview Unit 3

Federalism: U.S. v. the States

Learning Objectives

After completing this session, you will be able to:

Explain how the Constitution distributes power between the national and
state governments.

Describe the various types of federalism.

Explain the changes that have occurred in the federal system in the past 200
years.

Summarize the part played by state governments in the contemporary
federal system.

Discuss the role of grantinaid programs in the American federal system.

Describe the advantages and disadvantages of a federal system.

Unit 3 provides an overview of the workings of federalism in the United States. In
this unit, the complex and changeable relationship between the national and state
governments is explored. By focusing on the conflicts between national and state
powers, the unit develops a deeper understanding of nature of governmental power
in the American system.

Federalism is the division of powers between a central government and regional
governments. Most developed nations experience ongoing struggles over the
relative powers of their central and regional governments. The United States has a
federal system of government where the states and national government exercise
separate powers within their own spheres of authority. Other countries with federal
systems include Canada and Germany. In contrast, national governments in
unitary systems retain all sovereign power over state or regional governments. An
example of a unitary system is France.

The framers of the U.S. Constitution sought to create a federal system that
promotes strong national power in certain spheres, yet recognizes that the states
are sovereign in other spheres. In “Federalist No.

,” James Madison asserted
that the states and national government “are in fact but different agents and
trustees of the people, constituted with different powers.” Alexander Hamilton,
writing in “Federalist No.

,” suggested that both levels of government would
exercise authority to the citizens’ benefit: “If their [the peoples’] rights are invaded
by either, they can make use of the other as the instrument of redress.” However, it
soon became clear that Hamilton and Madison had different ideas about how the
national government should work in practice. Hamilton, along with other
“federalists” including Washington, Adams, and Marshall, sought to implement an
expansive interpretation of national powers at the states’ expense. Madison, along
with other “states’ rights” advocates including Thomas Jefferson, sought to bolster
state powers.

The U.S. Constitution delegates specific enumerated powers to the national
government (also known as delegated powers), while reserving other powers to
the states (reserved powers). Article VI of the Constitution declares the laws of
the national government deriving from the Constitution to be “the supreme law of
the land” which the states must obey. The Tenth Amendment to the Constitution, a
part of The Bill of Rights passed in 1

1, attempts to limit national prerogatives

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over the states by declaring: “The powers not delegated to the United States by the
Constitution, nor prohibited to it by the States, are reserved to the States
respectively, or to the people.”

While the Constitution carves out significant spheres of power for the states, it also
contains several potential powers for the national government. These potential
powers, also called implied powers, include Congress’s power under Article I,
Section 8, to make laws that are “necessary and proper” for carrying out its
enumerated powers. The president’s constitutional role as “commander in chief” has
allowed presidents, including Lincoln, Franklin Roosevelt, and now George W. Bush,
to claim emergency powers for the national government in times of national
emergency. Finally, the Supreme Court’s original delegated powers in Article III
were significantly enhanced in the case of Marbury v. Madison (1802), where
Chief Justice John Marshall first articulated the Court’s power to exercise judicial
review. Judicial review is the power to strike down as unconstitutional acts of the
national legislature and executive, as well as state actions.

A review of American history shows that the lines that divide power between the
national government and the states are blurry, and in practice the balance of
powers between the two levels of government is constantly in flux. At the same
time, certain periods of federalism can be identified, and are often associated with
creative (although not always precise) metaphors:

Dual federalism, also known as “layer cake federalism” involves clearly
enumerated powers between the national and state governments, and
sovereignty in equal spheres. This relationship predominated from the

90s
to

Cooperative federalism, also known as “marble cake federalism,” involved
the national and state governments sharing functions and collaborating on
major national priorities. This relationship predominated between 1930 and
1960.

Creative federalism, also known as “picket fence federalism,”
predominated during the period of 1960 to 1980. This relationship was
characterized by overloaded cooperation and crosscutting regulations.

Finally, new federalism, sometimes referred to as “on your own
federalism,” is characterized by further devolution of power from national to
state governments, deregulation, but also increased difficulty of states to
fulfill their new mandates. This period began in 1981 and continues to the
present.

There are other concepts of federalism that help describe the complicated
relationships between the national and state governments. Judicial federalism
involves the struggle between the national and state governments over the relative
constitutional powers of each, and over key constitutional provisions including the
Bill of Rights and the Fourteenth Amendment. With its power of judicial review, the
Supreme Court is the arbiter of what the Constitution means on various questions,
including federalism. Chief Justice John Marshall defended a nationalsupremacy
view of the Constitution in the 1819 case of McCulloch v. Maryland. In that case
the Supreme Court expanded the powers of Congress through a broad interpretation
of its “necessary and proper” powers, and reaffirmed national supremacy by striking
down Maryland’s attempt to tax the Bank of the U.S.

Not all judicial decisions favor national power. In the 1997 case, Printz v. United
States, for example, the court invalidated federal law that required local police to
conduct background checks on all gun purchasers. The court ruled that the law
violated the Tenth Amendment. Writing for the fivetofour majority, Justice Antonin
Scalia declared: “The Federal government may neither issue directives requiring the
states to address particular problems, nor command the states’ officers, or those of
their political subdivisions, to administer or enforce a Federal regulatory program….
Such commands are fundamentally incompatible with our constitutional system of
dual sovereignty.”

Fiscal federalism involves the offer of money from the national government to the
states in the form of grants to promote national ends such as public welfare,
environmental standards, and educational improvements. Until 19

, federal grants
were used only to support agricultural research and education. With the passage of
the Sixteenth Amendment in 19

, which legalized the federal income tax, the
national government gained a significant source of revenue that it used to shape
national policy in a variety of new policy areas.

Categorical grants, in which the national government provides money to the
states for specific purposes, became a major policy tool of the national government
during the New Deal era, and expanded rapidly during the 1960s’ Great Society. But
state and local officials began to criticize this method of national support because of
the costly application and implementation procedures. They also complained that it
was difficult to adapt the grants to local needs.

Beginning in the mid 1960s, block grants, which combined several categorical

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grants in broad policy areas into one general grant, became increasingly popular.
States prefer block grants because they allow state officials to adapt the grants to
their particular needs. Congress, however, is reluctant to use block grants because
they loosen Congress’s control over how the money is spent.

Revenue sharing was developed during the Nixon administration as a way to
provide monies to states with no strings attached. Using statistical formulas to
account for differences among states, the national government provided billions of
dollars to the states until the program was abolished in 1986.

There are several pros and cons associated with U.S.style federalism. Some
advantages include a greater degree of local autonomy, more avenues for citizens to
participate, and more checks and balances against concentrations of power. Some
disadvantages include increased complexity of government that can produce
duplication and inefficiency, and increased legal disputes between levels of
government.

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How Do Local Governments Decide on
Public Policy in Fiscal Federalism?
Tax vs. Expenditure Optimization

MARKO KOETHENBUERGER

CESIFO WORKING PAPER NO.

85
CATEGORY 1: PUBLIC FINANCE

SEPTEMBER 2008

An electronic version of the paper may be downloaded
• from the SSRN website: www.SSRN.com
• from the RePEc website: www.RePEc.org

• from the CESifo website: Twww.CESifo-group.org/wp T

CESifo Working Paper No. 2385

How Do Local Governments Decide on
Public Policy in Fiscal Federalism?
Tax vs. Expenditure Optimization

Abstract

Previous literature widely assumes that taxes are optimized in local public finance while
expenditures adjust residually. This paper endogenizes the choice of the optimization
variable. In particular, it analyzes how federal policy toward local governments influences the
way local governments decide on public policy. Unlike the presumption, the paper shows that
local governments may choose to optimize over expenditures. The result most notably
prevails when federal policy subsidizes local fiscal effort. The results offer a new perspective
of the efficiency implications of federal policy toward local governments and, thereby, enable
a more precise characterization of local government behaviour in fiscal federalism.

JEL Code: H7, H3, H1.

