Discussion Papers In Economics
And Business

Public Expenditure Composition and Economic Growth:
Optimal Adjustment by Using Gradient Method

Tatsuyoshi Miyakoshi, Yoshihiko Tsukuda, Tatsuhito Kono, Makoto Koyanagi

Discussion Paper 07-17

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPublic Expenditure Composition and Economic Growth:
Optimal Adjustment by Using Gradient Method

Tatsuyoshi Miyakoshi, Yoshihiko Tsukuda, Tatsuhito Kono, Makoto Koyanagi

Discussion Paper 07-17

May 2007
この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。
Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANPublic Expenditure Composition and Economic Growth:
Optimal Adjustment by Using Gradient Method †byTatsuyoshi Miyakoshi*, Yoshihiko Tsukuda**, Tatsuhito Kono***, Makoto Koyanagi**
Abstract:
Previous researches studied how the components of fiscal spending affect the economic
growth but did not explicitly enquire into how to adjust the components in order toachieve the highest rate of economic growth starting from the present shares ofcomponents. We investigate how to determine the optimal adjustment by introducing a
gradient method which explicitly takes account for the adjustment cost and incorporates
the constraint that shares of components are summed up to one. The resulting optimal
adjustment shares are proportional to the deviations from the average over elements of a
gradient vector and independent from the choice of regression equations. The optimal
adjustment share is completely estimated by using the linear regression with any choiceof omitted variable if the adjustment cost is given. The result is free frommulticollinearity problem but is considering all adjustment costs unlike most ofprevious researches. The paper also provides an illustrative example taken from theannual panel data for the Japanese prefectural governments.
JEL classification : E62;E23;H50
Keywords: Economic growth; Public expenditure composition; Adjustment†

Earlier versions of this paper were presented at 2006 East Asian Economic
Association Conference (in Beijing), 2006 All China Economic International
Conference (in Hong Kong), 2007 Western Regional Science Association
Conference (in Newport Beach) and others. We are grateful to Kazuo Mino, Koichi
Futagami, Komei Sasaki, Koichi Mera, Fred Kwan, Joseph Capuno, Mike Leu,andparticipants at the conferences for their helpful comments. The researches of thefirst and second authors were respectively supported by Grant-in-Aid 13630104 andGrant-in-Aid 18530153 from the Ministry of Education, Culture, Sport, Science andTechnology of Japan.
*
Osaka School of International Public Policy, Osaka University,** Graduate School of Economics, Tohoku University,*** Graduate School of Engineering, Tohoku University,
Correspondence: Tatsuyoshi Miyakoshi
Osaka School of International Public Policy, Osaka University
1-31, Machikaneyama-machi, Toyonaka, Osaka, 560-0043, JAPAN.
Tel:+81-6-6850-5638; fax:+81-6-6850-5656, E-mail: miyakoshi@osipp.osaka-u.ac.jp

1.

Introduction
Which component of government expenditure should be cut? Health?, security?,or education? Should tax rate be cut? These problems are important for advanced
countries (Levine and Rehelt (1992), Mendoza, Milesi-Ferretti and Asea (1997), Kneller,
Bleaney and Gemmell (1999) )and also for developing countries (Devarajan, Swaroop
and Zou (1996), Gupta, Clement and Mulas-Granades (2005)). Most of empiricalframeworks employ linear regressions in which the economic growth rate is regressedon the fiscal categories.
Davoodi and Zou (1998) and Xei, Davoodi and Zou (1999) have proposed an
economic growth rate maximization problem in terms of fiscal category shares. Theyfind out the optimal category share to maximize its growth rate. There are two
approaches to solve this optimization problem. The first one is to solve this problem
directly, given all necessary parameters. However, they cannot get sufficient kinds ofdata and then cannot estimate all necessary parameters to solve this problem. Therefore,
they use the second approach, i.e., the gradient method (but implicitly use) and solve theproblem starting from the existing category shares step by step. Then, in order to get thegradient vector at the existing category shares, they linearly approximate the objectivefunction of the growth rate around the existing category shares. Thus, as a result, they
have to estimate the coefficients for the fiscal category shares (which correspond to thegradient vector) in the linear regression, which coincides with the previous empiricalframework and provides a growth theory with its framework.
However, there is widespread non-robustness about results from the linear
regression analyses. The non-robustness may reflect two problems for amuliticolinearity of the estimated coefficients and for the adjustment cost of gradientmethod. For the first problem, they must omit one particular category share in the
regression due to the linear restriction on the fiscal category share: i.e., summing all1

shares equals one. Kneller, Bleaney and Gemmell (1999) have already pointed out thisissue. Moreover, if the category chosen to be omitted is altered, the estimatedcoefficients of the included categories will change. Based on the facts, they insist thatthe investigator must choose a ‘neutral’ omitted category. However, Zhang and Zou(1998), Davoodi and Zou (1998), Xei, Davoodi and Zou (1999), and Jin, Qian andWeingast (2005) have dropped one or two fiscal share variables in an ad hoc mannerwithout clearly stating any reason.1 The second problem is for the adjustment cost ofgradient method. One element of fiscal category shares must be omitted in the
regression equation in order to avoid perfect muliticolinearity, as mentioned above.
However, when we use the gradient method, we need to consider the adjustment cost
for all adjusted category shares. Because, the correct interpretation of the coefficient oneach fiscal category share is as the effect of a unit change in the relevant variable offset
by a unit change in the omitted category share, which involves the adjustment cost of
the omitted category. No paper pays attention on the adjustment cost for the omitted
category. The estimated coefficients will change, depending on the adjustment cost of
the omitted category.
The purpose of this paper is to propose a tractable theoretical framework linking
the share of fiscal expenditures to economic growth, and to produce an empiricalframework adjusting the present shares of component of fiscal spending toward the
optimal ones. Most of previous researches did not explicitly construct an empiricalframework for finding the optimal shares although they studied how the components ofgovernment expenditure affect the economic growth. We introduce a gradient method in

1

Correctly speaking, they investigated the devolution of fiscal power from the national

government to subnational governments to economic growth. They divided total
government expenditure into the three levels of federal, state, and local government
spending.
2

order to determine the optimal adjustment of shares. In this paper, the gradient method
explicitly takes account for the cost of adjusting the share of component andincorporates the constraint that shares of components are summed up to one.
The resulting optimal adjustment shares are proportional to the deviations fromthe average over components of a gradient vector and independent from the choice ofomitted variables from the regression equations. We can completely estimate theoptimal adjustment share by using the linear regression with any choice of omitted
variable if the adjustment size is given. Our result does not suffer from amulticollinearity but is considering all adjustment costs unlike most of the previous
researches. We provide an illustrative example taken from the annual panel data of theJapanese prefectural governments.
Section 2 employs a theoretical framework proposed by Davoodi and Zou (1998)
with a slight modification. Section 3 provides an empirical framework utilizing a
gradient method, and proposes how to adjust the component shares of fiscal spending
towards the optimal level. Section 4 gives an illustrative example taken from the annual
panel data for the Japanese prefectural governments. Section 5 states concluding
remarks.

