Discussion Papers In Economics
And Business

Inefficiency in fiscal policy:
A political economy of the Laffer curve
Yutaro Hatta
Discussion Paper 14-36

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANInefficiency in fiscal policy:
A political economy of the Laffer curve
Yutaro Hatta
Discussion Paper 14-36

December 2014

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANIneﬃciency in ﬁscal policy: A political economy of the Laﬀer
curve∗
Yutaro Hatta†

Abstract
This paper studies investment decisions by economic agents in cases where thetax rate is decided through voting. It will be shown that, in some cases, only aPareto-dominated tax policy on the wrong side of the Laﬀer curve is supported underrational expectations. Thus, the governments may collect revenue in an ineﬃcientway. To that end, a quite plausible assumption, the endogeneity of the return on
investment, is essential. Therefore this paper warns about the danger of ineﬃciency
in a wide variety of policies. Further, the model predicts that when the inequality
in an economy is low, the tax policy on the wrong side is likely to arise.

Keywords: Political economy; The Laﬀer curve; Ineﬃciency in ﬁscal policies
JEL Classification: E22, E62, H21∗
I am grateful to Tatsuro Iwaisako for his helpful comments and suggestions. The ﬁnancial support
from JSPS Research Fellowships for Young Scientists is greatly acknowledged. I am responsible for any
remaining errors.
†
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043,
Japan, Email address: mentonian.hatt@gmail.com

1

Introduction

It is believed that govenmental policy is responsible for economic development. Especially, the tax policy aﬀects citizens’ behaviors with regard to labor supply and capital
accumulation, and thereby determines the development level. Therefore, the institution
deﬁning how to decide economic policies is an important element in economic growth.
Recently, studies on political economy increasingly suggest that policies are not determined by a benevolent government, but that conﬂicts among self-interested agents are
somehow coordinated in a certain way and then policies determined. In this context, thisstudy uses a political economy model with investment decisions by citizens to show thatan ineﬃcient policy may emerge. Here, an ineﬃcient policy refers to the tax policy along
the wrong side of the Laﬀer curve, which is Pareto-dominated. Therefore, although thegovernment can raise the same level of revenue with a lower tax rate, it chooses a higher
tax rate and consequently suﬀers from less investment. The focus of the paper is to showthat such an ineﬃcient policy is a unique equilibrium.
The model used in this paper is a simple two-period general equilibrium model.
Individuals diﬀer in capital holdings at the beginning. They invest capital for next
period consumption, expecting the return on capital and tax rate chosen in the future.
In the model, the tax is levied only on capital income, which is politically determinedby majority voting. It is well known that endogenous policy determination coupled withinvestment decisions is likely to support multiple rational expectations on both sidesof the Laﬀer curve.1 Saint-Paul and Verdier (1997) analyze the case when a country
is subject to multiple equilibria and ascertain how these are sustained. They use amodel where agents have diﬀerent capabilities to access world capital markets. In theirmodel, multiple equilibria are likely to occur when the decisive voter on tax policy has
a higher elasticity for tax evasion than the average voter. However, their equilibria
consist of two scenarios: multiple equilibria with low and high taxes; and a uniquelow-tax equilibrium. The present paper oﬀers another scenario, that of a unique hightax equilibrium on the wrong side of the Laﬀer curve. For such analysis, this paperemphasizes that the endogeneity of capital return is essential for an ineﬃcient revenue
1

Persson and Tabellini (2000, Ch. 12) discuss the multiplicity problem, using a simple dynamic
taxation model.

