Inter‑regional redistribution of public
investment, differential taxation of regions, and economic growth: An analysis by a
two‑sector growth model
著者（英） Satoru Miyazaki
journal or
publication title
Keizaigaku‑Ronso (The Doshisha University economic review)
volume 56
number 3
page range 73‑86
year 2004‑11‑30
URL http://doi.org/10.14988/pa.2017.0000004669
【論 説】
Inter-regional redistribution of public invest- ment, differential taxation of regions, and eco- nomic growth: An analysis by a two-sector growth model*
Satoru Miyazaki†
1 Introduction
Recently, the concern about fiscal decentralization has been increasing in many countries, including Japan. The literatures on fiscal decentralization, such as Davoodi and Zou (1998), Xie, Zou, and Davoodi (1999), Akai and Sakata (2002), and Sato (2002), are receiving considerable attention.
Now, fiscal decentralization is progressing in Japan. About 60 percent of the taxes are paid to the central government, and a half of these are redistributed to local governments as intergovernmental subsidies. These subsidies are prepon- derantly distributed across the country so that the regional gap may not expand.
Therefore, the tax paid in the city is systematically spent in the country. The Japanese taxation system has been nationally uniform in principle. Many people have suggested that the tax paid to the central government should be turned to local governments, and that the subsidies should be reduced. The Japanese taxation system tends to vary, so that the local governments can independently decide on their policies.
The inter-regional redistribution of public investment has been an important
＊The author would like to thank Kazuo Mori, Tadashi Yagi, and Yoshio Itaba for their very helpful comments.
†Doshisha University Graduate School of Economics (E-mail: [email protected])
subject in the field of public economics. For example, Takahashi (1998)shows that centralized public investment is desirable based on a two-area, one-sector model.
Furukawa and Shimono (2002) extend this to a two-sector model that has an agriculture sector and an industry sector. Moreover, they suggest that the government should preponderantly distribute public investment in the country.
There are literatures on economic growth and the inter-regional redistribution of public investment, such as Davoodi and Zou (1998)and Xie, Zou, and Davoodi (1999). They focus on the shares of expenditure by three levels of governments, i.e., federal, state, and local and investigate the relation between fiscal decentral- ization and economic growth. Moreover, the literatures, which analyze the problem of economic growth and the regional gap by growth theory, such as Kondo (1998)and Martin (1999), have also been drawing attention lately.
This paper extends the study by Miyazaki (2003)based on a simple AK model and provides a two-sector growth model that has a production sector and an educational sector. We focus on the inter-regional redistribution of public invest- ment in Japan and analyze its effect on the growth rate and the regional gap.
Furthermore, we analyze the effect of differential taxation of regions.
2 Models and Analysis
We assume that the economy of a nation consists of two regions: the city and the country. We consider both the regions separately, as being the simplest closed economies without population growth or population movement. We define the city as the region with a larger output initially, and the country as the region with a smaller output. Moreover, initially, the city exceeds the country in capital, human capital, and consumption.
Applying the two-sector model, which has a production sector and an educational sector, used by Uzawa (1965)and Lucas (1988)1）, this paper internalizes the effect
of public investment with congestion of public services, as used by Barro (1990) and Barro and Sala-i-Martin (1992). At first, we assume nationally uniform taxation and analyze the effect of public investment separately in each sector2）. This paper provides two models: one with inter-regional redistribution of public investment and another without it. We compare these models on the basis of the regional gap, which is the ratio of both regional outputs and the national growth rate in a steady state. Furthermore, we consider the policy of differential taxation of regions.
The political variables, such as the tax rate and the ratio of inter-regional redis- tribution, do not change throughout the model.
2. 1 The model with public investment in the production sector At first, a momentary utility function U is
（1）
where 0 <θ< 1. The subscript idenotes a region i(i= 1, 2 ): the city is region 1 and the country is region 2. Ci denotes consumption in region i. We consider a proportionate income taxation system, and the tax rate τis given. The govern- ments of both regions act under balanced budgets as follows:
Gi＝Ti＝τYi （2）
Gidenotes public investment and increases the productivity. Tiand Yidenote taxation and output level, respectively. The production function is given by
（3）
1）This model is explained in detail as being the Uzawa-Lucas model by Barro and Sala-i-Martin (1995).
2）Although public investment affects both sectors in reality, this paper analyzes these effects separate- ly.