2.

A theoretical framework
Following Davoodi and Zou (1998), the growth model consists of a production

function with two kinds of inputs: private capital and public spending, where thefunction exhibits constant returns to scale in the two kinds of inputs.2 We depart a little

2

Davoodi and Zou (1998) basically follow Barro(1990) except for division of the

public goods of government into the three categories of central, state and local
governments. Barro (1990, p.107) discussed in detail the questions arising from the3from Davoodi and Zou’s model with three public goods consisting of central, state andlocal government. In our model, public goods consisting of the four fiscal categories ofgovernment expenditure: h (health), s (security), e (education) and r (the remainder(including industrialization and management)). Let k be private capital stock, and g be
total government spending. All variables are measured on a per capita basis andpopulation is constant:h+s+e+r = g.

(1)

The production function is a Cobb-Douglas:y = Ak α h β s γ e δ r λ ,(2)where y is per capita output, α, β, γ, δ and λ are all in [0,1] and α+β+ γ+ δ+ λ=1.3
The allocation of total government spending to each category takes the followingform:h= θ h g , s= θ s g , e= θ e g , r= θ r g ,and θ h +θ s +θ e+θ r = 1,(3)

where θ h, θ s ,θ e and θ r are respectively the shares of health, security, education and theremainder in the total spending on the interval [0,1]. The total government spending isfinanced by income tax at fixed rate τ:
g =τ y.

(4)Household behavior
We consider a long-lived household who chooses the consumption path {c(t): t≥0}

specification of public services as an input to production.
3

The production function in (2) is plausible if we think of k to include human capital,

or if we think of a production function Y=AL1-αKαG1-α proposed by Barro andSala-i-Martin (1995, p.153) where Y, L, K and G are aggregate output, labor force,
capital and government expenditure.
4

to maximize his discounted utility,U=∫∞ c 1−σ0− 1 −ρ te dt ,(5)1−σsubject to:(i) the dynamic budget constraint of the household4 5:•

k = (1 − τ ) ykk −1 − c = (1 − τ )τ (1−α ) / α k ( Aθ hβ θ sγ θ eδ θ rλ )1 / α − c ,(6)(ii) the value constraints:

0 ≤ c ≤ y , k0 given ,(7)

where σ is the inverse of the inter-temporal elasticity of substitution, and ρ is the rate oftime preference. After taking the government’s announcement on the tax rate τ and theshare of each fiscal category θ h ,θ s,θ e and θ r , the household chooses the consumption
path. We write the Hamiltonian:⎡ c1−σ − 1 ⎤ − ρ t(1−α ) / αH =⎢k ( Aθ hβθ sγ θ eδ θ rλ )1/ α − c .
⎥ e + φ (1 − τ )τ⎣ 1−σ ⎦()

(8)

The necessary conditions for the optimal path of consumption are given by

4

We formulate that the household is able to perform the production process since thesame equations as those in a decentralized economy emerge. Using (4) yields y=g/τ,
which together with (2) can rewrite k-1 by only g, τ, θ and A. We obtain (6) by insertingthese relations into y and k-1.
5

In Davoodi and Zou (1998, p.247), they consider k = (1 − τ ) y − c , where y isexogenously given for a household. The difference between (6) and theirs comes fromthe fact whether or not the household already knows a production function and the tax
to be financed for the government expenditure before the household decides a
consumption plan. The household knows them in our model. This difference leads toA1/α in (11) and αA1/α in Davoodi and Zou (1998, p.247), which has no influence on theanalytical results but makes the expression clear.
5∂H= e − ρ t c −σ − φ = 0∂c•∂H≡ φ (1 − τ )τ (1−α ) / α A1 / α θ hβ / α θ sγ / α θ eδ / α θ rλ / α = − φ .
∂k(9)Using the two equations above, we have6
φ(t)k(t) → 0 (The transversality condition),
•

•

[

(10)

]

y c 1
η ≡ = = (1 − τ )τ (1−α ) / α A1 / α θ hβ / α θ sγ / α θ eδ / α θ rλ / α − ρ ,(The Euler equation).
y c σ

(11)

Since the transition dynamics for φ does not exist, we can obtain the optimal pathdirectly, and the consumption growth coincides with the rate of the growth of output
and capital. The equations (6), (7), (10) and (11) determine the optimal path of
consumption and capital. We simply consider the Euler equation (11) as one ofnecessary condition for analysis.7

Government behavior

We assume that the government maximizes the growth rate of output in (11) bychoosing ω = (τ, θ h, θ s, θ e , θ r ), while the value of ω is fixed for the household,subject to the constraint on the shares for fiscal category:Max η ≡ F (ω ) ,

(12)

ω

subject to

6

of
7

By taking derivative of••∂H= 0 in terms of t, φ / φ = − ρ − σ c/ c . We insert the relation∂c∂H= 0 into this relation. We obtain (11).
∂k

We assume the following inequality:ρ > {(1 − σ ) / σ }{(1 − τ )τ (1−α ) / α A1 / α θ hβ / α θ sγ / α θ eδ / α θ rλ / α − ρ} .

The inequality assures the transversality condition.
6θ h + θ s + θ e + θ r = 1, 0 ≤ θ h , θ s , θ e , θ r ,τ ≤ 1 .