Keywords: tax vs. expenditure optimization, federalism, endogenous commitment, fiscal
incentives, policy interaction.

Marko Koethenbuerger

University of Vienna
Hohenstaufengasse 9

1010 Vienna
Austria

marko.koethenbuerger@univie.ac.at

First version: Aug 2007
This version: Aug 2008
Comments by Panu Poutvaara are gratefully acknowledged.

1 Introduction

Models of local public finance predominantly take taxes as being optimized while expenditures

are residually determined via the budget constraint. The view is one possible prescription of

how governments decide on fiscal policy. In making budgetary decisions, governments may

equally set expenditures optimally and let taxes adjust residually.1 Given the two options, a

natural question is why governments should prefer one or the other budgetary item as a policy

variable. A potential strategic motive is that each item differently influences the amount of

federal resources which flow to the jurisdiction. Incentives to attract federal transfers, either

intended or unintended by federal policy, are widespread in local public finance. Besides

responding to corrective grants to cash in on federal resources, local governments also adjust

their taxes in order to receive more formulaic equalizing transfer payments (Smart, 1998,

Buettner, 2006, and Egger et al., 2007). Similarly, local governments may well select inefficient

local policies to lure more discretionary federal transfers to the local budget, e.g., as part of

a bailout package (Wildasin, 1997, Qian and Roland, 1998, and Pettersson-Lidbom, 2008

Building on these insights, the goal of this paper is to analyze whether federal policy has

a bearing on the choice of the policy variable in local public finance. That is, we set up a

model where the choice of the policy variable is not imposed, but arises endogenously from

the fundamentals of the fiscal architecture of the federation. In so doing, we consider two

models of fiscal federalism: a model of formula-based equalization and a model of ex-post

federal policy. In either model local governments levy a tax on local residents and use the

proceeds along with federal transfers to provide a public good.

A presumption might be that expenditure and tax optimization yield identical policy

outcomes since taxes and expenditures are inherently related via the budget constraint. Th

presumption holds true when local economies are fiscally independent, i.e., when there exists

no fiscal interaction between policy choices by different local governments. The bias towards

tax optimization in the existing literature is innocuous in such a fiscal environment. We show

that the equivalence between tax and expenditure policy becomes invalid if local policies are

linked via transfer programmes. Key to the result is that tax and expenditure policy have

different effects on transfer payments and local governments thus strategically choose their
1Throughout the paper we interchangeably refer to tax (expenditure) optimization as tax (expenditure)

policy and to the optimization variable as policy variable.

policy variable in order to gain in transfers. To illustrate the incentives involved in the

equilibrium choice of policy variables, consider a two-state federation in which interstate

transfers are only conditioned on state taxes and a rise in one state’s tax rate lowers transfer

income in the tax-raising state and lures additional transfers to the neighboring state. Were

taxes optimized by the neighboring state, public expenditures would adjust residually to

the rise in transfer income with no consequences for interstate transfers. With expenditure

optimization, however, taxes decrease residually and this response lures more transfers to

the neighboring state, financed by a cutback of transfers to the tax-raising state. Given

the negative fiscal repercussion, the cost of raising taxes turns out to be higher when the

neighboring state sets expenditures instead of taxes.

Playing on this effect, state governments strategically influence the neighbor state’s cost

of taxation by its own choice of policy variable. To see the equilibrium implications, assume

that both states initially optimize over taxes and that one state (let’s say state 1) consid-

ers deviating to expenditure optimization. The deviation increases the cost of public good

provision in state 2 which induces the state to decrease its tax rate. The fiscal adjustment

lures more transfers to state 2, financed by a cutback in transfers to state 1. The transfer

retrenchment eliminates any incentive to deviate and, in equilibrium, governments set taxes

optimally and let expenditures adjust residually. A reversed type of reasoning applies when

the transfer scheme exerts a positive incentive effect on state policy.2 A deviation to expendi-

ture setting lures more transfers to the deviant state’s budget, and states choose to optimize

over expenditures in equilibrium.

A straightforward question relates to the sensitivity of the results to the channel through

which state policy interacts. To infer into it, we consider a different model of fiscal federalism

frequently invoked in the literature. In particular, we allow states to “see through” the federal

tax policy decision which opens up a second source of fiscal interaction.3 State policy not only

affects the amount of transfers, but also influences the amount of taxes state residents pay

to the federal government. In this setting, the divergent interaction of tax and expenditure

policy implies that states may not optimize over taxes even though transfers discourage state
2For instance, transfer schemes which equalize fiscal capacities are one type of transfers which exert a

positive incentive effect – e.g. Smart (1998). Transfer schemes which build on the notion of fiscal capacity
equalization are implemented in, e.g., Australia, Canada, Germany and Switzerland.

3The type of “seeing through” delineates a game of decentralized leadership – see, e.g., Wildasin (1997),
Qian and Roland (1998), Caplan et al. (2000), Akai and Sato (2005) and Koethenbuerger (2007).

fiscal effort. More precisely, provided the disincentive effect of transfers policy is sufficiently

pronounced, the equilibrium choices turn out to be asymmetric in the sense that one state

chooses to optimize over taxes, whilst the other state optimizes over expenditures.

The analysis allows for a more informed prediction as to the efficiency of pubic good

provision in fiscal federalism. For instance, when transfers encourage local taxation, the

prediction of both models considered is that state governments optimize over expenditures.

Public good provision is more severely downward distorted than when taxes are optimized

(as widely assumed in the literature). However, when transfers undermine taxing incentives,

the efficiency prediction as perceived in the literature turns out to be consistent with the

equilibrium choice in the first model considered in the paper. With ex-post federal tax policy,

the equilibrium only entails tax policy setting if the disincentive effect of transfer policy is

not too pronounced. Otherwise, one state sets expenditures and public good provision is

higher (either less underprovision or more-severe overprovision) relative to the widely held

conjecture in the literature.

The results are of relevance for the design of corrective policy. The prediction as to

the magnitude and even as to the sign of the inefficiency generically differs in models with

(exogenous) tax optimization and with an endogenous selection of policy variables, and so

does the appropriate matching component of the Pigouvian grant. Also, the analysis offers a

more nuanced perspective of the effects of federal policy on local public finance. For instance,

a reform of the federal transfer formula, which leaves more own-source tax revenues to local

governments, may not necessarily promote local spending incentives. Fixing the initial choice

of policy variables, public expenditures will indeed rise in response to the reform. However,

phasing in the endogeneity of the policy variables, the fiscal response may entail a reduction

in local public spending.

While the notion that expenditure and tax policy have different implications for local

public finance is well established in the literature, it lacks (to the best of our knowledge)

an analysis of when local governments opt for one or the other type of policy setting. In

particular, Wildasin (1988) and Bayindir-Upmann (1998) contrast expenditure and tax policy

in the presence of capital mobility among jurisdictions. Hindriks (1999) compares transfer

and tax competition when households are mobile. The papers do not endogenize the choice of

policy instruments over which local governments compete in fiscal competition. More related

to the present paper, Akai and Sato (2005) contrast expenditure and tax policy setting in

a two-tier federal system in which the federal government provides transfers ex-post. The

choice of the optimization variable is exogenous to the analysis.

Although this paper formally abstracts from tax base mobility, it is helpful in predicting

which policy scenario can be sustained as an equilibrium choice in models with tax base mo-

bility. For instance, applying the methodology of the paper to the setting in Wildasin (1988),

in which capital mobility is the only fiscal linkage between states, reveals that competition

over taxes is the equilibrium choice. The result is supportive for almost all papers on capital

tax competition to date which assume that states compete over taxes and expenditures adjust

residually.