1collection by the government.
As discussed above, the multiplicity of equilibria is pervasive in dynamic taxation
models with endogenous policy making. A Pareto-dominated equilibrium is often seen
as a coordination failure among economic agents. And this disastrous equilibrium issometimes ignored by assuming that economic agents can somehow coordinate their
expectations (Persson and Tabellini (1994)). However, it is shown in this paper that there
can be no room for coordinations and that the disastrous outcome may be inevitable.
Some literature in public economics show that the tax rate can be on the declining
portion of the Laﬀer curve (Yitzhaki (1987) and Slemrod and Yitzhaki(2002)). However,these studies diﬀer from this paper in the sense that the tax policy is exogenously given.
This paper is also related to Traxler (2012), who studies welfare consequences of thetax policy determined by majority voting in a model of tax avoidance. Traxler (2012)shows that the politically determined tax rate is generally ineﬃcient. However, theterminology “ineﬃcient” used in Traxler (2012) diﬀers from the one used in this paper.
Traxler (2012) says “ineﬃcient” when there is a divergence from the welfare-maximizing
policy according to a certain welfare function. On the other hand, this paper uses theterm “ineﬃcient” if the policy is worse for everyone, i.e. Pareto-dominated, compared toanother feasible policy. In fact, a Pareto-dominated tax policy is never an equilibrium inTraxler (2012). Buchanan and Lee (1982) explains how the tax rate on the wrong side of
the Laﬀer curve would be sustained with endogenous policy determination by politicaldecision makers. Their arguments depend on the assumption that the political decision
makers are more short-sighted than citizens. Short-sighted politicians do not take into
account the eﬀect that a high tax rate today distorts resource allocations in the future.
And they seek an increased government revenues with a high tax rate today, which isso high and beyond the peak of the long-run Laﬀer curve.2 The present paper provides
an alternative mechanism to end up with such a Pareto-dominated policy. Two criticalassumptions, but plausible, lead to ineﬃciency. The one is, as mentioned above, theendogeneity of the return on investment. The other is the timing of policymaking, or
2
Further, using a political contest model between politicians and citizens, Ihori and Yang (2012) pointout that a tax rate beyond the peak has an eﬀect to undermine political eﬀorts by citizens. And theyshow the possibility to have a tax rate on the right side of the Laﬀer curve. Wrede (2000) analyzes thecase where a number of governments put taxes on shared tax sources, which can result in overtaxationon the downward-sloping part of the Laﬀer curve.

2

put diﬀerently, a lack of commitment. Because the two conditions are common featuresin real-life economies, the model has a general applicability and can be extended toanalyze a wide range of policies.
For a positive analysis, the model predicts that when the inequality in an economy
is low, the tax policy on the wrong side of the Laﬀer curve is likely to occur. This result
is consistent with Trabandt and Uhlig (2011). They use a neoclassical growth model tocalibrate and show that Denmark and Sweden are on the wrong side of the Laﬀer curvefor capital taxation. On the other hand, countries with high inequality, such as U.S. and
U.K., are on the left side. Therefore this paper gives a rationale for their results.
The rest of paper is organized as follows. Section 2 sets up the model. The equilibrium policy is derived and investigated in Section 3. Section 4 discusses the assumptionsin the model and oﬀers some extensions. Finally, Section 5 concludes.

2

Model

Individuals with a mass of 1 are inhabitants of an economy. They live for two periods
and diﬀer with respect to capital holdings bi ≥ 0 at period 1. Its cumulative distributionfunction is denoted by G(·). Its mean is denoted by b̄. In the ﬁrst period, each individual
receives his income for supplying one-unit labor and renting their capital to a competitiveﬁrm. The consumer either consume or save his income. In the second period, he supplies
one unit of labor inelastically, and receive capital income from their savings.
An agent’s preference is deﬁned over consumption in both periods and public goodsin the second period, and given by

U (ci,1 , ci,2 ) = u(ci,1 ) + β(u(ci,2 ) + v(x))= log ci,1 + β(log ci,2 + γ log x),(1)

where ci,1 , ci,2 respectively represent the ﬁrst and second period consumption for an agent
i and β denotes a discount factor. The level of public goods provided by government is3

represented by x ≥ 0. The budget constraint of individual i is given byci,1 + si = w1 + r1 bici,2 = w2 + (1 − τ )r2 si,(2)where si , wt , rt and τ denote the saving of individual i at period 1, the wage rate atperiod t, the return on capital at period t and the capital income tax rate, respectively.
Note that capital fully depreciates in the second period. Also, for simplicity, it is assumed
that the government can levy a tax only on capital income.
In each period, competitive ﬁrms maximize their proﬁt according to the following
production function,

F (Kt , Lt ) = AKtα L1−α
.
t

(3)

In the second period, the government collects its revenue from capital income tax toprovide public goods. The government budget is∫x=τ r2 si dH(si ),(4)where H(·) is the cumulative distribution function of saving from period 1, which is
generated through the ﬁrst-period decisions by individuals. It is assumed that the government can use the technology which translates one ﬁnal good into one public good.
As in Persson and Tabellini (2000, Ch. 12) and others, the timing of events is asfollows; investment decisions followed by voting on the tax rate.
1. In the ﬁrst period, each individual decides how to allocate income between consumption and saving.
2. At the end of the ﬁrst period, the tax rate on capital income is decided through amajority voting.
3. In the second period, given the tax rate, individuals consume their income andgain beneﬁts from public goods provided by the government.
4Note that individuals make their decisions before the tax policy is determined. Therefore, they must expect which level of capital tax will be imposed in the second period.3
This timing of events is crucial for the following results. The wrong side of the Laﬀer
curve is never chosen if the ﬁrst and second events are replaced. It is because, if thetax policy chosen is on the wrong side, there must exist a better way to collect the
government revenue without any loss. However, when the timing is as above, there canbe an equilibrium with the right side of the Laﬀer curve.