The F is continuous, the constraints are compact sets, and then the maximum andminimum exist. The F is also a strictly quasi-concave (while the constraint for θ isconvex) and then the maximum is unique. Then, the Lagrange equation and the
Kuhn-Tucker conditions hold (assuming that there is no corner solution):
L = F (ω ) + ϕ (1 − θ h − θ s − θ e − θ r ) ,
(13)
∂L
∂L
= 0,
=0.
∂ω
∂ϕ
The solutions for the shares of health, security, education and the remainder, and for thetax rate are given by
θh* =

β

, θs * =γ

β + γ +δ + λ
β +γ +δ + λλθr * =, τ* = β + γ + d + λ .
β +γ +δ + λ, θe* =

δβ +γ +δ + λ

,

(14)

If the actual expenditure for each share and the actual tax rate do not coincide with thesolutions of (14), reallocation of resources will enhance growth rate. We need allparameters in (14) in order to numerically determine the solutions. In order to estimate
those parameters, we need a data set of either (θ, τ , ρ, σ, y) or (k, h, s, e, r, y).
Howeever, it is not easy to obtain the data of (ρ, σ, k) in any country.8 It is reasonable
that Davoodi and Zou (1998) and Xei, Davoodi and Zou (1999) linearly approximatedthe function F (ω ) at the present level or the average of data for ω. They came to use thegradient method as a result to solve this problem step by step and sought the gradientvector, i.e., the estimated coefficients for ω in the linear regression. They proposed anempirical framework in which growth rate is linearly regressed on ω, which is the same
framework as the previous one without the growth theory.

8

In Appendix 1, we explain how to estimate (β, γ, δ, λ) by using these data.

7

3 An empirical framework
3.1 An optimal adjustment of sharesThis section investigates how to adjust the shares of fiscal expenditure startingfrom the present level of ω = (ωτ , ωθ ) = (τ ,θ h ,θ s ,θ e ,θ r ) to the optimal one in which therate of economic growth is maximized. We introduce a gradient method for solving (12)step by step which provides the optimally adjusting shares, and relate this method to theempirical framework of regression analysis.9 The gradient method extends the previous
researches by Davoodi and Zou (1998), Xei, Zou and Davoodi (1999), and Kneller,
Bleaney and Gemmell (1999) among others in the sense that it brings them a framework
to optimally adjust the shares of fiscal components.
We formulate a gradient method for maximizing a linearized functionof η ≡ F (ω ) in (12) at the present level of ω as follows:
Max dF = Γ' Z

(15)

z

subject toZ ′Z = ξ 2 for ξ > 0,

and J 4′ Z θ = 0,

where the partial differential coefficients Γ ′ = (Γτ , Γθ′ ) = ( Fτ , Fθ h , Fθ s , Fθ e , Fθ r ) withFν =∂F
for ν = τ , θ h , θ s , θ e and θ r are called a gradient vector, the differentials∂νZ ′ = ( Z τ , Z θ′ ) = ( dτ , dθ h , dθ s , dθ e , dθ r ) evaluated at the present level ofω = (ωτ , ωθ ) denotes the adjustment of shares, the constant scalar value ξ represents a

norm of Z, and J4 = (1,1,1,1)′ is a 4 × 1 unit vector. The second constraint in (15)indicates that the components of the differentials of share variables add up to zero. Thiscondition is necessary because both the present and optimal shares must

9

The gradient method is a standard tool for stepwise maximizing a nonlinear function.

See, for instance, Bazaraa, Sherali, and Shetty (1993) for detailed explanation.
8

satisfy θ h + θ s + θ e + θ r = 1 .
The norm ( ξ ) of Z can be interpreted as a size of adjustment. If the cost foradjusting the shares of fiscal spending is proportional to the size of adjustment, the sizeof ξ can be a measure of adjustment cost. The adjustment cost consists of equallyweighted components in Z. For any given ξ > 0, the problem of (15) determines the
optimal adjustment in the sense that the solution provides the highest growth rate
differential of dF(= dη) in (12). For easy of expositions, we introduce a notation(γ 0 , γ 1 , γ 2 , γ 3 , γ 4 ) ≡ (Γτ , Γθ′ ) which plays an important role in the following discussion.

The problem in (15) is maximized at
⎛Z* ⎞
⎛Γ ⎞
−1 / 2
1/ 2
Z * = ⎜⎜ τ* ⎟⎟ = ξ ⎜⎜ τ* ⎟⎟ Γτ2 + Γθ* ' Γθ*with its maximum dF * = ξ Γτ2 + Γθ* ' Γθ* , (16)⎝ Γθ ⎠⎝ Zθ ⎠()

where Γθ* = (γ 1 − γ ,… , γ 4 − γ )' , and γ =

(

1
4

)

∑ γ . The derivation of (16) is given in
4

j =1

j

Appendix 2. The solution Z θ* is proportional to Γθ* which represents the partial
differential coefficients of shares measured from their averaged value. The vector Z*
determines the highest economic growth rate differential at the present level of ω. The
share of government expenditure is adjusted from ω to ω∗ ≡ ω + Z* and the economic
growth rate is adjusted from F (ω ) to F (ω*) ≅ F (ω ) + dF * .10
Figure 1 illustrates how the gradient method works started from the present levelof ω at the point P. The objective function dF = Γ' Z of (15) indicates the tangent surface
to η = F (ω ) at the point of P. The optimal vector Z* is indicated by the arrow startingfrom the point P.11

10

We can choose any value of ξ as long as ω∗ satisfies the constraint for the problem of

(12).
11

Unlike the standard gradient method, the optimal differential vector Z* is notnecessarily orthogonal to the tangent line q because of the constraint J 4′ Z θ = 0 .
9[Insert Figure 1]

In order to estimate the gradient vector from the observed data of ω = (ωτ , ωθ ) ,following Davoodi and Zou (1998), and Kneller, Bleaney and Gemmell (1999) welinearly approximate the function F(ω) and set up the linear regression:ηit = A 0 + γ 0τ it + γ 1θ h, it + γ 2θ s, it + γ 3θe, it + γ 4θ r, it + B′ Di + C′ Tt + uit(17)

where i = 1,…, I and t = 1,..., N refer to prefecture i and time t; A0 , γ 0 , γ 1 , γ 2 , γ 3 , andγ 4 are the parameters for constant term, tax rate, health, security, education, and theremainder respectively. B and C are (I-1) × 1 and (N-1) × 1 coefficient vectors; prefecture
dummy Di is a (I-1) × 1 vector of prefecture fixed-effects, time dummy Tt is a (N-1) × 1
vector of time fixed-effects, and uit are the iid random errors with normal distribution
N(0, σu2).The coefficient vector of (γ 0 , γ 1 ,..., γ 4 ) in (17) represents the gradient vector in(15). We cannot estimate the parameters by using the ordinary least square (OLS)
because the regression (17) has a perfect multicollinearity problem due to the restrictionof θ h +θ s +θ e+θ r = 1. Kneller, Bleaney and Gemmell (1999) pointed out that previous
researches had suffered from this difficulty.12
An alternative formulation of (15) is to directly incorporate the restriction on theshare variables by deleting one of dθ j ( j = h, s, e, and r ) . We delete dθ r by inserting theconstraint dθ r = - dθ h - dθ s - dθ e into (15). The maximization problem becomes
Max dF = ∆'W
W

