1. Introduction
This paper examines the condition under which the neutrality of income transfer policyon the allocation of goods holds (hereafter referred to as the transfer neutrality). It uses amodel with privately provided public goods, under the presence of two types of costdifferentials. Warr (1982, 1983) formulated the transfer neutrality theorem, which statesthat the income transfer is independent of the equilibrium allocation of goods in thestandard model of the private provision of a public good. Following Warr, the theorem has
been highlighted because it is important in the discussion of income transfer policy.1 Some
researchers derived the result that income transfer will not be completely neutral undersome general assumptions.2 In all such papers, it was implicitly assumed that contributors
faced the same cost when they increased the amount of public goods marginally. However,contributors may face different costs of contributions. When agents contribute to the public
good, costs usually differ between agents. It is important to analyze the effect of income
transfers under this general situation. Following this point of view, some recent papers(Boadway et al., 1989; Buchholz and Konrad, 1995; and Ihori, 1996) focus on costdifferentials among contributors.
However, the existing analysis of cost differentials is not sufficient to obtain general andclear results, as the neutrality result has not been consistent. Boadway et al. (1989) has
shown that the transfer neutrality holds in the presence of the cost differentials. In contrast,Buchholz and Konrad (1995) and Ihori (1996) have shown that the neutrality will not hold.
It is important to clarify the differences between these inconsistent conclusions so that theeffect of income transfers can be clearly discussed.
The main difference underlying these seemingly inconsistent conclusions is the way thatthe cost differential is created. In the model developed by Boadway et al. (1989),
government can subsidize the private provision of public goods and the cost differential iscreated by tax deductibility or matching grants. On the other hand, in the models of
Buchholz and Konrad (1995) and Ihori (1996), a productivity differential exists before
intervention by the government. It is important to clarify the characteristics of each costdifferential, which is the purpose of this paper.
This paper is organized as follows. Section 2 presents the model and considers thenon-cooperative Nash equilibrium of voluntary contributions to a pure public good. To
compare existing models, section 3 derives the welfare effect of the income transfers anddiscusses the condition required to hold the neutrality.
2. The Model
Consider an economy with two types of heterogeneous regions, each of which
1See Bergstrom and Varian (1985) for the general theorem.
For example, consider the case where some agents do not contribute at all or the case where altruism is notpure. See Bergstrom et al. (1986) for the case with non contributors and Andreoni (1989) for the case ofimpure altruism.

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contributes to a pure public good, G .3 We call this a privately provided public good.
Therefore, the quantity of public goods in this economy becomes G .
Region i obtains utility from consumption, x i , and public goods. Then the utility inregion i is represented by the following utility function:(1)
U i = u i ( x i , G) ,where both x i and G are assumed to be normal goods. Each local government, by
setting the local lump sum tax, divides fixed endowments or incomes in each region into
the contribution to pure public goods and consumption goods. It is assumed that eachgovernment in each region has a linear production frontier associated with public goods
and consumption goods. Finally, each local government faces the following budget
constraint:(2)
x i + c i g i = y i − Tax i ,iwhere g is the level of public goods contributed by the local government in region i ,c i is an original cost, representing the productivity of public goods in region i before theintervention of the central government, y i is exogenously given income in region i , andTax i is an effective tax burden. Since the public good contributed by the local government
is pure, G becomes equal to the sum of the quantity of contributions in each region. Thatis:(3)
G ≡ g1 + g 2 .
Next, consider the tax and subsidy schemes of the central government. The central
government is assumed to control the income tax level and the subsidy rate on eachcontribution of public goods (or the grants tied to privately provided public goods). Thenthe effective tax burden in region i is represented as follows:(4)
Tax i = ty i − β i c i g i ,iwhere t and β represent an income tax rate and a rate of subsidy to the public good
provided by the local government i , respectively. We assume that 0 ≤ β < 1 . As shown
below, when this subsidy rate is different between regions, the cost differential is createdartificially.
Then the budget constraint of the central government becomes:(5)
ty 1 − β 1c1 g 1 + ty 2 − β 2 c 2 g 2 = 0 .
If the subsidy rate is assumed to be exogenous, this equation determines the level of theincome tax endogenously.
Substituting equation (4) into (2), we obtain:(2)'
x i + c i g i = y i − ( ty i − β i c i g i ) .
Rewriting this, we have:(6)
x i + (1 − β i )c i g i = (1 − t ) y i ,3Although we present an inter-regional transfer in the model, the same results would hold for a transfer
between agents or an international transfer.