U(Ci)＝ ,Ci 1−θ−1 1−θ
Yi＝AiKiα(uiHi)1−αGi Yi
β
Ai, Ki, Hi, and ui, respectively, denote the technical level given exogenously, capital, human capital, and the ratio of the human capital in the production sector.
In addition, ui,α, andβare constants in the range [0, 1].
Aggregate demand is defined as follows:
Yi＝Ci＋K・
i＋δKi＋Gi （4）
where δis the depreciation rate.
Then, the equilibrium condition is given by Ci＋K・
i＋δKi＝(1−τ)Ai Kiα(uiHi)1−ατβ． （5）
Moreover, the educational sector, where human capital Hi accumulates, is defined as follows:
H・
i＋δHi＝Bi (1−ui)Hi （6）
where Biis the educational level given exogenously.
From the above setup, we derive a growth rate. We define ωi = Ki / Hiand χi= Ci/ Ki. From equations (5) and (6), the growth rates of Kiand Hi, denoted by gKiand gHi, are obtained as follows:
gKi＝K・
i/ Ki＝(1−τ)Ai ui1−αωiα−1τβ−χi−δ, （7）
gHi＝H・
i/ Hi＝Bi(1−ui)−δ. （8）
Then, the growth rate of ωi, gωiis obtained as gωi＝gKi−gHi＝(1−τ)Aiui1−αωi
α−1τβ−Bi(1−ui)−χi. （9）
Hamiltonian Jito the utility maximization problem of this model is defined by Ji＝U(Ci) e−ρt＋νi {(1−τ)AiKiα(uiHi)1−ατβ−Ci−δKi}
＋μi {Bi(1−ui)Hi−δHi}. （10）
From the first-order conditions, the following equations are obtained:
Ui′(Ci)＝νi・eρt （11）
＝ (1−τ)(1−α)ui−αωi （12）
ατβ Ai
Bi μi νi
（13）
（14）
From equations (1) and (11),
U′(Ci)＝Ci−θ＝νi・eρt. （15）
On taking the logarithm of this equation, differentiating it with respect to time t, and using equation (13), the growth rate of Ci, gCi, is obtained as
（16）
The growth rate of χi, gχi, is obtained as
（17）
Moreover, differentiating equation (12) with respect to t, we obtain the following equation:
（18）
Then, from equations (9), (13), and (14), the growth rate of ui, guiis given by
（19）
Whenχ・
i＝u・i＝ω・i＝0 (i. e., gχi＝gui＝gωi＝0), the growth path of the economy is in a steady state, and all the growth rates of Yi, Ci, Ki, and Hibecome uniform. The combination of parameters is defined as follows:
（20）
Then, ωi, χi, and uiare obtained as follows:
＝−(1−τ)αAi ui1−αωα−1i τβ＋δ νi
νi
・
νi μi
＝− (1−τμi )(1−α)Ai ui1−αωαiτβ−Bi(1−ui)＋δ＝−Bi＋δ μi
・
gCi＝C・ i/Ci＝ .(1−τ)αAi ui1−αωα−1i τ−δ−ρ θ
gχi＝gCi−gKi＝ ＋χ(1−τ)(α−θ)Ai ui i− .
1−αωi α−1τ θ
δ(1−θ)＋ρ θ
＝ − ＝−α ＋α .μi μi
・ νi νi
・ ui ui
・ ωi ωi d(μi/νi) ・
dt
gui＝u・ i/ui＝ ＋B(1−α)Bi i ui−χi・ α
Φi＝ .δ(1−θ)＋ρ Bθ i
（21）
（22）
（23）
As a result, the growth rate gi＊, which is equal to all the growth rates of Yi, Ci, Ki, and Hi, in the steady state is obtained as follows:
（24）
This shows that the growth rate is determined only by Bi. Although the output is determined by the tax rateτand public investment Gi, the growth rate is not determined by these. Therefore, when the educational level, Bi, of both the regions are different and the public investment is redistributed, the regional gap reduces instantaneously but increases later. In this case, in order to keep the regional gap constant, governments must continue to increase the redistribution;
but they cannot keep it constant because of a limit in the actual redistribution.
The same can be said in the case of differential taxation of regions. Therefore, policies such as redistribution and differential taxation do not have a long-term effect.
Now, the short-term national growth rate, g*, is as follows:
（25）
where Y1＋Y2=Y.