Finally, the choice of optimization variable determines with which policy variable state

governments commit toward other states’ fiscal policy.4 The endogenous choice of com-

mitment relates the paper to the Industrial Organization literature on endogenous timing

of moves, and hence commitment, in models of firm competition (e.g., Van Damme and

Hurkens, 1999, and Caruana and Einav, 2008). Therein, the sequence of decisions is deter-

mined endogenously, while the choice of optimization variables is exogenous. In this paper

it is reversed: the sequence of moves is exogenous while the choice of optimization variables

(for state governments) is endogenous.5

The outline of the paper is as follows. Section 2 introduces a model of formula-based

equalization and Section 3 characterizes the choice of the optimization variable. Section 4

extends the basic model by allowing for ex-post federal tax policy, i.e. local governments

have the capacity to “see through” the federal choice of taxes, and characterizes the selection

of policy variables. Section 5 summarizes and concludes.

2 Model

Consider two states which may differ with respect to preferences and endowments. The rep-

resentative household in state i (i = 1, 2) derives utility from private and public consumption,

ci and gi, according to the utility function U i(c, g) = ui(c) + ui(g) where uik > 0 and u
i
kk < 0,

4Optimizing over, e.g., the tax rate implies that other states perceive the tax rate to be held fixed, and
thus to be pre-committed, when they choose their policy simultaneously.

5Note, the two types of commitment, i.e. the sequencing of moves and the choice of optimization variables,
are not equivalent. The former relates to sequential games while the latter already exists in simultaneous move
games. Also, with the former type it is the best response of players which determines the value of commitment.
With the latter it is the residual variation of fiscal variables (determined by the budget constraint rather than
first-order conditions) which is primarily decisive for the choice of the commitment strategy.

k = c, g.6,7 Households have an endowment Ii which is subject to taxation. The private

budget constraint is

ci = Ii − T i, (1)

where T i ∈ [0, Ii] are taxes levied by state government i. State governments finance public
expenditures {gi}i=1,2 by locally collected taxes {T i}i=1,2 and interstate transfers {zi}

i=1,2

gi = T i + zi. (2)

The transfer to state i, zi, is conditioned on the level of locally collected tax revenues T i and

tax revenues of the neighbor jurisdiction T j ,

zi = γ(T i, T j ), where zi + zj = 0. (3)

Given the generality of the transfer formula, we impose three reasonable assumptions:

∣∣zi

T i

∣∣ < 1, sign{ziT i} = sign{z

T j

}, and sign{ziT i} = const.

First,
∣∣zi

T i

∣∣ < 1 such that changes in taxes do not imply an over-proportional change in transfers.8 The marginal tax or subsidy on own-source tax revenues hence does not exceed

100 percent. Second, states are symmetrically treated by transfer policy in the sense that

sign{

zi
T i

} = sign{zj

T j
}. The transfer formula may still be non-linear in taxes and, thereby,

the slope {zi
T i
}i=1,2 may differ in magnitude over the range of feasible taxes. Third, sign{ziT i}

is non-reversal, i.e., it is the same for all feasible levels of taxes.

The transfer scheme (3) embeds different types of formulaic transfers which most notably

differ w.r.t. the sign of the transfer response zi
T i

. As an example, transfers which share

locally collected tax revenues across states typically respond negatively to a rise in own-

source tax revenues (e.g., Baretti et al., 2002), whilst fiscal capacity equalization transfers

rise in response to a hike in own-state tax rates (Smart, 1998).9

State governments are benevolent and maximize utility of the representative household,

while the federal government maximizes the sum of utilities.
6Additive preferences are without loss of generality in Section 3. In Section 4 the identified equilibria

extend to non-additive preferences when private and public consumption are complements (U icg < 0) or weak substitutes, i.e. U icg is not too positive.

7As long as confusion cannot arise we omit the state-specific superscript for consumption levels.
8Subscripts denote partial derivatives throughout.
9To firmly model fiscal capacity equalization transfers we would have to introduce a tax-sensitive tax base.

One possibility is to allow for endogenous labor supply which negatively responds to higher state taxes on
labor – e.g. Smart (1998). The extension would complicate the exposition of the paper, without affecting the
main results of the analysis.

Straightforwardly, the (first-best) efficient public expenditure level in state i satisfies

uig
uic

= 1. (4)

The marginal rate of substitution between public and private consumption has to equal the

social marginal rate of transformation (normalized at unity).

Unless otherwise stated, the sequence of fiscal decisions is:

Stage 1: States simultaneously choose whether to optimize over taxes or expenditures.

Stage 2: States simultaneously optimize over the policy variable chosen at the first stage.

Stage 3: Transfers {zi}i=1,2 are paid, taxes {T i}i=1,2 are collected, and households con-
sume {ci, gi}i=1,2.

We solve for the subgame-perfect equilibrium (in pure strategies) by applying backward

induction.

3 Equilibrium Analysis

To isolate the incentive effects inherent to federal policy, it is instructive to first characterize

local decision-making in the absence of transfers, i.e. zi ≡ 0. In this case changes in taxes
yield a one-to-one change in expenditure levels. The tax price of marginal public spending is

unity irrespectively of whether the change in taxes is fixed and expenditures adjust residually

or vice versa. Thus, solving

max U i(c, g) s.t. Eqs. (1) and (2) (5)

either by differentiating w.r.t. T i (with gi being residually determined) or w.r.t. gi (with T i

being residually determined) yields the first-order condition

uig
uic

= 1. (6)

Public goods are efficiently provided. Noting (1), (2), and (6), optimal state policy is inde-

pendent of the neighbor state’s policy and so is utility in each state. The implications for the

equilibrium choice of policy variables at stage 1 of the game are straightforward. Given the

absence of fiscal interaction between states and the equivalence between tax and expenditure

optimization within a state, we conclude that any choice of policy variable yields the same

level of state utility and, hence, is an equilibrium of the policy selection game.

Proposition 1: In the absence of transfers (zi ≡ 0) any pair of policy variables selected
by state governments is a subgame-perfect equilibrium of the policy selection game.

We now re-introduce federal transfer policy and analyze the extent to which the equiva-

lence result in Proposition 1 is preserved. Consider first that state government j optimizes

over taxes T j . Optimal policy in state i follows from

max U i(c, g) s.t. Eqs. (1), (2) and (3), (7)

taking T j as given. Differentiating w.r.t. T i or gi gives

uig
uic

1 + zi
T i

. (8)

The equivalence between tax and expenditure policy is formally shown in the Appendix.

When zi
T i

< 0 (> 0) state i anticipates a loss (gain) in transfers in response to a rise in

taxes with the consequence that public goods are underprovided (overprovided) relative to

the first-best solution (4).

When state j optimizes over expenditures, its taxes adjust residually to a change in trans-

fer income which in turn affects state i’s transfer payment. State i realizes the fiscal feedback

mediated via the transfer system and, hence, perceives transfer income to be implicitly given

zi∗ = γ(T i, gj − zj∗), (9)

which follows from inserting (2) into (3), to substitute T j by gj − zj∗.

State i solves

max U i(c, g) s.t. Eqs. (1), (2) and (9), (10)

taking gj as given. Differentiating w.r.t. T i or gi, the optimal policy satisfies (see the

Appendix)

uig
uic
=
1

1 + zi∗
T i

. (11)

Straightforwardly, public consumption is underprovided (overprovided) relative to the first-

best rule when zi∗
T i

< 0 (> 0).

Comparing (8) and (11) reveals that state i’s policy is inefficient for any choice of opti-

mization variable by the neighboring state, but the scope of inefficiency depends on which

variable the neighboring state optimizes. Implicitly differentiating (9) and using the self-

financing requirement zi
T j

= −zj
T j

yields

zi∗T i =
zi
T i

+ zj
T j

⇒ zi∗T i < ziT i . (

)

The tax-induced change in transfers is more favorable for state i (either more inflow or less

outflow) when state j optimizes over taxes.10 The rationale is that tax and expenditure

policy interact differently through the transfer scheme. More precisely, assume higher taxes

reduce entitlement payments, zi
T i

< 0. A rise in state i’s taxes reduces transfers to state i

and, in order to balance the budget, increases transfers to state j. When state j optimizes

over taxes, the additional transfer income increases public expenditures in state j. Transfer

payments are unaffected by the residual adjustment. Differently, when state j optimizes over

expenditures, the rise in transfer income reduces state i taxes. In response, more transfers

go to state j, financed by a further cutback of transfers to state i – hence (12) follows. The

negative repercussion on state i’s budget renders public good provision more costly. Analo-

gously, when zi
T i

> 0 a higher tax in state i decreases state j’s transfer income. When state

j optimizes over expenditures, state j’s taxes rise residually which yields a budget-balancing

retrenchment of state i’s transfer income. The retrenchment dilutes state i’s incentives to

spend on public consumption. We can thus summarize:

Lemma 1: State i’s incentives to provide public goods are less pronounced when state j

optimizes over expenditures rather than taxes. In particular, provided zi
T i

< 0 (> 0) public

goods are more severely underprovided (less severely overprovided) in state i when public ex-

penditures rather than taxes are subject to optimization in state j.