2where (1− β i )c i is the effective and actual cost of the public good, which is constructedby two costs, the artificial cost, 1− β i , and the original cost, c i . Further, this equationmeans that the effective cost of the public good is artificially lowered by the subsidy fromthe central government. Using equation (3), the above equation can be rewritten as:
x i + (1 − β i )c i G = (1 − t ) y i + (1 − β i )c i g − i .
Then each local government determines the public good provision and the level ofconsumption in each region, treating the effective cost of the privately provided public
good, the income tax rate and the contribution in the other region, as given.4
In order to describe the Nash equilibrium model, the utility maximization behavior ofeach region is defined by using the following expenditure function:
Minimize E i ≡ x i + (1 − β i )c i G,Sub ject to u i ( x i , G ) ≥ U i
Then, the expenditure level is represented as the function of the effective cost and utility asfollows:
E i = E i (U i , (1 − β i )c i ) .
Noting normalities of goods, the expenditure function has the following sign andcharacteristics under the framework of this paper:
EUi = xUi + (1 − β i )c i GUi > 0 、 GUi > 0 、 xUi > 0 .
Here G i (U i , (1 − β i )c i ) and x i (U i , (1 − β i )c i ) are the compensated demand function forthe public good and consumption in each region, respectively. Using the definition of E i ,equation (6) becomes:(1 − t ) y i = E i − (1 − β i )c i G i + (1 − β i )c i g i .
Then the budget constraints in each region are formulated as follows:(7-1)(1 − t ) y1 = E 1 − (1 − β 1 )c1G 1 (U 1 , (1 − β 1 )c1 ) + (1 − β 1 )c1 g 12222 222222 2(1 − t ) y = E − (1 − β )c G (U , (1 − β )c ) + (1 − β )c g .
(7-2)
iiii
12Since G = G (U , (1 − β )c ) = g + g , equation (7-2) gives:
1g1 =( E 2 − (1 − t ) y 2 )22(1 − β )cSubstituting this into equation (7-1) and subtracting g1 , we have:(1 − β 1 )c1 2(1 − t ) y1 = E1 − (1 − β 1 )c1G1 +( E − (1 − t ) y 2 ) ,(1 − β 2 )c 2
which becomes:(1 − β 2 )c 2 (1 − t ) y1 + (1 − β 1 )c1(1 − t ) y 2 = (1 − β 2 )c 2 E1 + (1 − β 1 )c1 E 2 − (1 − β 1 )(1 − β 2 )c1c 2G1 .
In addition, from equations (2)' and (5), we have the following resource constraint:
4

In order to focus on the role of cost differentials in the transfer neutrality theorem, we assume the case of
the inner solutions, where all agents are contributors. In the case of the corner solution, where there exists
non-contributor in the model, transfer neutrality does not necessarily hold. See Bergstrom et al. (1986).

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y 1 + y 2 − x1 − x 2 − c1 g 1 − c 2 g 2 = 0 .
Noting that G i = g 1 + g 2 and g 2 =1

( E 1 − (1 − t ) y 1 ) , we obtain:(1 − β )cy + y − x (U , (1 − β )c ) − x 2 (U 2 , (1 − β 2 )c 2 )12112111c1 − c 2( E 1 (U 1 , (1 − β 1 )c1 ) − (1 − t ) y1 ) = 0.
11c (1 − β )Finally, the model reduces to the following three equations:(1 − β 2 )c 2 E 1 (U 1 , (1 − β 1 )c1 ) + (1 − β 1 )c1 E 2 (U 2 , (1 − β 2 )c 2 )− c1G1 (U 1 , (1 − β 2 )c1 ) +− (1 − β 1 )(1 − β 2 )c1c 2 G 1 (U 1 , (1 − β 1 )c1 ) = (1 − β 2 )c 2 (1 − t ) y1 + (1 − β 1 )c1 (1 − t ) y 2(8-1)
11112222(8-2)
G (U , (1 − β )c ) = G (U , (1 − β )c )1211212222y + y − x (U , (1 − β )c ) − x (U , (1 − β )c ). (8-3)c1 − c 2− c1G1 (U 1 , (1 − β 2 )c1 ) + 1( E 1 (U 1 , (1 − β 1 )c1 ) − (1 − t ) y1 ) = 01c (1 − β )Equation (8-1) represents the budget constraint in region 1. In particular, the right hand sideof this equation represents the income, taking into consideration the externality effect ineach region, referred to as an effective income. Equations (8-2) and (8-3) represent thepublic good constraint and the resource constraint, respectively. From these three equations,the utility level in each region and the level of the income tax rate are determined. 5