If B1= B2, then g1
＊=g2
＊, and the regional gap is fixed at the initial level. If B1> B2, then g1
＊>g2
＊, and the regional gap consistently expands. The long-term national χi＝Bi Φi＋ − ,1
α 1 θ ui＝Φi＋ ,θ−1
θ
ωi＝ . Φi＋ .θ−1 θ (1−τ)αAτi
Bi 1 1−α
g＊i＝ .Bi−δ−ρ θ
g＊＝ − ,δ＋ρ θ Y1B1＋Y2B2
θY
growth rate converges to g1
＊, because the output of the city becomes relatively large. When B1< B2, then g1
＊<g2
＊, and the regional gap expands in reverse after contraction at first. The long-term national growth rate converges to g2
＊.
2. 2 The model with public investment in the educational sector but without its redistribution
Next, we suppose that governments invest in the educational sector instead of the production sector. The utility function and the balanced budget condition of governments are the same as in equations (1) and (2). The tax rates of both regions are alsoτ. The production function is changed as follows:
Yi＝AiKiα(uiHi)1−α （26）
Aggregate demand is the same as equation (4), and the equilibrium condition is obtained as
Ci＋Ki＋δK・
i＝(1−τ)AiKi (uiHi)1−α. （27）
The educational sector with public investment is defined as follows:
（28）
In the same procedure as mentioned above, the growth rate, giN
＊, in the steady state is obtained as follows:
（29）
Unlike the preceding subsection, the growth rate is determined by both Biandτ.
Tax rate, τ, (or public investment, Gi) directly affects the growth rate. Now, the short-term national growth rate, gN
＊, is as follows:
Hi＋δHi＝Bi(1−ui)Hi Gi Yi
・ β
giN＊＝ .τβBi−δ−ρ θ
（30）
The long-term movements of both the regional gap and the growth rate, which are caused by the relation between B1and B2, are the same as above.
2. 3 The model with public investment in the educational sector and its redistri- bution.
In this subsection, we consider the policy that makes both the growth rates identical by inter-regional redistribution of public investment. This policy fixes an output ratio z = Y1/Y2, and prevents the expansion of the regional gap3）. We assume that a part of the city s public investment, λY1(λ<τ), is transferred to the country.
The utility function and the balanced budget condition of governments are also the same as in equations (1) and (2). The production function is defined by equa- tion (26). The aggregate demands of both regions are changed as follows:
Y1＝C1＋K・
1＋δK1＋(G1−λY1). （31）
Y2＝C2＋K・
2＋δK2＋(G2＋λY1). （32）
The equilibrium conditions are obtained as C1＋K・
1＋δK1＝(1−τ)(1＋λ)A1K1α(u1H1)1−α, （33）
C2＋K・
2＋δK2＝(1−τ)(1−λz)A2K2α(u2H2)1−α. （34）
Moreover, the dynamic equations in the educational sector are changed as follows:
（35）
3）As stated above, the regional gap expands in reverse after contraction, in the case of B1<B2. However, we focus on the size of the long-term gap, regardless of its direction.
G1−λY1
Y1
H1＋δH1＝B1(1−u1)H1 ＝B1(1−u1)H1(τ−λ)β,
・ β
δ＋ρ gN＊＝ − .τβ(Y1B1＋Y2B2) θ
θY
（36）
By the procedure stated above, the growth rate in the steady state, giT＊, is obtained as follows:
（37）
（38）
Since gT
＊= g1T
＊= g2T
＊, λis as follows:
（39）
Since zis a fixed ratio, λis also a constant. Moreover, since this fraction is less than 1, the condition λ<τis fulfilled.
In equation (39), when B1=B2, λ= 0, and there is no redistribution. When B1> B2, thenλ> 0, and there will be a natural transfer from the city to the country. However, when B1<B2, thenλ< 0, and there will be an opposite transfer from the country to the city.
2. 4 The effect of inter-regional redistribution
Here, we only consider the case that the educational levels of both regions are different. At first, we compare the national growth rates of each case−whether there is inter-regional redistribution of public investment or not. The short-term growth rate without redistribution is gN
＊; the long-term growth rate is g1N
＊when B1>B2or g2N
＊when B1<B2. On the other hand, the growth rate without redistribu- tion is always gT
＊. A comparison of the short-term growth rate is based on the sign of the following equation:
G2＋λY2
Y2
H2＋δH2＝B2(1−u2)H2 ＝B2(1−u2)H2(τ＋λz)β.