3

Equilibrium Analysis

This section provides the equilibrium of this economy. It is shown, in some cases, that
only the wrong side of the Laﬀer curve is an equilibrium. Note that economic agents havea perfect foresight. And an equilibrium refers to a rational expectations equilibrium.

3.1

Equilibrium

The return on each production factor for t = 1, 2 is given byrt = αAKtα−1 L1−αtwt = (1 − α)AKtα L−α
t

.

(5)

By maximizing utility (1) with the budget constraint (2), the consumers’ saving
function is obtained assi (τ e ) =1w2
β(w1 + r1 bi ) −,1+β
1 + β (1 − τ e )r2(6)

where τ e denotes the expected capital tax rate. Note that the saving of each agent
is decreasing in the expected capital tax rate (∂si (τ e )/∂τ e < 0).
Because the labor supply is ﬁxed to the number of population, the labor market

3

It can be interpreted as follows: the government promises a tax policy at the beginning of the ﬁrst
period, but cannot commit to that. The actual policy is chosen at the end of the ﬁrst period by taking
private agents decisions as given.

5clearing condition is

Lt = 1 for t = 1, 2.

(7)

The amount of capital supplied to the market in the ﬁrst period is ﬁxed and given by
the total amount of capital holdings at the beginning. However, its amount in the second
period is endogenously determined through saving decisions by agents. Therefore, using(6), the conditions to clear the capital market are obtained as follows.





K1




K2








= b̄
=
≡

∫
∫

.

si dH(si )

(8)

β
w2
1
1+β (w1 + r1 bi ) − 1+β (1−τ e )r2 dG(bi )At the voting stage, as each individual’s capital holding for period 2 is already predetermined, the tax policy does not distort capital accumulation from then on. Therefore,
the wage rate and the return on capital are ﬁxed at the voting stage. The preferred tax
policy for the agent with the saving of si solves the following

max β(u(ci,2 ) + v(x)),τ
ci,2 = w2 + (1 − τ )r2 si
subject to∫x = τ r2 si dH(si ),given w2 , r2 and [si ]1i=0 .

The ﬁrst-order condition for this problem is
−

r2 si
γ
+ i = 0,i
w2 + (1 − τ )r2 si τ(9)where τ i is deﬁned as the best policy for the individual with si .4 The ﬁrst term represents
the utility loss as the tax rate marginally increases. The second term is the marginal
Here τ i ≥ 1 is allowed. However, any policy with τ ≥ 1 is excluded as an equilibrium, since it isunrealistic.
4

6

beneﬁt from an increased amount of public goods. Note that because of the log utility,
the preferred policy is irrelevant to redistributive concern and does not depend on otheragents’ savings. In this economy, the policy is determined by a majority voting. The
equilibrium policy never loses against any policy in a pair-wise voting, which is called theCondorcet winner. Because agents’ preferences are single-peaked, there exists a uniqueCondorcet winner, the policy which the median voter prefers, τ med .5 For expositional
reason, assume that the median has more wealth in the ﬁrst period than the mean.6 This
assumption is justiﬁed by the observation that rich individuals have a high propensity
to give political contributions. When rich voters engage in political activities more thanpoorer ones, the “politically median” voter should be richer than the actual medianvoter, and could be richer than the mean. Rosenstone and Hansen (1993) presents thedata which shows the propensity to participate in every form of political activity rises
with income.7
Equations (5), (6), (7), (8), (9) and the rational expectations requirement τ med = τ ejointly determine the equilibrium. Hereafter, the equilibrium tax rate is represented by
τ ∗.

3.2

the Laﬀer curve

In order to evaluate the equilibrium policy obtained in the previous section, this section
provides the conﬁguration of the Laﬀer curve. The Laﬀer curve is the relationshipbetween the tax rate and the government revenue.
The government’s revenue Π is deﬁned as∫Π≡τ r2 si dH(si ).

Combining (5), (6), (7) and (8), Π can be calculated as follows.