12

The fiscal variables in our model are h (health), s (security), e (education), r (theremainder), and π( ≡ τy, amount of tax). The budget constraint is satisfied since theequation τ (=π/y) = (h + s + e + r)/y for defining the tax rate is employed in (17) and inan empirical study in Section 4. The formulation in (17) has no multicollinearity due tothe budget constraint for fiscal variables, but has another muliticolliniearity due to theshare variables.
10

subject to

0′⎛1⎞⎟⎟W = ξ 2 ,W ′⎜⎜⎝ 0 I 3 + J 3 J 3′ ⎠

ξ >0,

(18)

where ∆ ′ = (∆τ , ∆ θ ' ) = (γ 0 , γ 1 − γ 4 , γ 2 − γ 4 , γ 3 − γ 4 ) , W ′ = (Wτ , Wθ′ ) = (dτ , dθ h , dθ s , dθ e ) ,and I 3 is a 3 × 3 unit matrix. We note that ∆ θ represents the partial differential
coefficients measured from the value of γ 4 , while Γθ in (16) represents the coefficientsmeasured from their averaged value. Though the formulation of (18) has a different
expression, it is equivalent to that of (15).
The problem is maximized at−1 / 2⎛W ** ⎞⎛ ∆ ⎞⎛
9⎞
W * * = ⎜⎜ τ** ⎟⎟ = ξ ⎜⎜ *τ* ⎟⎟⎜ ∆2τ + ∆*θ* ' ∆*θ* + ∆2θ ⎟16 ⎠⎝ ∆θ ⎠⎝
⎝Wθ ⎠

9
⎛
⎞
with dF * * = ξ ⎜ ∆2τ + ∆*θ* ' ∆*θ* + ∆2θ ⎟
16 ⎠
⎝

1/ 2

, where ∆*θ* = ∆ θ −

(19)

1
3
J 3 ∆θ , and ∆θ = J 3 ' ∆ θ .
3
4

The derivation is given in Appendix 2. Plugging W** into the constraint dθ r = - dθ h dθ s - dθ e , we have the optimal adjustment of Z asZ**

⎛ ∆τ ⎞−1 / 2⎜⎟⎛ 2⎛ Z τ** ⎞
9 2⎞
**
**
**
⎜
⎟
.
= ⎜ ** ⎟ = ξ ⎜ ∆ θ ⎟⎜ ∆ τ + ∆ θ ' ∆ θ + ∆ θ ⎟16⎠⎝ Zθ ⎠⎜ − J ′ ∆** ⎟⎝
3 θ ⎠
⎝

(20)

The solution in (20) is identical to that of (16), and independent from the choice ofomitted variables as proved in Appendix 2. The solution Wθ** is proportional to ∆*θ* . An
advantage of the expression in (20) over (16) is that we do not need each parameter ofγ j ( j = h, s, e, and r ) since Z** is expressed only in terms of(∆ 0 , ∆ θ ' ) = (γ 0 , γ 1 − γ 4 , γ 2 − γ 4 , γ 3 − γ 4 ) .

In order to estimate the values of ∆ 0 and ∆ θ , we can use the linear regression
omitting θ r , it from (17):ηit = A 0 + γ 0τ it + (γ 1 − γ 4 )θ h, it + (γ 2 − γ 4 )θ s, it + (γ 3 − γ 4 )θe, it + B′Di + C′Tt + uit .

11

(21)

Unlike the regression of (17), the multicollinearity does not exist in (21) and the OLS
method is applicable. We can compute the solution (19) by using the estimates of ∆ anddetermine the optimal adjustment Z**. The share of government expenditure is adjusted
from ω to ω∗∗ ≡ ω + Z**. We can say that when a particular component of Z** ispositive, the share of its fiscal category is below the optimal level and we should
increase it towards the optimal level.
We can apply the other three regressions analogous to equation (21), depending on
what variables we omit from the regression equation. In fact, Kneller, Bleaney andGemmell (1999) examined the regression analyses with changing omitted variables.
However, once we choose a particular equation, say (21) for example, the additional useof other regression equations does not provide further information in the sense that theestimates of all other regressions can be computed by using only the estimates of theoriginally chosen regression equation.13 This fact will be empirically illustrated in
Section 4.

3.2 Comparison with the framework of previous researches

We review the previous researches of Davoodi and Zou (1998), Xei, Zou andDavoodi (1999), and Kneller, Bleaney and Gemmell (1999) from the view point of
optimal adjustment to maximize the differential rate of economic growth and clarify thesignificance of their contributions. Most of previous researches understood the problem
along the following line of arguments. The estimated positive coefficient of a particularvariable means that the present level of its category is below the optimal and the present

13

The claim above mentioned is proved in Appendix 3. This claim may provide a

caution against some previous researches. Kneller, Bleaney and Gemmell (1999), forexample, seem carried out the regression analyses with changing omitted variables
without being aware of this fact.
12

level of the variable should be increased toward the optimal with the same amount ofdecrease in the omitted variable as illustrated in Figure 2. However, they did not
explicitly examine what combination of shares induces the maximum rate of economic
growth.
[Insert Figure 2]

We formulate the maximization problem:
Max dF = ∆ 'W

(22)

W

subject to W ′W = ξ ′ 2 , ξ ′ > 0 ,where ξ ′ is positive constant but not necessarily equal to ξ in (18). This problem is
identical to (18) except for the constraint. The equation (22) imposes a constraint
directly on the norm of the vector W after deleting one of the share variables. On theother hand, (18) essentially imposes a restriction on the original vector Z. The solution
to (22) is as follows:
W

Z

***

***

⎛W *** ⎞⎛ ∆τ ⎞= ⎜⎜ τ*** ⎟⎟ = ξ ′⎜⎜ ⎟⎟(∆2τ + ∆ θ ' ∆ θ ) −1 / 2 , with dF *** = ξ ′(∆2τ + ∆ θ ' ∆ θ )1 / 2 ,⎝ ∆θ ⎠⎝Wθ ⎠(23)⎛ ∆τ ⎞⎟⎜⎛ Zτ*** ⎞= ⎜⎜ *** ⎟⎟ = ξ ′⎜ ∆ θ ⎟(∆2τ + ∆ θ ' ∆ θ ) −1 / 2 .
⎝ Zθ ⎠⎜− J '∆ ⎟⎝ 3 θ⎠The derivation is given in Appendix 2. The optimal adjustment of Wθ *** = (dθ h, dθ s, dθe )′ is proportional to the vector of ∆ θ .