・ β
g1T
＊ ＝ ,B1(τ−λ)β−δ−ρ θ
g2T
＊ ＝ .B2(τ＋λz)β−δ−ρ θ
λ＝ ・τ.(B1/B2)1/β−1 (B1/B2)1/β＋z
（40）
The sign of this equation cannot be determined because of the lack of conditions. In the long term, the growth rate without redistribution will always be higher than the other, regardless of the relative size of Bi. This can be checked from the following equation. When B1>B2, the following equation is positive because 0 <λ<τ:
（41）
When B1<B2, the following equation is positive because 0 <λ.
（42）
Therefore, when there is no redistribution, the regional gap is expanded. On the other hand, when there is redistribution, it is always constant.
2. 5 The model with differential taxation of regions
Differential taxation may also make both the regional growth rates equal. It is clear from equation (29)that the difference between both growth rates is caused by the difference in the educational level and the tax rate. Here, we assume that governments adopt differential taxation so that the following formula may hold.
τ1βB1＝τ2
βB2 （43）
There are many combinations in the tax rates of both regions. The growth rate can be increased by changing the combination of tax rates. For example, when only the government of the lower educational region raises its tax rate, the national growth rate becomes higher.
When differential taxation is adopted, the tax rate of the lower educational region becomes relatively high. The higher educational region helps the lower one when inter-regional redistribution of public investment is adopted. However,
gN
＊−gT
＊＝ z＋ z＋ −τβB1 (1＋z)1＋β . θ(1＋z)
B2
B1 B2
B1
−1/β β
g1N＊ −gT
＊＝ {B1 τβ−(τ−λ)β}>0.
θ
g2N＊ −gT＊＝ {B2 τβ−(τ＋λz)β}>0.
θ
the lower one pays a relatively high cost when differential taxation is adopted.
We have considered the educational level, Bi, as a constant that is given exoge- nously in advance. If this setting is changed and the educational level, Bi, rises due to efforts of the region, the following conclusions will be obtained. When there is no redistribution, the incentive for raising the educational level by differential taxation will be stronger. On the other hand, when there is redistribution, there will be no incentive for raising the educational level.
3 Conclusion
In this paper, we consider inter-regional redistribution of public investment and differential taxation of regions and analyze their effects on the growth rate.
At first, when governments invest in the production sector, only the educational level affects the growth rates of both regions. When the educational level of the city is higher, the regional gap consistently expands. On the contrary, when that of the country is higher, the gap contracts at first, and then it expands in reverse.
When both educational levels are equal, it is consistently constant. The growth rate is not affected, but the output is affected by the tax rate (or public investment). Therefore, both the redistribution and the differential taxation do not have a long-term effect on the growth rate and the regional gap.
Next, when governments invest in the educational sector, the growth rates of both regions are determined by the educational levels and the tax rates. When both the tax rates are the same and there is no redistribution, the long-term movement of the regional gap and the growth rate are the same as before.
When both the tax rates are the same and the public investment is redistributed to keep the regional gap constant, the national long-term growth rate is smaller than the one without redistribution. In turn, when differential taxation is adopted in order to keep the gap constant, some combinations of tax rates may raise the
national growth rate. Since the lower educational region pays a relatively high cost, the incentive for raising the educational level will be stronger.
As mentioned above, when public investment affects the educational sector, the redistribution may keep the regional gap constant but decrease the national growth rate. The differential taxation of regions may keep the gap constant and raise the national growth rate. Therefore, differential taxation is more desirable.
However, this paper cannot examine the incentive in detail, because the educational level is given exogenously. Furthermore, negative effects due to the rise in tax-rates and population movement, which actually exist, are not consid- ered. These problems remain to be studied further.
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The Doshisha University Economic Review Vol.56 No.3 Abstract
Satoru MIYAZAKI, Inter-regional redistribution of public investment, differential taxation of regions, and economic growth : An analysis by a two-sector growth model
This paper provides a two-sector growth model and considers the policy of inter-regional redistribution of public investment and differential taxation of regions. We then analyze their effects on economic growth and regional gap.
Public investment in the production sector does not affect the growth rate.
Therefore, these policies do not have a long-term effect on the growth rate and regional gap. However, public investment in the educational sector affects the growth rate. Inter-regional redistribution may keep the regional gap constant but decrease the growth rate. Differential taxation may fix the regional gap and raise the growth rate.