We will now turn to the subgame-perfect choice of the optimization variable at stage 1

of the game. This involves a comparison of stage-2 utilities for any possible combination of
10Thus, in case zi

T i
< 0 the first-order condition (11) only holds provided zi

T i
is not too negative. Otherwise,

state i will select a zero tax rate. To save on notation we abstract from corner solution in what follows.

optimization variables by both states. Utility in the different stage-2 games may differ because

the tax price in the neighboring state depends on the own choice of policy variable. States

understand the effect and are inclined to manipulate the neighbor state’s policy in order to

qualify for more transfers. Before proceeding, note that in any stage-2 subgame, the states’

best responses are implicitly defined by the first-order conditions (8) and (11), respectively.

Equilibrium existence follows from standard fixed point theorems. We assume uniqueness

and stability of the stage-2 equilibrium throughout.11 Equilibrium stability implies that a

change in a state’s tax price translates into a change in the stage-2 equilibrium tax to the

opposite sign; an implication which simplifies the analytical exposition in what follows.

We will first analyze the case of zi
T i

< 0. Consider both states initially optimize over taxes.

When state i deviates from tax to expenditure optimization, state i’s public good level, and

thus taxes, stay the same unless state j changes its policy in response.12 In fact, state j faces

a distinct interaction of policy variables via the transfer scheme when state i optimizes over

gi rather than T i. As observed in connection with Lemma 1, the tax price of marginal public

expenditures rises and T j will be set at a lower level in the ensuing equilibrium. To infer the

induced change in state i’s well-being, we compute state i’s utility when it optimally sets its

policy for given state j’s policy. To this end, let’s define vi(T j ) as the utility evaluated at

taxes T i satisfying the optimality condition (8) for given T j .

Invoking the envelope theorem

and rearranging (all these steps are relegated to the Appendix), the change in utility vi(T j )

in response to a hike in state j’s taxes is

dvi(T j )
dT j

= −uigziT i . (13)

Since T j drops following the deviation, state i experiences a loss in transfers and thereby

in utility. By symmetry of policy incentives, neither state has an incentive to switch to

expenditure optimization given that the neighbor state optimizes over taxes. A reversed type

of argument applies when zi
T i

> 0 with the consequence that state i’s utility increases when

setting expenditures rather than taxes.
11In particular, we need to impose global stability rather than local stability since changes in the policy

variable at stage 1 trigger discrete changes in states’ best responses at stage 2. See, e.g., Vives (2000) for a
formal definition of global and local stability.

12Recall, for a given level of taxes in state j, state i’s utility is independent of whether it optimizes taxes or
expenditures – see (8).

13Given the equivalence between tax and expenditure optimization by state i, vi(T j ) applies prior and after
state i’s deviation.

state i’s best-response

state j’s best-response

Ti
Tj
state i’s best-response
state j’s best-response
A
B

Figure 1: Best Responses for zi
T i

< 0 (left panel) and zi T i

> 0 (right panel).

To graphically illustrate state i’s best response at stage 1, consider zi
T iT j

≡ 0. In this
case it is straightforward to show that states’ taxes are unambiguously strategic substitutes

(complements) if zi
T i

< 0 (> 0). The left panel in Figure 1 depicts the best responses by both

states when zi
T i

< 0. Point A is the initially prevailing equilibrium of the stage-2 game.

The

switch to expenditure policy by state i shifts state j’s best-response function inwards. The lo-

cus of state i’s best-response function stays the same (implicitly defined by (8)), and the new

tax choices are illustrated by point B. From (13), the change in state i’s utility, when moving

along state i’s best-response function from A to B, is negative. Hence, state i becomes worse

off due to the deviation. The right panel in Figure 1 depicts both states’ best responses for

zi
T i

> 0. The initial equilibrium of the stage-2 subgame is point A. Following state i’s devia-

tion to expenditure policy state j’s tax price rises, and state j’s best-response function shifts

inwards. Taxes T j decrease (see point B). Following (13), state i’s utility rises when moving

along state i’s best response from the initial equilibrium A to the new equilibrium B. In sum:

Lemma 2: Assume that state j optimizes over taxes. State i’s best response is to opti-

mize over taxes (expenditures) iff zi
T i

< 0 (> 0).

Suppose now that state j initially optimizes over gj and state i over T i. From Lemma 1

we observe that a deviation to expenditure optimization by state i increases state j’s tax price

which incentivizes state j to spend less on public consumption. Defining vi(gj ) as state i’s
14For simplicity, best responses are drawn as linear functions.

utility evaluated at public policy which satisfies the optimality condition (11), the envelope

theorem implies (see the Appendix)

dvi(gj )
dgj

= −uig
zi
T i

1 + zj
T j

. (14)

As the drop in state j’s expenditures translates into a lower tax, state i’s transfer income

and thus utility drops (rises) in response to the lower tax in state j when zi
T i

< 0 (> 0).15

Lemma 3: Assume that state j optimizes over expenditures. State i’s best response is to

optimize over taxes (expenditures) iff zi
T i

< 0 (> 0).

Combining Lemma 2 and 3, we conclude for the case zi
T i

< 0 that state i loses in utility

when optimizing over expenditures instead of taxes. The result holds irrespective of whether

state j optimizes over taxes or expenditures. Hence, in the subgame perfect equilibrium of the

policy selection game both states optimize over taxes. When zj
T j

> 0 state i gains in utility

when optimizing over expenditures rather than taxes. Again, the finding holds irrespective

of whether state j optimizes over taxes or expenditures. Consequently, the subgame perfect

equilibrium of the policy selection game entails states to optimize over expenditures.

Proposition 2: The subgame perfect equilibrium of the policy selection game entails tax

(expenditure) optimization when transfers undermine (strengthen) state fiscal incentives, i.e.,

zi
T i
< 0 (> 0).

States compete for transfers and choose the optimization variable strategically so as to

lure more funds to the public budget. Although the results have been derived in the absence of

resource mobility, the logic underlying the results equally applies when states do not compete

for transfers, but for mobile resources. To be more precise, consider a model of symmetric

capital tax competition in which states tax capital at source and decide on the tax system

to attract more capital (Zodrow and Mieszkowski, 1986, and Wildasin, 1988). Therein, the

interstate flow of resources is in capital rather than in transfers, and the formal analog to

the tax-induced flow of transfers, zi
T i

, is how the capital tax base responds to own-state tax

15The diagrammatic exposition of the choice of policy variables is analogous to Figure 1. We hence refrain
from a graphical illustration (as we do for all other games analyzed in the sequel).

hikes. As governments spend tax revenues on a public consumption good, which in itself has

no bearing on the return to capital, the tax base response is negative. The conclusion is that

states choose to compete for mobile capital by optimizing over taxes.16 A formal proof of the

result is relegated to Appendix B.