The result of (23) implies that the share of anelement of Wθ *** should be increased when the sign of corresponding element of ∆ θ ispositive. Though the argument illustrated in Figure 2 is consistent with the optimal
adjustment of Wθ *** as far as the sign of adjustment for each component is concerned, theprevious researches did not discuss how to obtain the optimal share of adjustment even
within the framework of (22). And the framework of (22) is itself not preferable to that13

of (15) as will be explained later in this section.
The result of (23) implies that the optimal adjustment vector depends on the
choice of omitted variable, and it gives a different vector from (23) if we choose analternative omitted variable in the regression. For instance, if the variable θ e is dropped~

instead of θ r , ∆ θ = (γ 1 − γ 3 , γ 2 − γ 3 , γ 4 − γ 3 )′ is the gradient vector for the corresponding~~

maximization problem and the solution Wθ *** = (dθ h , dθ s , dθ r )′ is proportional to ∆ θ .
The optimal adjustment of components (dθ h, dθ s ) can be different in sign among thetwo regressions. Kneller, Bleaney and Gemmell (1999) tried to find the regression
which would provide a plausible solution of Z by selecting appropriate omitted
3
4variables. On the other hand, the optimal differential ∆*θ* = ∆ θ − J 3 ∆θ for the problemof (18) is not necessarily equal to ∆ θ . Even the sign of element of ∆*θ* could be
opposite to that of the corresponding element of ∆ θ . The solution W*** is notnecessarily justified from the analysis of (18).
An underlying key idea that distinguishes the two problems lies in how tomeasure the cost for adjusting the fiscal shares. The maximization problem in (15) orequivalently in (18) measures the norm of the original variables before omitting one ofthe variables while the problem in (22) does it after omitting one variable. The former
formulation equally reflects the adjustment of each component of the shares in the cost,
while the latter one takes account for only the components in the regression but ignoresthe cost for adjusting the omitted variable. The solution to the former problem does notdepend on which variable we omit from the regression equation. But, the latter actually
does because the ignored costs differ among the regressions. We believe that the former
solution is more appropriate than the latter one.

14

4.

An illustrative example by using the panel data for Japanese prefectures

We provide an empirical example from the annual panel data for Japanese
prefectures for the purpose of illustrating the discussion in section 3.14

4.1 Data sources

The data are take from “Annual Report of Local Finance” published by theInstitute of Local Finance (Chihou-zaimu -kyokai, in Japanese), and “Annual Report ofPrefectural Account” published by the Cabinet Office of the Japanese government. The
data for the 47 prefectures are available during the fiscal years 1981-2002. Eachvariable of the government expenditures are compiled from “Annual Report of Local
Finance” (Settlement of Expenditure by Purpose) and defined as:h: (Health) = welfare + hygiene;s: (Security) = police + fire fighting;e: (Education) = education;r: (Remainder) = Commerce and manufacturing + Civil engineering works+ Agriculture, forestry and fisheries + the others;g : (Total government spending) = total value of the items above.

14

Davoodi and Zou (1998) work with time-averaged data since the benefits of fiscal

decentralization are not expected to affect year-to-year fluctuations in growth. The
growth regression is estimated on data averaged over five- and ten-year periods.
Accordingly, the dependent variable is the average growth rate over these two periods.
However, we use annual data as Zhang and Zou(1998), Xie, Zou and Davoodi (1999)
and Gupta, Clements, Baldacci and Mulas-Granados (2005) did. For the sake ofsimplicity, without implementing the Hausman test with a null hypothesis of nocorrelation among the individual effects and the error term, we adopt the fixed effects
model in (21).
15The shares of h, s, e and r to the g are given by θ h ≡ h/g , θ s ≡ s/g , θ e ≡ e/g , θ r ≡ r/g .
The output and tax are taken from “Annual Report of Prefecture Account” (Summary
Tables)15:
y: (Output) = Gross Prefecture Domestic Product at constant prices;τ : (Tax rate) ≡ g*/y, g* = total government expenditure at constant prices;
η: (Growth rate of real output) = the growth rate of y.
These variables are measured in per capita base at constant prices and the population
data are compiled from the latter data base.

4.2 Estimated results and illustration

Table 1 indicates descriptive statistics for the data set. During the period of
1981-1990, the GDP of prefectures grew, on average, at the rate around 3.78% per
capita per annum, and the average tax rate was 12.85%. Among the expenditures offiscal variables, the share of health, education, security, education, remainder categories
accounted for 9.38, 5.63, 27.21, and 57.78%, respectively. On the other hand, during theperiod of 1991-2002, the growth rate of GDP dropped down to 0.83% while the tax rateincreased to 13.8%. The share of health increased to 10.15%, but the shares of security
and education decreased. Examining Table 1, we suppose that there exists a structural
break between the periods 1981-1990 and 1991-2002, and divide the sample periods
into the two parts.
[Insert Table 1]

Table 2 summarizes the results of estimating equation (21). The values in the15

The data of 1981-1990 are based on 1963SNA standard and the data of 1991-2002

based on 1993SNA standard, and measured at 1997 prices. The Cabinet Office of theJapanese government does not provide the re-estimated data prior to 1990 based on the
1993SNA standards.
16

columns 1 to 4, respectively, show the estimates with the omitted variables, h, s, e, and r.
The estimated coefficients are different depending what variable is dropped from theregression equation. More precisely, the estimated coefficients in each panel are
symmetric about the diagonal elements as expected from proposition 1 in Appendix 3.
For example, if we take m = 4 and n = 1 in Appendix 3, we have γˆ1, 4 = −0.372 (thecoefficient of health when the remainder is omitted), and γ~4,1 = −γˆ1, 4 = 0.372 (thecoefficient of the remainder when health is omitted). Similarly, we have γˆ 2, 4 = 0.534
and γ~2,1 = γˆ 2, 4 − γˆ1, 4 = 0.534 − (−0.372) = 0.906 for m = 4 and n = 1. According to theframework of Section 3.2, the adjustment shares of W** in (23) are proportional to thevector of ∆ θ , which are given by the estimates in each column depending on the omitted
variable. Kneller, Bleaney and Gemmell (1999) tried to find the regression which would
provide a plausible solution of differentials by selecting the variable to be omitted.
[Insert Table 2]