5

See Persson and Tabellini (2000).
It ensures that the equilibrium policy is unique (see the Appendix).
7
Benabou (2000) employs the assumption that voters with more wealth have more votes. Bourguignon
and Verdier (2000) also uses this kind of assumption in a theoretical model. They suppose that individuals
with enough income (or education) have a right to vote. This is motivated by historical examples thatvoting rights were restricted to individuals with enough property.
67(Π = αAτβAb̄α1(1 + β) + 1−α
α 1−τ

)α
.

(10)

The Laﬀer curve in this model is unimodal, as usual. The tax rate which attains the
maximal government revenue, τ max , is obtained from the ﬁrst order condition for (10).
That tax rate satisﬁes τ max ∈ (0, 1) and the following equation,∂Π/∂τ = 0 ⇐⇒(1 + β)(τ

3.3

[

]
1−α
1−α
) − 2(1 + β) +(1 + α) τ max + (1 + β) +
= 0.
α
α

max 2

Ineﬃciency in ﬁscal policy

This subsection investigates whether the tax rate determined through voting is on theleft or right side of the Laﬀer curve. To see this, compute the equilibrium tax rate.
Combining (5), (7) and (8), the aggregate capital holdings at period 2 is obtained,
depending on the expected tax rate.

K2 =

βAb̄α
1 ≡ Φ(τ ).
(1 + β) + 1−α
α 1−τ

(11)

Φ(τ ) reveals the intuitive relationship that the capital accumulation is decreasing in
the tax rate. Next, using (5), (6), (7) and (9), the median-preferred tax policy can beobtained, relating to the aggregate capital.

K2 =α βA[(1 − α)b̄α + αb̄α−1 bmed ](τ − γ(1 − τ ))(1 − τ )≡ Ω(τ ).
1−αβγ(1 − τ ) + τ(12)Figure 1 shows the equilibrium policy determination at the intersection of the twofunctions. It can be shown that the equilibrium tax rate τ ∗ is unique and interior.8
If the condition below is satisﬁed, the politically-determined tax rate is beyond the8

See the Appendix.

8Φ(τ )

Ω(τ )

O

γ
1+γ

τ max τ ∗τ1Figure 1: Equilibrium policy on the wrong side of the Laﬀer curve
peak of the Laﬀer curve and gets trapped into ineﬃciency.9 If this is the case, the tax
policy is Pareto-dominated and there exists an alternative that every agents prefers.

Φ(τ max ) > Ω(τ max ).

(13)

In what situations, this ineﬃcient tax collection is likely to arise? First, when theinequality of the economy, relative richness of the median to mean, is small (low inequality), the ineﬃciency in ﬁscal policy tends to occur. It is because the median with alower capital holdings prefers a higher tax rate as the tax burden of less wealth is small(see equation (9)). Trabandt and Uhlig (2011) calibrates a neoclassical growth modeland argues that Denmark and Sweden are on the “wrong” side of the Laﬀer curve forcapital income taxation, while other high-inequality countries like U.S. are on the left
side. Their results are consistent with the observation in the model that countries with
low inequality are likely to suﬀer from ineﬃcient ﬁscal policies. Second, the preference
for public goods is also important. The reason is quite simple. When the preference
for public goods is strong, the median demands a higher tax rate to obtain more public
goods.
However, why is it impossible to have the lower tax rate which attains the same9

The parameter set which satisﬁes the assumptions and equation (13) is reasonably large. For aquantitative example, A = 1, α = 0.3, β = 0.5, γ = 2, b̄ = 1, bmed = 2.

9government revenue as τ ∗ ? That tax policy, say τ̂ , must be better for every agent(Pareto-dominating τ ∗ ), since it achieves the same revenue (consequently, the same levelof public goods) and incurs less distortion to economic activities. Consider the situationthat the tax policy is committed to be τ̂ and that every individual believes it. Then theaggregate investment in the economy would increase and the return on capital would be
low. Observing this, which tax rate would the median choose? Because the tax burden
is now small due to the low return on capital, the median would never prefer τ̂ and adopt
another higher tax rate. Therefore the commitment for the tax rate τ̂ is not credible.
It is the reason why the Pareto-dominating policy, τ̂ , is not supported under rational
expectations.

4

Discussion

This section discusses a few assumptions which are crucial to get the main results in thispaper. Also, some extensions will be provided.