3. Welfare effects of disposable income transfers
We examine the welfare effects of disposable income transfers. The income tax rate is
assumed to be endogenous to hold the budget constraint. Totally differentiating (8-1), (8-2)
and (8-3) gives:(1 − β 2 )c 2 xU1 (1 − β 1 )c1 EU2 (1 − β 2 )c 2 y1 + (1 − β 1 )c1 y 2   dU 1  2 − GU20GU1
 dU 12
 1
  dt c −c1 1
− xU2y1
 − xU − c GU

11c (1 − β )


(9)


2211
(1 − β )c − (1 − β )c  dT=0c1 − c 2c1 (1 − β 1 )5Note that budget constraint in region 2 is omitted by the Warlas Law.

4where dT is defined as d (1 − t ) y1 = − d (1 − t ) y 2 , which represents the transfer ofdisposable income from region 2 to region 1, and the determinant of(1 − β 2 )c 2 xU1 (1 − β 1 )c1 EU2 (1 − β 2 )c 2 y1 + (1 − β 1 )c1 y 2 
1− GU2GU10 , which is defined as ∆ , 1c1 − c 2 11 12−−−xcGxyUU U
11c (1 − β )becomes negative when we assume c1 ≥ c 2 without loss of generality.
Summarizing each term of equation (10), we have:fsdU 1 GU2 1(10)= 1 ( y + y 2 )( c1 − c 2 ) .6
∆dT
The artificial cost differential does not appear in equation (10) and the sign of equation (10)crucially depends on original cost differentials. The neutrality holds as long as an original
cost differential does not exist. This corresponds to the result of Boadway et al. (1989). Inaddition, this corresponds to the results of Buchholz and Konrad (1995) and Ihori (1996),
which prove that the income transfer is effective in the presence of the original costdifferential.
First, the intuition of the reason why the original cost differential affects isstraightforward. If c1 > c2 , the income transfer from region 2 to region 1 means theresource allocation from the low cost region to the high cost region inefficiently. Therefore,
the utility changes (decreases).
Next, to understand the intuition of the reason why the artificial cost differential does notaffect, we first examine the effect of the transfer on the tax rate. Setting c1 = c 2 , we have:dt
β1 − β 2,(11)=dT (1 − β 2 ) y1 + (1 − β 1 ) y 2
which shows that the sign of the effect on the tax rate directly depends on the sign ofβ 1 − β 2 . Assuming that β 1 > β 2 , then the effective income in the economy increases by
the transfer from region 2 to region 1. Therefore, the provision of public goods increases,
which in turn increases the subsidy amounts. Through the budget constraint, the income tax
rate increases in order to keep revenue constant. Since this increase of the income tax ratecompletely offsets the increase of the effective income in the economy, the effect of theartificial cost differential vanishes.
We can summarize the results as follows.

Proposition 1
Even if the actual mixed cost which contributors face is the same among contributors, thetransfer may not be neutral. Original cost equalization, that is, productivity equalization
6dU 2 GU1 1
Similarly, we have
= 1 ( y + y 2 )( c1 − c 2 ) .
dT∆5before intervention by the central government ( c1 = c 2 ), is essential to hold the transferneutrality.

4. Conclusion
This paper has examined the condition to hold the transfer neutrality in an economy withprivately provided public goods under the presence of both artificial and original costdifferential. In the standard model of private provision of a public good, if the cost is thesame, the transfer neutrality holds. However, when there exists two types of costdifferentials (artificial cost differential and original cost differential), the neutrality is
ambiguous and should be examined. In this point of view, this paper shows that even if theactual mixed cost that contributors face is the same among contributors, the transfer may
not be neutral, and original cost equalization is essential to hold the transfer neutrality. The
results of this paper indicate that the central government should focus only on the original
cost differential, rather than on the actual cost. The policy of income redistribution should
be reexamined, considering the relationship between the original cost and the artificial cost.

References
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Boadway, R., P. Pestieau and D. Wildasin (1989) “Tax transfer policies and the voluntaryprovision of public goods” Journal of Public Economics 39, 157-176.
Buchholz W. and K. A. Konrad (1995) “Strategic transfers and private provision of publicgoods” Journal of Public Economics 57, 489-505.
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