In order to evaluate the optimal adjustment share of Z**, we assume that the
government wants to adjust the shares of component by 2% points as a total from thepresent level, that is ξ = 0.02. Moreover, we assume the present level is equal to thesample means for each period. Table 3 shows ∆ ′** = (∆τ , ∆ θ′** ,− J 3′ ∆*θ* ) , ω (present levelsof the share), Z** (optimal adjustment vector), and ω ** (adjusted vector of the shares)for each sample period. The entries of Z** in Panels (a) and (b) satisfy the restrictiondθ h +dθ s+ dθ e +dθ r = 0 in (15), and are independent from the choice of omitted
variables. For the first period (1981-1990), we obtain the largest increase (dF** =1.70% points) of the growth rate from the level at the sample means of ω if we choose
the adjustment vector of Z ′** = (-0.79, -0.69, 1.40, -0.87, 0.17) % points. The vector ofZ** neither coincide with nor are proportional to the columns (1) through (4) on Panel17

(a) in Table 2.
[Insert Table 3]

5. Concluding remarks.

The previous researches proposed theoretical frameworks for analyzing the
optimal share of fiscal spending and attempted to implement them into empirical studies
by using the linear regressions, while they did not explicitly construct an empiricalframework for finding the optimal share or the optimal adjustment starting from thegiven fixed share. They only argue that the present level of a particular category is
below the optimal and the level of this category should be increased toward the optimal
with the same amount of decrease in the omitted variable if the estimate of coefficientfor the corresponding variable is positive. The result from this discussion cruciallydepends on what the omitted variable is.
This paper proposes a tractable theoretical framework linking the share of fiscal
expenditures to economic growth and an empirical framework adjusting the present
shares of component of fiscal expenditures toward the optimal ones. We introduce a
gradient method in order to determine the optimal adjustment of shares. The gradientmethod explicitly takes account for the cost of adjusting the share of component andincorporates the constraint that shares of components are summed up to one. The
optimal adjustment shares are proportional to the deviations from the average overcomponents of a gradient vector and independent from the choice of omitted variablesfrom the regression equations. The discussion of previous researches is not necessarily
conformable to our result. We can completely estimate the optimal adjustment share byusing the linear regression with any choice of omitted variable if the adjustment cost ofξ is given. In this sense, our result does not suffer from a multicollinearity in the linear18

regression but is considering all adjustment costs unlike most of the previous researches.
An empirical example from the annual panel data for Japanese prefectures illustrates
how the procedure proposed in this paper numerically works.

19

Appendix 1:

Direct methods to find the optimal solution of (14)We need all parameters in (14) to find the optimal solution. The first way to obtain themis to estimate the regression for (11) with a slight modification. After taking thelogarithm of (11), we move the first term of the right hand side to the left and estimate
the regression:⎛y ρ⎞1G ≡ ln⎜⎜ + ⎟⎟ − ln (1 − τ )
σ
⎝y σ⎠
1−α
1
β
γ
δ
λ
lnτ + ln A + lnθ h + lnθ s + lnθe + ln 1 − θ h − θ s − θe
=

α

α

α

α

α

α

(

)

(A1)

by using the data of θ h , θ s , θ e , θ r , τ , y and ρ, σ.
The second way is to estimate the following function:y / k = A(h / k ) β ( s / k ) γ (e / k ) δ (r / k ) λ .

(A2)where the first order homogeneous restriction α+β+ γ+ δ+ λ=1 is incorporated in the
Cobb-Douglas production function (2).The second way needs the data of h, s, e, r, y andk. The main difference between the two formulations lies in the point whether werequire the data of k in (A2) or (ρ, σ) in (A1). In general, the data of k is so muchdifferent in each prefecture, though the data of (ρ, σ) is not. The reliable data of k foreach prefecture is necessary but not available for the most countries. The
prefecture-basis capital stock k is not published in Japan. The second way may not be
useful under this circumstance of data availability in Japan.

Appendix 2 : Derivations of (16), (19), (23)
Derivations of (16): We can solve the problem of (15) by using the Lagrange multipliermethod:1
L = Γ ′Z + ϕ1 (ξ 2 − Z ′Z ) − ϕ 2 (0, J 4′ ) Z ) ;
2
∂L
∂L
∂L
= Γ − ϕ1 Z − ϕ 2 (0, J 4′ )′ = 0 , (ii)= ξ 2 − Z ′Z = 0, (iii)= J 4′ Z θ = 0, (A3)
(i)
∂Z
∂ϕ1
∂ϕ 2

20

where Γ ′ = (Γτ , Γθ′ ) and Z ′ = ( Z τ , Z θ′ ) are evaluated at the present level ofω ′ = (ωτ , ωθ′ ) . From (i), we have
⎛ Zτ ⎞ 1 ⎛⎞⎛ 0 ⎞ ⎞ 1 ⎛ Γτ⎟⎟ .
Z = ⎜⎜ ⎟⎟ = ⎜⎜ Γ − ϕ 2 ⎜⎜ ⎟⎟ ⎟⎟ = ⎜⎜⎝ J 4 ⎠ ⎠ ϕ1 ⎝ Γθ − ϕ 2 J 4 ⎠⎝ Z θ ⎠ ϕ1 ⎝Using (iii) and (A4), we obtain 0 = J 4′ Z θ =ϕ 2 = ( J 4′ J 4 ) −1 J 4′ Γθ =

1
4

1

ϕ1

J 4′ (Γθ − ϕ 2 J 4 ) , and∑ γ =γ .
4

j =1

(A4)

(A5)

j

Substituting ϕ 2 in (A.4) by γ , we have
Zθ =

1

ϕ1

(Γθ − γ J 4 ) ≡

1

ϕ1

Γθ* .

(A6)

Then, due to (A4) and (A6), we have
⎛ Zτ* ⎞ 1 ⎛ Γτ ⎞Z * = ⎜ * ⎟ = ⎜⎜ * ⎟⎟ ,⎜ Z ⎟ ϕ1 Γ
⎝ θ⎠
⎝ θ⎠

and ϕ1 =

1

ξ

(A7)

(Γτ2 + Γθ′* Γθ* )1 / 2 from (ii) and (A7). Finally, we obtaindF * = Γ ′Z * = ξ (Γτ2 + Γθ′* Γθ* )1 / 2 .