4 Ex-Post Federal Tax Policy and Formula-Based Transfers

In the sequel we analyze the robustness of the results derived in the last section. In so doing,

we resort to an alternative, frequently invoked model of fiscal federalism in which the federal

government has taxing authority, but cannot commit to tax policy, i.e., it sets federal policy

after states have determined their policy. This type of vertical interaction is referred to as

decentralized leadership and lies at the root of the soft budget constraint syndrome in fiscal

federalism – see, e.g., Wildasin (1997), Qian and Roland (1998), Caplan et al. (2000), Akai

and Sato (2005) and Koethenbuerger (2007).17 In particular, consider the federal government

has access to taxes {ti}i=1,2.18 The budget constraint of the household in state i becomes

ci = Ii − T i − ti. (15)

For simplicity, we retain the assumption that transfers are budget-balancing, zi + zj = 0.

The federal budget constraints thus read

t1 + t2 = 0 and z1 + z2 = 0. (16)

Federal taxes redistribute private income across states, while transfers redistribute public

funds across states. The budgetary dichotomy simplifies the analysis without affecting the

qualitative insights.19

The sequence of decisions becomes:

16In particular, when both states initially optimize over taxes, a deviation to expenditure optimization by
one state leads to lower capital taxes in the neighboring state and, thereby, to an outflow of capital in the
deviant state. Anticipating the negative effect on own-state revenues, states choose to compete over taxes.

17See Kornai et al. (2003) and Vigneault (2007) for a review of the related literature.
18A non-uniform tax scheme can be implemented by a uniform federal tax scheme and state-specific subsidies

to address disparities in private consumption. Thus, the net tax revenue the federal government collects in each
state differs (as allowed for here). With the richer set of instruments, we could allow the federal government
to pre-commit toward the common tax scheme and to set subsidies ex-post, i.e., after state governments have
moved (as, e.g., in Caplan et al., 2000). Importantly, the results of the analysis would be preserved.

19The identified equilibria of the policy selection game equally exist in the absence of the assumption. Details
of the calculations are available upon request.

Stage 1: States simultaneously choose whether to optimize over taxes or expenditures.
Stage 2: States simultaneously optimize over the policy variable chosen at the first stage.

Stage 3: The federal government selects {ti}i=1,2 for given state policy.
Stage 4: Transfers {zi}i=1,2 are paid, taxes {ti, T i}i=1,2 are collected, and households

consume {ci, gi}i=1,2.

Solving backwards, the federal government solves

max
{ti}i=1,2

∑

i=1,2

ui(c, g) s.t. Eqs. (2), (3), (15) and (16), (17)

taking the states’ policy choices as given. In so doing, the federal government sets taxes in

order to equalize the marginal utility of private consumption, i.e.

uic = u
j
c, i 6= j. (18)

At stage 2, state government i anticipates the effect its policy has on the federal govern-

ment’s choice of tax rates. Assume first that both states optimize over taxes. Differentiating

the federal first-order condition (18) and the federal budget constraint w.r.t. ti, tj , and T i and

inserting −T i
ti

= T i
tj

in the differentiated first-order condition, to eliminate the tj derivative,

yields

tiT i = −
uicc

uicc +

j
cc

∈ (−1, 0). (19)

More locally collected tax revenues, T i, reduce private consumption in state i. To equalize

the marginal utility of consumption across states (see (18)), the federal government reduces

the federal tax rate ti.

Replacing (1) by (15) in state i’s optimization problem (7) and additionally taking (16)

and (19) into account, public good provision satisfies

uig
uic
=

(
1 − u

i
cc

uicc +

u
j
cc

)
1

1 + zi
T i

. (20)

The first-order condition equally applies when state i optimizes w.r.t. gi rather than T i (see

the Appendix). Evident from (20), the federal tax-transfer policy influences the perceived

tax price of marginal public expenditures in two ways: ex-post federal tax policy provides a

subsidy on state i’s taxing effort as depicted by the bracketed term, while the transfer scheme

imposes a tax (subsidy) on state i’s taxing effort when zi
T i

< 0 (> 0). Federal tax-transfer

policy hence renders public good provision inefficiently high or low.

Differently, consider state i still optimizes w.r.t. T i, but conjectures state j to set expen-

diture levels optimally. Starting at (18), iterating the same steps involved in deriving (19)

(now differentiating w.r.t. gj rather than T j ) and noting (9), the marginal adjustment in ti

following a rise in T i is

tiT i = −
uicc

uicc + u
j
cc

−
u

j
ccz

j∗
T i

uicc + u
j
cc

. (

)

The first term coincides with (19); representing a subsidy on state i’s marginal tax revenues.

As to the second term, zj∗ denotes transfer payments for state j when state i conjectures

state j to optimize over expenditures as defined in (9). A rise in state i’s taxes translates into

a change in entitlement payments which depends on the sign of zj∗
T i

. As state j’s taxes adjust

residually to the change in transfers, its private consumption rises (drops) when zj∗
T i

> 0

(< 0). To restore (18), the federal government offsets the imbalance in private consumption

by reducing (increasing) ti.

Substituting (1) by (15) in state i’s optimization problem (10) and additionally taking

(16) and (21) into account, state i chooses a level of public goods which satisfies

uig
uic

= 1 − u
i
cc

uicc + u
j
cc

. (

)

The first-order condition holds irrespectively of whether state i optimizes over taxes or ex-

penditures (see the Appendix). It might be intuitive that state i chooses an inefficient policy

in the pursuit of a favorable treatment under federal policy. What might be less intuitive

is that the tax price of public expenditures is independent of how federal transfers respond

to state policy and, thus, that the inefficiency is only related to the federal tax response.

The rationale is that the adjustment in federal taxes insulates state governments from the

incentive effects of transfers; albeit the federal government aligns the marginal utility of pri-

vate consumption which is not directly affected by transfer payments. However, when a rise

in state i’s tax, for instance, lowers state i’s entitlement payments, the induced inflow of

transfers in state j reduces the amount of taxes state j levies on its residents. Private con-

sumption in state j consequently rises and the federal government offsets the imbalance in

private consumption across states by lowering state i’s tax as captured by the second term

in (21). The positive effect on state i’s utility turns out to be proportional to the negative

effect of transfers, thereby nullifying the latter. In sum, ex-post federal tax policy neutralizes

the incentive effect of transfer policy and subsidizes state-financed spending. Consequently,

the state i’s tax price is unambiguously below the social price.

Lemma 4: Public good provision in state i is either inefficiently high or low under tax

policy setting by state j. When state j sets expenditures public good provision in state i is

inefficiently high. In particular, provided zi
T i

< 0 (> 0) state i’s incentives to spend on public

goods are more pronounced (weaker) when state j optimizes over expenditures instead of taxes.

The divergence of policy incentives is due to the fact that state policy instruments in-

teract differently through federal tax-transfer policy. To illustrate the interaction, suppose a

higher tax rate in state i leads to more transfers in state j, i.e. zi
T i

< 0. Under tax policy

setting by state j, the residual adjustment in public expenditures does not spill back through

federal tax-transfer policy to state i. Transfers are formally conditioned only on taxes and

federal tax policy seeks to align private consumption levels. In contrast, when state j sets

expenditures, the higher transfer income lowers state j’s tax rate (residually determined).

Private consumption cj rises which yields a lower federal tax rate in state i – a repercussion

which strengthens state i’s fiscal incentives.20

To infer how federal tax-transfer policy influences the choice of the optimization variable

at stage 1 of the game, we proceed by characterizing incentives of each state to unilaterally

deviate from a given combination of policy variables. As in the previous section, we will

assume the stage-2 equilibrium to be unique and globally stable throughout. Assume first that

states initially optimize over taxes and that transfers undermine taxing incentives

(zi
T i

< 0).

As inferred from Lemma 1, a change to expenditure policy on the part of state i reduces state

j’s tax price, and the new equilibrium entails a higher tax T j . The impact on state i’s utility

is evaluated using state i’s best-response function as implied by the first-order condition (20).

Define vi(T j ) as the associated utility level for a given level of T j . Applying the envelope

theorem, deriving the tax-transfer responses and rearranging (all the steps are relegated to

the Appendix), we get

dvi(T j )
dT j

= −uic
1 + zi

T i
+ zj

T j
1 + zi
T i
u
j
cc
uicc + u
j
cc

. (23)

20The associated transfer changes are absorbed by the federal tax response – see (22).