4.1

Discussion on assumptions

There are two critical assumptions in the model. First, it is the timing of events; thepolicy determination after investment decisions. If the policy is decided at the beginning,
the tax rate would never be on the wrong side.10 The second is the endogeneity of returnon capital. With exogenous return on capital, Pareto-dominating tax rate τ̂ is also an
equilibrium policy. Whichever the median voter commits τ̂ or τ ∗ , both are credible. And
two tax rates can be rational expectation equilibria, because the median does not change
preferred tax rate with exogenous return.11 There are some previous studies whichshow two possible tax rates on both sides of the Laﬀer curve with exogenous return on
capital (Saint-Paul and Verdier (1997) and Persson and Tabellini (2000, Ch. 12)). The
distinct feature of this model is that only the wrong side can be a rational expectation
equilibrium, incorporating the endogenous return on investment. The multiplicity issue
is well known in dynamic taxation models with endogenous policy making. A Paretodominated equilibrium is often seen as a coordination failure among economic agents.
1011Another interpretation is that a lack of commitment results in the ineﬃciency in ﬁscal policy.
When endogenous return on capital, recall the argument in the last paragraph in Section 3.

10

And this disastrous equilibrium is sometimes ignored by assuming that economic agentscan somehow coordinate their expectations (Persson and Tabellini (1994)). However, thispaper emphasizes that there can be no room for coordinations and that the disastrous
outcome may be inevitable.

4.2

Extensions

For a very simple model, several extensions would be available. The same structure
applies for many policy-determining situations. For instance, a child allowance may be
a good example. Families decide how many children to have before the ﬁscal policy
on a child allowance is adopted. Because the government budget is usually approved
annually, while the decision for the quantity of children is a long term decision. And thereturn on a child in terms of the wage earned by each child is endogenous, because of
the wage rate determination through the labor market in the future. Therefore there isa possibility that a child allowance policy can be on the “wrong side” among possible
policies. Another example is education. Consider an income tax and agents’ education
decisions. Younger voters have to decide whether to receive higher education before
their voice in political spheres is empowered compared to older voters. Therefore thepolicy choice led by young voters is postponed long after education. Of course, thereturn on education depends on relative scarcity of skilled workers to unskilled workers,
which is endogenous. Hence, as is the case with child allowance, a politically determined
income tax rate may be Pareto-dominated and too high so that it excessively distorts
education decisions. Calvo (1988) shows that multiple equilibria may arise in a model of
public debt repayment. There exist a Pareto-eﬃcient equilibrium where there is no debt
repudiation and a Pareto-ineﬃcient equilibrium in which debt is partially repudiated.
This is analogous to multiple equilibria on both sides of the Laﬀer curve in Persson andTabellini (2000, Ch. 12). By endogenizing the relevant variable in Calvo (1988),12 it
may lead to a unique ineﬃcient outcome.

12

In his model, the interest factor of the public debt is exogenous. However, it seems natural to thinkthat the interest factor is increasing in the total amount of outstanding bonds.

11

5

Conclusion

This paper shows that an endogenous tax policy may generate an extremely ineﬃcient
government activity, the wrong side of the Laﬀer curve. To that end, there are two
critical, but totally plausible, assumptions; the timing of policy making and endogeneityof the return on capital. Because some tax policies are determined after agents’ decision
makings, the government’s performance can often get trapped into ineﬃciency. This
point is repeatedly emphasized in previous studies which argue multiple policies on theboth sides of the Laﬀer curve can be possible. However, together with the endogeneityof the return on capital, citizens may suﬀer from the wrong side of the Laﬀer curve, theunique possible outcome. The mechanics of the model have a general applicability andcan be extended to analyze a wide range of policies.

Appendix
Uniqueness of the interior equilibrium tax rate
To obtain the equilibrium tax rate, solve the following equation,

Φ(τ ) = Ω(τ ),
⇐⇒

βAb̄α
α βA[(1 − α)b̄α + αb̄α−1 bmed ](τ − γ(1 − τ ))(1 − τ )
=
.
1
1−α
βγ(1 − τ ) + τ(1 + β) + 1−α
α 1−τ
(A.1)

Because (A.1) is a third degree equation, the number of solutions is at most three.
Also note that τ = 1 is an obvious solution. Thanks to bmed > b̄, the following inequality
is straightforward,

0>

d
d
>
,
Φ(τ )
Ω(τ )
dτ
dτ
τ =1
τ =1

which implies that the slope of Ω(·) is steeper than that of Φ(·) at τ = 1, as shownin Figure 1. Note that Φ(0) > 0 > Ω(0). By the continuity of both functions, it follows

12□that the interior solution must be unique.

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14