Derivations of (19): In a similar manner to (15), we solve the problem of (18):1
L = ∆ ′W + ϕ1 (ξ 2 − W ′AW ) ;2∂L∂L= ∆ − ϕ1 AW = 0 , (ii)= ξ 2 − W ′AW = 0,
(i)
∂ϕ1
∂W
⎛1

0'

(A8)

⎞

⎟⎟ and I 3 is a 3 × 3 unit matrix. Then, due to (i), the solution is,where A = ⎜⎜⎝ 0 I 3 + J 3 J 3′ ⎠∆⎞⎞ 1 ⎛⎜ τ
1 ⎛⎜ ∆τ⎟ 1 ⎛ ∆τ ⎞⎟W = ⎜= ⎜ ⎟,= ⎜3−1⎟
ϕ1 ⎝ ( I 3 + J 3 J 3′ ) ∆ θ ⎠ ϕ1 ⎜ ∆ θ − ∆θ J 3 ⎟⎟ ϕ1 ⎜⎝ ∆*θ* ⎟⎠4⎝⎠

where a relation ( I 3 + J 3 J 3′ ) −1 = I 3 −1
J 3 J 3′ is used.
4

Due to (ii) and (A9), we have ξ 2 = W ′AW =

1

ϕ

21

2
1

(∆20 + ∆ θ′** ( I 3 + J 3 J 3′ )∆*θ* ) and

(A9)

1⎛

1/ 2

⎞

9

ϕ1 = ⎜α 02 + ∆*θ* ' ∆*θ* + ∆2θ ⎟
ξ⎝
16 ⎠

(A10)

The solution in (19) follows from (A.9) and (A.10).The last element of Z** in (20) is
obtained from dθ r ** + J 3′ ∆*θ* = 0 .
The equivalence of Γθ * = ∆*θ* follows from the fact: for j = 1, 2, 3∆*θ*, j = (γ j − γ 4 ) −31∆θ = γ j −
4
4

∑ γ

and ∆*θ*, 4 = −∑ j =1 ∆*θ*, j =γ 4 −4i =1 i

3

1
4

∑ γ .
4
i =1 i

Derivations of (23): The problem of (22) can be understood as a special case of (18) inwhich the matrix A is replaced by a unit matrix:1
L = ∆ ′W + ϕ1 (ξ ′ 2 − W ′W ) ;
2

(i)

∂L
= ∆ − ϕ1W = 0 ,
∂W

(ii)

(A11)

∂L
= ξ ′ 2 − W ′W = 0 .
∂ϕ1Then, in an analogous way to (18) we obtain the solutionW=

1 ⎛ ∆τ ⎞⎜ ⎟,ϕ1 ⎜⎝ ∆ θ ⎟⎠Due to (ii) and (A12), we have ϕ1 =

(A12)

(

)

1/ 2
1 2
∆τ + ∆ θ ' ∆ θ. The last element of Z*** in (23)ξ′

is obtained from dθ r *** + J 3′ ∆ θ = 0 .

Appendix 3 : Effects of omitted variables on the estimates

This appendix examines the effect of omitted variables on the estimates in aregression with perfect collinearity, and shows that any additional use of regressions
with different omitted variables can not extract further information from the given data
set since the estimates of all other regressions can be computed from the estimates ofthe originally chosen regression equation. We use a model and notation of Kneller,
Bleaney and Gemmell (1999) in this appendix, which are not necessarily conformableto the main body, for the purpose of clarifying their discussion. Let a linear regression

22

bey it = β 0 +∑ β Y +∑ γ X
k

j =1

m

j

jit

j =1

j

jit + u it ,i = 1,..., I , t = 1, … , N(A13)with a linear constraint among the variables: ∑ j =1 X jit = 0 , where Y jit are non-fiscalmvariables and X jit denote fiscal variables . If we omit the variable X mit from (A13), theequation is expressed as
y it = β 0 +∑ β Y +∑ γ
m −1

k

j =1

j

jit

j =1

j , m X jit + u it ,(A14)

where γ j ,m = γ j − γ m , j = 1, …, m − 1 . On the other hand, if we omit anothervariable X n, it , the equation has the form:y it = β 0 +∑ β Y +∑
k

j =1

j

jit

m
j =1 ( j ≠ n )γ j , n X jit + u it ,(A15)

where γ j , n = γ j − γ n ; j = 1,…, m ( j ≠ n) .T he following relation among the coefficients of(A14) and (A15) holds:
γ j , n = γ j ,m − γ n,m j = 1,…, m − 1( j ≠ n) ; and γ m, n = −γ n, m .

(A16)

We can utilize the OLS method to estimate the parameter of either (A14) or (A15),
while it is not applicable for (A13). The next proposition indicates that the estimates of(A14) completely determine those of (A15).
Proposition 1: Let the OLS estimates of (A14) and (A15) be γˆ j ,m and γ~ j , n respectively.

Then, the estimates γ~ j , n are completely determined by the following equation:
γ~ j , n = γˆ j ,m − γˆ n, m and γ~m, n = −γˆ n, m .

(A17)

Proof: We can assume n = m-1 without loss of generality by arranging the order ofvariables. In matrix notations, equation (A14) and (A15) are respectively expressed as:23y = Y∆ β ∆ + X ∆ γ ∆ ,m + X m −1γ m −1,m + u = Wβ + u ,Y∆ = (Y0 , Y1 ,… , Yk ), X ∆ = ( X 1 , … , X m − 2 ), W = (Y∆ , X ∆ , X m −1 ),(A18)

β ′ = ( β ∆′ , γ ∆′ ,m , γ m−1,m ),and similarlyy = Y∆ β ∆ + X ∆ γ ∆ ,m−1 + X m γ m,m −1 + u = Zδ + u ,Z = (Y∆ , X ∆ , X m ), δ ′ = ( β ∆′ , γ ∆′ ,m , γ m −1,m ).

(A19)

We have the relation among the matrices of independent variables W and Z:⎛ I k +1⎜
W = ZQ, Q = ⎜ 0
⎜ 0
⎝

0
I m−2
0

0 ⎞
⎟
− J m−2 ⎟ ,− 1 ⎟⎠(A20)

where the constraint X ∆ J m−2 + X m−1 + X m = 0 is used. From (A18), (A19) and (A20),
we have y = ZQβ + u , implying δ = Qβ . Plugging W = ZQ in the normal equation~−1
W ′Wβ̂ = W ′y for (A18), we have Qβˆ = (Z ′Z ) Z ′y = δ . This proves Proposition 1.