Provided zi
T i

+ zj
T j

< −1 (∈ (−1, 0)), utility rises (drops) following state i’s change in the optimization variable.21 The intuition is that both the federal tax ti and the transfer payment

zi rise in response to state j’s tax hike, exerting counteracting effects on state i’s utility. The

disincentive effect of transfer policy amplifies the positive transfer response in two ways:

First, the stronger the disincentive effect zj
T j

the larger the inflow of transfers to state i in

response to a higher tax rate T j . Second, the more pronounced zi
T i

the higher the inflow of

transfers to state i which results from the residual reduction in its own tax rate subsequent

to the first-round inflow of transfers. If the joint effect is sufficiently strong (as measured by

zi
T i
+ zj
T j

) the positive transfer response dominates the rise in federal taxes ti.

The result changes when transfers promote taxing incentives (zi
T i

> 0). In this case, state

j decreases its tax rate in response to state i’s deviation from tax to expenditure optimiza-

tion which exerts two unidirectional effects on state i’s utility: the federal tax ti decreases,

to level off differences in private consumption, and transfers increase. In response, state i

unambiguously enjoys a higher level of utility in the new equilibrium.

Lemma 5: Consider state j optimizes over taxes. State i has an incentive to deviate

from tax to expenditure policy if either zi
T i

> 0 or zi
T i

< 0 and zi T i

+ zj
T j

< −1.

Finally, assume state j optimizes over expenditures. Provided transfers discourage state

fiscal effort, a switch to expenditure optimization by state i increases expenditures, and thus

taxes, in state j. To unravel the impact on state i’s utility, define vi(gj ) as state i’s utility

evaluated at state i’s best-response function which follows from (22). Invoking the envelope

theorem, computing the responses in transfers and federal taxes, and rearranging yields (the

derivation is dealt with in the Appendix)

dvi(gj )
dgj

= −uic
u

j
cc
uicc + u
j
cc

< 0. (

)

The federal government taxes state i’s household at a higher rate in response to the deviation,

just to counteract the imbalance in private consumption which results from the higher tax

T j .22 Accordingly, state i becomes worse off. Following a related line of arguments, state i

becomes better off following the deviation when transfers encourage state fiscal effort.
21For expositional simplicity, we omit the special case zi

T i
+z

T j
= −1, in which dvi(T j )/dT j = 0, throughout.

22Note, the federal tax response absorbs the effect of transfer policy on state i – see (22).

Lemma 6: Consider state j optimizes over expenditures. State i has an incentive to

deviate from tax to expenditure optimization iff zi
T i

> 0.

What are the equilibrium implications of both Lemmata? When zi
T i

< 0, Lemma 6 rules

out expenditure optimization as an equilibrium since both states have an incentive to deviate.

Lemma 5 shows that tax optimization is an equilibrium if transfer responses zi
T i

+ zj
T j

are not

too pronounced. Otherwise, we observe an asymmetric equilibrium, i.e., one state optimizes

over taxes whilst the other state does so over expenditures. When zi
T i

> 0 Lemma 5 rules out

tax policy as an equilibrium outcome, and the only policy which is immune to a unilateral

deviation is expenditure optimization – see Lemma 6. To summarize,

Proposition 3: (i) When transfer policy undermines state fiscal incentives (zi
T i

< 0) the

subgame perfect equilibrium of the policy selection game entails both states to optimize over

taxes if the overall disincentive effect of transfer policy, as measured by zi
T i

+ zj
T j

< 0, is

sufficiently weak, i.e., zi
T i

+ zj
T j
∈ (−1, 0). Otherwise, states choose to optimize over different

policy variables in equilibrium. (ii) When transfer policy strengthens state fiscal incentives

(zi
T i

> 0) the subgame perfect equilibrium involves both states to optimize over expenditures.

Common to the analysis in Section 3, states choose their policy variable such that the

induced adjustment in fiscal policy of the neighboring state yields a favorable response in

federal policy. However, different to the preceding model, tax optimization is not the unique

equilibrium outcome when transfers discourage states from relying on own-source public

funds. In fact, states select themselves into different policy regimes if the disincentive effect

is sufficiently strong.23

The finding that states may not optimize over the same policy variable opens up the

possibility that the tax price of marginal public expenditures drops when transfer policy

exerts a stronger disincentive effect on state policy. For illustrative simplicity, assume the

transfer system (3) to be linear and zi
T i

= zj
T j

. A reform of the transfer formula such that

23The required strength of the disincentive effect is not implausibly large. Empirical estimates of the
“marginal tax” federal transfer programmes impose on own-source tax revenues may well exceed 0.5 (e.g.,
Baretti et al., 2002, and Zhuravskaya, 2000).

the transfer slope changes from a pre-reform level above -0.5 to a post-reform level below -0.5

leaves fewer own-source tax revenues to the state. Conditional on the initial choice of policy

variables (here taxes – see Lemma 6), incentives to spend on public goods are predicted

to be diluted (see (20)). Taking the endogeneity of the policy variable into account, one

state changes from tax to expenditure policy while the other state still engages in tax policy.

Implied by Lemma 4, spending incentives are strengthened in the non-switching state and,

as a consequence, its outlays on public consumption rise.

5 Conclusion

Previous literature predominantly assumes taxes to be optimized and expenditures to adjust

residually. The paper endogenizes the choice of policy variables by state governments and,

in particular, explores how federal policy toward state governments influences the choice.

Albeit the equilibrium choice turns out to be sensitive to the way federal tax-transfer policy

targets state policy, a common finding in both models considered is that governments choose

to optimize over expenditures when federal transfers subsidize state fiscal effort. The paper’s

results are of relevance for the design of corrective policy and for evaluating the impact of

federal tax-transfer schemes on the efficiency of state policy. Specifically, the conditional

response (i.e., for a given choice of policy variables) and the unconditional response (i.e.,

accounting for the adjustment of policy variables) of local policy to changes in federal policy

differ not only quantitatively, but may also differ qualitatively. A reform of the federal tax-

transfer system, which discourages state spending conditional on the choice of policy variable,

may in fact promote spending incentives when accounting for the endogeneity of the choice.

The main part of the paper builds on the assumption that state budgets are only fiscally

linked through federal policy. In practice, budgets are also fiscally connected through the

mobility of taxable resources such as capital or households. As shown for the canonical capital

tax competition model (see Appendix B), the logic underlying the results straightforwardly

applies to models with capital mobility. Having said this, we believe that the paper’s insights

will likewise provide a helpful starting point for analyzing the choice of policy variables in

different models of local public finance; e.g., nesting fiscal interaction through both tax base

mobility and federal policy.

Finally, a natural question is whether the strategic incentives pertaining to the choice of

policy variables may not be more multi-faceted than suggested in the paper. For instance,

decomposing the expenditure side of the budget into consumption outlays and infrastructure

investment, states are equally in a position to compete for transfers by optimizing over, e.g.,

consumption expenditures and taxes and by letting infrastructure spending adjust residually.

Which pair of policy variables ultimately constitutes an equilibrium choice and how the choice

influences the efficiency of the local public sector are interesting questions which are left to

future research.

A Appendix

A.1 Derivation of (8)

Tax policy A rise in state i’s tax rate dT i implies a change in transfers by zi
T i

dT i, and

public expenditures increase by dgi = (1+zi
T i

)dT i. The tax price of marginal public spending

is dT i/dgi = 1

/(1 + zi
T i

). Thus, the first-order condition is (8).

Expenditure policy Inserting (2) into (3), to express transfers to state i as an implicit

function of policy variables in state i, gi, and state j, T j , we have z̄i = γ(gi − z̄i, T j ) with
slope z̄i

gi
= zi

T i
/(1 + zi

T i
). A rise in expenditure by dgi now leads to a change in transfers

by z̄i
gi

dgi. In response to it, taxes have to increase by dT i = (1 − zi
gi

)dgi. Hence, 1 − zi
gi

the tax price of marginal public spending under expenditure optimization and the first-order

condition is
uig
uic

= 1 − z̄igi . (

)

Inserting the explicit form of the slope term z̄i
gi

into the tax price shows that it coincides

with the tax price under tax optimization. The first-order condition (25) is hence equivalent

to (8).

