Tax Audit and Vertical Externalities
Hideya Kato
Mitsuyoshi Yanagihara※

Nagoya Keizai University
Nagoya University

1. Introduction
The vertical fiscal externalities arise when the different levels of governments, such as thefederal and state governments, imposes taxes on the same tax base. When a state increases its tax
rate, it causes the shared tax base to shrink. However, the state governments ignore the reduction
in federal tax revenues caused by the shrinkage effect. As a result, the state tax rates tend to behigh.
Boadway and Keen (1996) and Dahlby (1996) consider intergovernmental transfer tointernalize the vertical externalities. They analyze the equilibrium outcomes in a federation system
where both the state and federal governments impose tax on the same tax base to finance theirown public good. They conclude that the fiscal gap which represents the direction of the transferfrom the federal government to state government is negative.
So far, the theoretical analyses of tax competition with vertical externalities have not taken tax
evasion and tax audit scheme into account. However, in reality, there is tax evasion in the world.
Tax audit concept can vary between different forms of organizations and also between countries.
In fact, as indicated by Stowhase and Traxler (2005, p. 529) and Bordignon, Manasse andTabellini (2001, p.719), there are three audit systems: audit system implemented by both state andfederal level; audit system by federal level; and audit system by state level. From the standpoint ofthese, this paper incorporates tax evasion and tax audit systems into a vertical externality model by
Boadway and Keen (1996). In the model, we focus on the question how the tax audit systemshould be implemented.
By considering the tax audit, we can show the following vertical fiscal externalities which arefocused on by the existing studies. When the different levels of governments audit the same tax
base, if a state government increases tax audit expenditure, tax evasion decreases. The state
governments ignore the increase in the federal tax revenues caused by this effect, so that the state
tax audit expenditures will be low.
In this paper, we show a unitary nation in which a federal government can choose all
governments’ variables. The unitary nation is the second-best optimum. We compare thefollowing three different systems with the unitary nation case. The first case is a decentralization
system where both the federal and state governments implement tax audit. The second case is asystem of tax audit centralization where only the federal government implements tax audit. The
third system is a system of tax audit decentralization where only the state governments implement
tax audit. In the first and third systems, if rent is not allocated to the federal government, thefederal optimal tax rate and the fiscal gap should be zero and negative, respectively, to replicate
the unitary second-best optimum. In the second case, the federal optimal tax rate should benegative to replicate the unitary second-best optimum.
2. ModelThere are identical k regions in a federal economy and n inhabitants in each region.
Households are identical and immobile among states. The utility function of the households is thefollowing form:(1)
U ( x, l ) = u ( x, l ) + b( g ) + B(G ) ,where g and G are state and federal public goods, respectively. We assume that u ( x, l ) isincreasing in x , decreasing in l and quasi-concave, and b(g ) and B(G ) are increasing andconcave functions.
The households are endowed with one unit of time and divide their time between labor andleisure. They earn labor income, wl , where w and l are wage rate and labor supply,respectively. Per unit tax levied on labor is τ ≡ t + T , where t and T denote state tax rate andfederal tax rates, respectively. The households can evade the tax by failing to a part of labor e(called “undeclared labor”). On the other hand, the governments pay per capita audit expendituresα ≡ a + A to preserve tax base, where a and A are per capita audit expenditure by the state
governments and per capita audit expenditure by the federal government, respectively. In order toevade tax, households must bear the costs of C (e,α ) , where Ce > 0 , Cee > 0 , Cα > 0 , andCeα < 0 .1 The budget constraint faced by each household is
x = ( w − τ )l + τe − C (e,α ) ,(2)where x is a private good and numeraire.
The households choose e and l to maximize the utility function (1) subject to budget
constraint (2). Solving this problem, we obtain the following first-order conditions:u x ( w − τ ) + ul = 0 ,(3)

τ − Ce (e,α ) = 0 .

(4)

From these first-order conditions, labor supply function and undeclared income function arerepresented by l = l ( w − τ ) , e = e(τ ,α ) . As is the case with Boadway and Keen (1996), we
assume that l ′( w − τ ) > 0 .2 Undeclared income function has the following properties.
eτ =12

Subscripts denote partial derivatives.
A prime denotes the derivative of a function.

C (e,α )1> 0 , eα = − eα<0Cee (e,α )Cee (e,α )Substituting (5) and (6) into (1), an indirect utility function can be represented asV ≡ v(( w − τ ),τ ,α ) + b( g ) + B(G ) . By using the envelope theorem, we get the following results:∂v(( w − τ ),τ ,α )= u xl ,∂(w − τ )(9)∂v(( w − τ ),τ ,α )= uxe ,∂τ(10)∂v(( w − τ ),τ ,α )(11)= −u xCα .
∂αFirms maximize profits f (nl ) − wnl and therefore choose labor demand that is given by themarginal productivity condition: f ′(nl ) = w . Substituting labor supply function into thiscondition, we implicitly write the equilibrium wage rate and rent.
w = f ′(nl ( w − τ )) .
r (τ ) = f (nl ( w− τ ) ) − nl ( w − τ ) f ′(nl ( w− τ ) ) .