24

References

Barro, J. R. (1990), “Government Spending in a Simple Model of Endogeneous
Growth,” The Journal of Political Economy, 98, 5(2), 103-125.
Barro, J. R. and Sala-i-Martin,X.(1995), Economic Growth,McGraw-Hill,Inc. New
York.
Bazaraa, M. S., Sherali, H. D., and Shetty, C., M. (1993), “Nonlinear Programming:
Theory and Application”, John Wily & Sons, Inc., New York.
Davoodi, H., and Zou, H. (1998), “Fiscal Decentralization and Economic Growth:
A Cross-Country Study,” Journal of Urban Economics, 43, 244-257.
Devarajan, S., Swaroop, V., and Zou, H. (1996), “1996 The Composition of Public
Expenditure and Economic Growth,” Journal of Monetary Economics, 27,
313-344.
Gupta, S., Clements, B., and Mulas-Granados, C. (2005), “Fiscal Policy, Expenditure
Composition, and Growth in Low-Income Countries,” Journal of International
Money and Finance, 24, 441-463.
Kneller, R., Bleaney, F. M., and Gemmell, N. (1999), “Fiscal Policy and Growth:
Evidence from OECD Countries,” Journal of Public Economics , 74, 171-190.
Levine, R.and Renelt, D. (1992),” A Sensitivity Analysis of Cross-Country Growth
Regressions”, American Economic Review, 82, 942–963.
Mendoza, E., Milesi-Ferretti, G., and Asea, P. (1997),”On the Effectiveness of Tax
Policy in Altering Long-Run Growth: Harberger’s Superneutrality Conjecture”,
Journal of Public Economics, 66, 99–126.
Xei, D., Zou, H., and Davoodi H. (1999), “Fiscal Decentralization and Economic
Growth in the United States,” Journal of Urban Economics, 45, 228-239.
Zhang, T., and Zou, H. (1998), “Fiscal Decentralization, Public Spending, andEconomic Growth in China,” Journal of Public Economics, 67, 221-240.

25

Table 1. Descriptive statistics

Panel (a) (1981-1990)
Mean

Standard

Minimum

Maximum(%)

deviation(prefecture/year)(prefecture/year)

Growth rate of prefectural GDP

3.78

2.26

-8.97(Wakayama/1983)

9.92(Kagawa/1988)

Share of health

θh

9.38

1.64

6.24(Niigata/1987)

15.29(Fukuoka/1984)

Share of security

θs

5.63

2.05

1.53(Shimane/1984)

19.11(Tokyo/1986)

Share of education

θe

27.21

4.52

15.08(Tokyo/1989)

40.43(Kanagawa/1984)

Share of the remainder

θr

57.78

6.32

38.06(Kanagawa/1982)

72.92 (Shimane/1984)

Tax rate

τ

12.85

4.50

5.33(Osaka/1984)

26.23(Shimane/1984)

Mean

Standard

Minimum(%)

deviation(prefecture/year)(prefecture/year)

Growth rate of prefectural GDP

0.83

2.13

-6.31(Iwate/2001)

11.70(Shiga/1991)

Share of health

θh

10.15

1.72

6.68(Nagano/1995)

18.18(Hyogo/1994)

Share of security

θs

5.57

2.00

3.35(Shimane/1999)

13.75(Tokyo/2002)

Share of education

θe

23.47

4.15

14.49(Tokyo/1992)

36.45(Kanagawa/2002)

Share of the remainder

θr

60.81

6.12

38.54(Kanagawa/2002)

70.82(Shimane/2000)

Tax rate

τ

13.87

6.27

5.31(Kanagawa/1991)

38.75(Shimane/1999)

Panel (b) (1991-2002)

26

Maximum

Table 2. Estimation results

Panel (a) (1981-1990)(No. of observations=423)

Omitted variables

health (1)security (2)education (3)the remainder (4)Constant-12.61677.979**-20.735*

24.549***(17.367)(35.806)(10.677)(7.073)___-0.906**0.081-0.372*(0.412)(0.231)(0.200)___0.987**

0.534(0.437)(0.365)___-0.453***

Share of health
Share of securityθhθs0.906**(0.412)

Share of education θe

Share of remainder θr
τ

Tax rate

Adjusted R-square

Panel (b)-0.081-0.987**(0.231)(0.437)0.372*-0.5340.453***(0.200)(0.365)(0.129)-0.343-0.343-0.343-0.343(0.231)(0.232)(0.232)(0.231)

0.408

0.408

0.408

0.408

(1991-2002)(0.129)___(No. of observations=564)

Omitted variables

health (1)security (2)education (3)Constant-2.857

20.616-4.1020.586(8.078)(27.680)(7.212)(2.314)___-0.2350.012-0.034(0.295)(0.124)(0.089)

___

0.247

0.200

(0.331)(0.278)___-0.047

Share of health

Share of security

θh

θs

0.235
(0.295)

Share of education θe

Share of remainder θr

Tax rate

Adjusted R-squareτthe remainder (4)-0.012-0.247(0.124)(0.331)0.034-0.2000.047(0.089)(0.278)(0.085)

0.015

0.015

0.015

0.015

(0.017)(0.017)(0.017)(0.017)

0.438

0.438

0.438

0.438

(0.085)___

Note: All regressions include prefecture-specific dummies and time-specific dummies(fixed effect) although not reported here. Standard errors are in parentheses. Asterisks
indicate variables significant levels; 10%(*), 5%(**) and 1%(***).

27

Table 3. Optimal adjustment of Z ** in (20)Panel(a)(1981-1990)(b)(1991-2002)∆**

Z**(%)

ω(%)

ω**(%)∆**

Z**(%)

ω(%)

ω**(%)Tax rate-0.343

-0.79

12.85

12.06

0.015

0.15

13.87

14.02

Health

-0.299

-0.69

9.38

8.69

-0.064

-0.64

10.15

9.51

Security

0.607

1.40

5.63

7.03

0.171

1.71

5.57

7.28

Education-0.380

-0.87

27.21

26.34

-0.076

-0.77

23.47

22.7

Remainder

0.073

0.17

57.78

57.95

-0.030

0.30

60.81

61.11

**

**

Growth rate dF = 1.70%

dF = 0.40%

Note: The government is assumed to adjust the shares by 2% points as a total( ξ = 0.02). ∆** = (∆τ , ∆*θ* ,− J 3′ ∆*θ* ) , ω ** = ω + Z ** .

28

Figure 1. Gradient Method1
F (ω ) = η

θs

M

Z*
P
q
0

θh

1

Note: M: maximum point of η = F (ω ) , P: present level of ω,
q: tangent line to the contour of F (ω ) = η at the present levelof ω, which lies on the tangent surface of dF ≡ Γ' Z . The
arrow starting from the point P stands for the optimal
differential vector Z*.

29

Figure 2. Linear approximation of the optimal sharesηdF = ∆ ' W

η = F(ω)γ1-γ4 > 0

0

θ h1

θ h*

θh

Note: θ h* : element of ω * at the maximum of η * = F (ω * ) .
θ h1 : present level, θ h1 < θ h* .

30