A.2 Derivation of (11)

Tax policy Given (9) a rise in taxes by dT i yields a change in transfers equal to zi∗
T i

dT i.

Expenditures residually rise by (1+zi∗
T i

)dT i which gives a tax price of marginal public spending

of 1/(1 + zi∗
T i

). The first-order condition thus becomes (11).

Expenditure policy Inserting (2) into (3), to express transfers to state i as an implicit

function of policy variables in state i and state j, gi and gj , z̃i = γ(gi − z̃i, gj − z̃j ). Now, a

rise in expenditures by dgi requires an adjustment in taxes at an amount (1 −

z̃i
gi

)dT i. The

tax price is 1 − z̃i
gi

and the first-order condition reads

uig
uic

= 1 − z̃igi . (26)

Implicit differentiation of z̃i and using z̃i
T j

= −z̃j
T j

, we obtain z̃i
gi

= zi
T i

/(1 + zi
T i
+ zj
T j
).

Inserting the slope term into (26) and inserting (12) into the first-order condition under tax

policy (11) shows that conditions (11) and (26) coincide.

A.3 Derivation of (13)

A rise in T j when state i sets expenditures yields a change in utility vi(T j ) of

dvi(T j )
dT j

= uicz
∗i
T j , (

)

where the term has been simplified by invoking the envelope theorem. Inserting (8) and (12)

into (27) and noting that transfer payments must balance the budget, the expression (27)

simplifies to

dvi(T j )

dT j
= −uigziT i . (28)

A.4 Derivation of (14)

Marginally increasing T j when state i optimizes over expenditures implies a change in utility

vi(gj ) which is equal to
dvi(gj )

dgj
= uicz̃

i
gj . (29)

z̃i = γ(gi − z̃i, gj − z̃j ) implicitly defines transfer payments to state i as a function of
expenditure levels in both states. Implicit differentiation and using z̃i

T j
= −z̃j

T j
yields

z̃i
gi
= zi
T i
/(1 + zi
T i
+ zj
T j

). Inserting the expression and (11) into (29) and noting that

transfers are self-financing
dvi(gj )

dgj
= −uig

zi
T i
1 + zj
T j

. (30)

A.5 Derivation of (20)

Tax policy Increasing the state tax by dT i, the combined effect on state and federal taxes,

T i + ti, is (1 + ti
T i

)dT i. Expenditures change by (1 + zi
T i

)dT i which yields a tax price equal

to (1 + ti
T i

)/(1 + zi
T i

). The first-order condition is (20).

Expenditure policy As a first step insert (2) into the federal first-order condition (18), to

substitute T i for gi − z̄i, where z̄i = γ
(
gi − z̄i, T j

)
. Differentiating the modified optimality

condition w.r.t. ti, tj , and gi yields the response ti
gi

= −uicc(1−z̄igi )/(uicc +u
j
cc). The change in

the combined tax burden to a rise in expenditures thus is [1−z̄i
gi
−uicc(1−z̄igi )/(uicc + u

j
cc)]dgi.

The bracketed term represents the tax price of marginal public expenditures. Hence,

uig
uic

= 1 − z̄igi − uicc(1 − z̄igi )/
(
uicc + u

j
cc

)
. (31)

Noting that z̄i
gi

= zi
T i

/
(
1 + zi

T i

)
, the first-order condition (31) equals the first-order condition

under tax policy (20).

A.6 Derivation of (22)

Tax policy Given (21) the effect of a rise in taxes by dT i on the combined tax burden

ti + T i is [ujcc(1 − zj∗T i )/(uicc + u
j
cc)]dT i. Expenditures change by (1 + zi∗T i )dT

i. Since transfers

are budget-balancing, zi∗
T i

= −zj∗
T i

the tax price of marginal public expenditures simplifies to

1 − uicc/(uicc + ujcc) and the resulting first-order condition is (22).

Expenditure policy Inserting (2) into (3), to express transfers to state i as an implicit

function of policy variables in state i and state j, gi and gj , z̃i = γ(gi − z̃i, gj − z̃j ). Next,
insert (2) into the federal first-order condition (18), to substitute T i and T j for gi − z̃i and
gj−z̃j . Differentiating the modified optimality condition w.r.t. ti, tj , and gi, while accounting
for (16), yields the response ti

gi
= −uicc/(uicc + ujcc) + z̃igi . Now, a rise in expenditures by dgi

yields a change in the tax burden of (1 + ti

gi
− z̃i

gi
)dgi. The tax price is 1 + ti

gi
− z̃i

gi
and

consequently, the first-order condition becomes

uig
uic

= 1 + tigi − z̃igi . (32)

Finally, inserting the tax response ti
gi

into (32) shows that the first-order conditions under

tax and expenditure optimization, (22) and (32), coincide.

A.7 Derivation of (23)

Invoking the envelope theorem, the change in utility when state i optimizes over expenditures

is
dvi(T j )
dT j

= −uic(tiT j + z∗iT j ). (33)

Noting, following (16), that ti
T j

= −tj
T j

and zi
T j

= −zj
T j

and inserting (12) and (19) into (33),

the utility change becomes

dvi(T j )
dT j
= −uic
1 + zi
T i
+ zj
T j
1 + zi
T i
u
j
cc
uicc + u
j
cc

. (34)

A.8 Derivation of (24)

When state i optimizes over expenditures, the envelope theorem implies

dvi(gj )
dgj

= −uic(tigj + z̃igj ). (35)

To characterize both federal responses, insert (2) into (3), to express transfers to state i as an

implicit function of policy variables in state i and state j, gi and gj , z̃i = γ(gi − z̃i, gj − z̃j ).
Next, insert (2) into the federal first-order condition (18), to substitute T i and T j for gi − z̃i

and gj − z̃j . Differentiating the modified federal optimality condition w.r.t. ti, tj , and gj ,
while accounting for (16), yields the response tj

gj
= −ujcc/(uicc +ujcc)+z̃jgj . Plugging tigj = −t

j
gj

into (35) and noting that transfers are self-financing yields

dvi(gj )
dgj
= −uic
u
j
cc
uicc + u
j
cc

< 0. (36)

B Appendix: Capital Tax Competition

Consider 2 symmetric states. In each state i (i = 1, 2), the representative household is en-

dowed with capital K and a fixed factor (e.g. inelastically supplied labor) which is normalized

to unity. The income of the household is given by ci = wi + rK where wi is the wage rate

in state i and r is the interest rate determined in the international capital market. House-

holds derive utility from private consumption ci and from a local public good gi. The utility

function is24

U (ci, gi) = u(ci) + u(gi), (37)

where uik > 0 and u
i
kk < 0, k = c, g. Regional output can be transformed on a one-to-one

basis either in a private good ci or a local public good gi. Output is produced using the

technology f (ki) which exhibits (i) constant returns to scale and (ii) a positive and declining
24We adopt the same utility function as in the paper. The additive structure is without loss of generality.

marginal productivity of capital. Firms are assumed to be profit-maximizer. Profits are given

πi = f (ki) − wi − (r + T i)ki. (38)

Capital employment ki is taxed at source at a rate T i. Firms maximize (38) with respect to

ki, which leads to the first-order condition

f ′(ki) = r + T i. (39)

(39) implicitly defines regional capital employment as a function of the tax rate T i and

the interest rate r. The wage rate wi equals f (ki) − f ′(ki)ki. Hence, noting (39), private
consumption is given by

ci = f (ki) + r(K − ki) − T iki. (40)

Capital is perfectly mobile across states and locates in the state which offers the highest

net-of-tax rate of return. The capital market equilibrium is characterized by the first-order

condition (39) and the capital market clearing condition

k1 + k2 = 2K. (41)