(12)

(13)
The rents are allocated to the federal government at a constant rate θ ∈ (0,1) , and to the state
governments at the rate of 1 − θ . Differentiating (12) and (13) with respect to τ , we obtain thefollowing result:− f ′′nl ′wτ =∈ (0,1) ,1 − f ′n′ l ′rτ = (1 − wτ ) f ′n′ 2l ′l =n 2 f ′l′ ′l< 0.
1 − f ′n′ l ′(14)

The state governments collect the state tax revenues which consist of the tax, rent, and transferfrom the federal government to the state government S . The state governments use the tax
revenue for the state public good and the state audit cost. Therefore, the state budget constraint isgiven by(16)
g = ntl ( w(τ ) − τ ) − nte(τ ,α ) + S + (1 − θ )r (τ ) − na .
The federal public good and the transfer between the governments are supplied by the federal
governments and are financed by the federal’s revenue which consists of the tax revenue and rent.
Therefore, the federal government’s budget constraint is given by:(17)
G = knTl ( w(τ ) − τ ) − knTe(τ , α ) − kS + kθr (τ ) − knA .
3. Optimal policies under unitary nation
In unitary nation, the federal government chooses τ , α , g , and G to maximize socialwelfare (individual utility). Solving this problem, we obtain the necessary conditions:l−enb′( g ) knB′(G )− Cα
1 − f ′′nl ′
===.
luxux
 τl ′   τeτ  nτeα + 11 −  − e1 +1 − f ′′nl ′ l  e (29)This conditions means that the sum of the marginal rate of substitution between both the state andfederal goods and the private good is equal to the marginal cost of public funds (MCPF) and themarginal cost of tax audit (MCTA) which is the cost of raising a unit of revenue by increasing thetax audit expenditure. MCPF and MCTA are greater than 1. (29) together with the budget
constraint in (16) characterizes the unitary second-best optimum.
4. Optimal policies under decentralized system
4.1. The state government’s behavior
The state governments choose t , a and g to maximize the indirect utility function V
subject to (16), taking as given the decision variables of the federal government, T , A , S andG . The necessary conditions of this maximization problem are given by[( wτ − 1)l + e]u x + b′g t = 0 ,(31)− Cα u x + b' g a = 0 .

(32)

As indicated by Appendix, we obtain the following necessary conditions

l−e
− Cα
1 − f ′n′ l ′nb′( g ) ==.
l
 τl ′   τeτ  Gt nτe + 1 − Ga
1 −  − e1 +−αknl  e  kn
1 − f ′n′ l ′ (33)Comparing the denominator in (29) to (33), we can see that Gt and Ga are important elements.
If Gt < 0 and Ga < 0 , then the MCPF and MCTA for the second best case is greater than thatfor the state governments under decentralized system. On the other hand, if Gt > 0 and Ga > 0 ,it lead to the opposite result. These elements represent the effects of the state tax rate and the stateaudit expenditure on the federal government revenue. The state governments neglect these effects(vertical externalities).
4.2. The federal government’s behavior
In this subsection, we consider the federal government behavior. The federal governmentschoose T , A , S and G to maximize the indirect utility function V subject to (17), taking
into account the reaction of the state government. The necessary conditions for this problem arewritten as follows:knB′(G ) = nb′( g )l −eG (1 + tT ) + Ga aT
l −e+ tknknB′(G ) = nb′( g )1.
Gt t A + (1 + a A )Ga
1−kn(38)(39)

(nb′( g ) − knB′(G )) + nB′(Gt tS + Ga aS ) = 0 .

(40)

The optimal federal policies are to satisfy the equations (38), (39) and (40) simultaneously.
Therefore, the federal optimal tax rate and audit expenditure are characterizing by the conditions

Gt = 0 and Ga = 0 . To achieve these conditions yields that MCPF for the federal government isequal to that for the state government and replicates the second-best optimum.
From Gt = 0 , the optimal federal tax is given by
T* =− θrτ
< 0.
n( wτ − 1)l ′ − neτ(41)

Substituting (41) into Ga , the sign of Ga becomes negative. This means that the decentralized
system cannot replicate the second-best equilibrium.
Here, suppose that the federal government does not get revenue from rents: θ = 0 . In this case,the federal government has no revenue from tax and rents. Therefore, the federal public goodshould be financed by the transfer from the state governments. This implies that the fiscal gap isnegative.
We provide the following proposition.
Proposition 1
Under the decentralized system, if the federal government does not get revenue from rents,
the federal government can replicate the second-best. In this case, the federal government
set the tax rate at zero, and therefore, the fiscal gap is negative to finance the federal
public good.
5. Alternative tax audit systems
5.1. Tax audit centralization
We consider a case that only the federal government can implement tax audit: In this case,α = A . Here, we assume that the state governments receive accurate information about tax audit
from the federal government.
In this case, there is no solution because the federal government only has three policy tools (T ,A , S , G ) to attain the four second-best variables ( τ * , α * , g * and G * ).

Proposition 2
Under the tax audit centralized system, the federal government can replicate thesecond-best. In this case, the federal government should set the tax rate at negative.

5.2. Tax audit decentralization
We turn to show a tax audit decentralization such that only the federal government implement
tax audit: in this case, α = a . In this case, there is no solution because the federal government
only has three policy tools (T , S , G ) to attain the four second-best variables ( τ * , α * , g * andG * ). Therefore, the tax audit decentralization cannot replicate the second-best optimum. However,if θ = 0 , the tax audit decentralization can replicate the second-best optimum. These results are
the same as those under the decentralization system.
6. Conclusion
In vertical externality models, it is typically assumed that there are no tax evasion and no cost oftax collection. This assumption does not reflect the realities. Therefore, this paper extends thevertical externality model in Boadway and Keen (1996) by considering tax evasion behavior andtax audit expenditure. Thereby, we obtained the significant results relevant to tax policy, fiscal
transfer policy and tax audit policy. Concretely, the results of this paper are as follows.
When the state governments implement tax audit, if rent is not allocated to the federal
government, the federal optimal tax rate and the fiscal gap should be zero and negative,
respectively, to replicate the unitary second-best optimum. On the other hand, when only thefederal government implement tax audit, the federal optimal tax rate should be negative toreplicate the unitary second-best optimum.
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