(39) and (41) define capital employment and the interest rate as a function of both states’

tax rates, ki(T i, T j ) and r(T i, T j ). The responses of ki and kj to a change in ti are given by

kiT i = ∆
−1 and kj

T i
= −∆−1 with ∆ := f ′′(ki) + f ′′(kj ) < 0. (42)

In each state tax revenues, T iki, are recycled by providing a public consumption good, gi,

whose price is normalized at unity:

gi = T iki. (43)

B.1 Tax Optimization by State j

Assume first that state i optimizes over taxes. State i solves

max
T i

U i(c, g) s.t. Eqs. (40), (42) and (43), (44)

taking T j as given. Differentiating w.r.t. T i the first-order condition is

uic(f
′(ki)kiT i + rT i (K − ki) − rkiT i − ki − T ikiT i ) + uig(T ikiT i + ki) = 0 (45)

Evaluated at a symmetric equilibrium (ki = K) and inserting (39) yields

−uicki + uig(T ikiT i + ki) = 0. (46)

Consider state i optimizes over expenditures. Inserting the public budget constraint into

ki(T i, T j ) to express capital employment and the interest rate as a function of expenditures

gi and taxes T j gives k̃i(gi/k̃i, T j ) and r̃(gi/k̃i, T j ). The slope of k̃i w.r.t. gi is

k̃igi =
ki

T i
/k̃i

1 +

ki
T i

gi/(k̃i)2
< 0. (47)

State i solves

max
gi

U i(c, g) s.t. Eqs. (40), (43) and (47), (48)

taking T j as given. Differentiating (48) w.r.t. gi gives

uic(f
′(k̃i)k̃igi + r̃gi (K − k̃i) − r̃k̃igi − 1) + uig = 0. (49)

Evaluated at a symmetric equilibrium (ki = K) and inserting (39), the first-order condition

simplifies to

uic(T
ik̃igi − 1) + uig = 0. (50)

Inserting (47) and noting (43), the first-order condition (50) reduces to (46). Thus, given

that state j optimizes over taxes, expenditure and tax optimization by state i yield identical

policy incentives.

B.2 Expenditure Optimization by State j

Consider state i optimizes over taxes. Inserting the public budget constraint by state j into

ki(T i, T j ) to express capital employment in state i and the interest rate as a function of taxes

T i and expenditures gj gives k∗i(T i, gj /k∗j ) and r∗(T i, gj /k∗j ). The slope of k∗i w.r.t. T i is

k∗iT i =
ki

T i

1 − ki
T j

gj /(k∗i)2
. (51)

State i solves
max
T i

U i(c, g) s.t. Eqs. (40), (43) and (51), (52)

taking gj as given. Differentiating (52) w.r.t. T i gives

uic(f
′(k∗i)k∗iT i + r

∗
T i (K − k∗i) − r∗k∗iT i − k∗i − T ik∗iT i ) + uig(T ik∗iT i + k∗i) = 0. (53)

Evaluated at a symmetric equilibrium (k∗i = K) and inserting (39) yields

uic(−k∗i) + uig(T ik∗iT i + k∗i) = 0. (54)

Finally, suppose state i optimizes over expenditures. Inserting the public budget con-

straint by state j into ki(T i, T j ) to express capital employment in state i and the interest rate

as a function of expenditures gi and expenditures gj gives k̄i(gi/k̄i, gj /k̄j ) and r̄(gi/k̄i, gj /k̄j ).

The slope of k̄i w.r.t. gi is

k̄igi =
ki

T i
/k̄i

1 + ki
T i

gi/(k̄i)2 − ki
T j

gj /(k̄j )2
. (55)

State i solves
max
gi

U i(c, g) s.t. Eqs. (40), (43) and (55), (56)

taking gj as given. Differentiating (56) w.r.t. gi

uic(f
′(k̄i)k̄igi + r̄gi (K − k̄i) − r̄k̄igi − 1) + uig = 0. (57)

Evaluated at a symmetric equilibrium (k̄i = K) and inserting (39) the first-order condition

reduces to

uic(T
ik̄igi − 1) + uig = 0. (58)

Inserting (55) into (58) and rearranging gives

−uic(ki + kiT i ) + uig(ki + T ikiT i − T j k
j
T j

) = 0. (59)

To compare the first-order condition (59) with the first-order condition (54), insert the capital

response (51) into (54) which reveals that both first-order conditions coincide. Thus, given

that state j optimizes over expenditures, expenditure and tax optimization by state i yield

identical policy incentives.

The analyze how policy incentives differ when state j sets expenditures rather than taxes,

we compare (46) and (54). In symmetric equilibrium (k∗i = ki) the conditions differ by the

response of capital to changes in the policy variable. Using the public budget constraint (43)

and the fact that the capital market clears, i.e. ki
T j

= −kj
T j

the response (51) becomes

k∗iT i =
ki
T i

1 + kj
T j

T j /kj
.

The capital response k∗i
T i

differs from ki
T i

by the term 1/(1 + kj
T j

T j /kj ) which is state j’s

tax price of marginal public expenditures when state j and state i optimize over taxes – see

(46). It exceeds unity and thus the capital response k∗i
T i

is larger in absolute value relative to

ki
T i

. We conclude that the tax price of marginal public expenditures is higher when state j

optimizes over expenditures rather than taxes.

Lemma A1: State i’s incentives to provide public goods are less pronounced when state

j optimizes over expenditures rather than capital taxes.

The intuition for the result is that following a rise in state i’s tax rate capital moves from

state i to state j. The inflow of capital leads to a rise in expenditures when state j sets taxes,

but leads to a reduction in taxes when state j sets expenditures. In the latter case even more

capital will move to state j in response to a tax hike in state i and, hence, state i’s tax price

of marginal public expenditures is higher. The finding is in line with Wildasin (1988).

B.3 Equilibrium Choice of Policy Variable

To infer into the choice of policy variable, we first characterize the best response of state i

given that state j optimizes over taxes. We assume the stage-2 equilibrium to be unique and

globally stable throughout. Consider state i initially optimizes over taxes and switches to

expenditure optimization. Invoking Lemma A1, the tax price of marginal public expenditures

in state j rises and, given global stability of the equilibrium, state j’s tax rate will be lower in

the new second-stage equilibrium. The effect on state i’s utility can be computed by defining

vi(T j ) as state i’s utility evaluated at state i’s policy which satisfies the first-order condition

(46). Using (39) the response in utility to a rise in state j’s tax rate is

dvi(T j )
dT j

= uicT
ik∗iT j . (60)

The sign of the utility change is positive since k∗i
T j

= −k∗j
T j

> 0 – see (51). Since T j decreases,

state i experiences a loss in utility when deviating. Thus,

Lemma A2: Assume that state j optimizes over capital taxes. State i’s best response is

to optimize over capital taxes.

Differently, assume state j optimizes over expenditures. When state i switches from

tax to expenditure optimization state j’s tax price of marginal public expenditures goes up.

Owing to global stability, state j’s level of expenditures and thus taxes will be lower in the

new symmetric equilibrium. Denoting state i’s utility evaluated at the policy satisfying the

first-order condition (54) by vi(gj ), the change in utility emanating from a rise in gj is

dvi(gj )
dgj

= uicT
ik̃igj . (61)

The term has already been simplified by using (39). Since gj is set at a lower level and,

following (47), k̃i
gj

= −k̃j
gj

> 0, state i’s utility drops following the deviation.

Lemma A3: Assume that state j optimizes over expenditures. State i’s best response is

to optimize over capital taxes.

Combining Lemma A2 and A3, state i has a dominant strategy: optimizing over taxes is

a best response irrespective of state j’s choice of policy variable. Consequently,

Proposition A1: The subgame-perfect equilibrium of the policy selection game entails

both states to optimize over capital taxes.

References

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CESifo Working Paper Series
for full list see Twww.cesifo-group.org/wp T
(address: Poschingerstr. 5, 81679 Munich, Germany, office@cesifo.de)

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