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On the theory of interregional tax incidence
Regional Science and Urban F;conomics7 (1977) 377-392. d North-Holland
ON TWE THEORY OF
EGIONAP, TAX INCIDENCE’
Masaaki HOMMA Departmenr of Economies, Osaka UniversiZy,Topnuka, Osaka 560, Japan Received June 1976 The state of factor income distribution prevailing in a local jurisdiction may be greatly influenced by the change in tax policies carried out by some other local jurisdictions throlggh the market mechanism. The theory of interregional tax incidence is concerned with the fb otlem of interactive distributional effects among local jurisdictions when each local govgrnment executes independently tax policies. This paper develops a new metho.d of con@cting a comparative static analysis on the current issue by making use of a general equilibrium model of interregional tax incidence characterized by the assumption that labor is perfsctly immobile between different jurisdictions.
More than a decade has passed since Harberger (1962) established the theory of tax incidence in a two-sector g/:neral equihbrium setting. In rhe intervening years, a number of other authorci have developed the logical ,jtruoture of the incidence model under the assumption t!zat factors of production are perfectly mobile between different sectors [Mieszkowski (1967), MeLure (1970a), Shaven and Whalley (1972)&l An interrelated extension of the general equilibrium a.pproach is to the' fieid of interregional incidence characterized by the assumption that Iabd.rr is perfwtly immobile between different regions. There has been a rapid gawth of interest in interregional incidence as one of the problems of inter-governmental finance. Despite the importance of the problem, there has been virtually no cumparative static analysis done on the current issue except McLure’s theoretical investigations (1959, 1970b, 1971, 1974). While the McLure’s works are excellent, his direct appiicaticm of the Harberger model to this problem would seem to limit tbc scope and generaiity of his results because of the concomitant shortcomings with the Harberger model.’
M. Homma, Y?tetheory of rTnterregkmaI tax hi&me
3?8
In view of the present state of the method of conducting a comparative comprehensive results of interregional the sta&lard procedure as described
study, this paqer aims to follow a new static analysis, and then to obtain more incidence. In so doing, we shall follow
below. Section 2 formulates the interregional model associated with a set of tax policy instruments in two different regions. !&zction3 is devoted to the study of causality of the system relative to a given tax structure in wh region. It is an indispensable step to conduct a comparative static analysis that causality of the system is shown to be guaranteed. Section 4 establishes some new propositions of inter-regional tax incidence. Not only the wage/rental ratio in each region but the labor incomes share relative to capital incomes share in the economy as a whole will be examined as measures of tax incidence resulting from the once and for all changes of various taxes imposed by one local government. Section 5 briefly mentions the limitations of our analysis and concluding remarks for further study.
The economy consists of two regions hbelled by 1 and 2, respectivtily. Without losing of its generality, we assume that the region I (2) is ‘lerfectly specialized in the production of the investment goods (the consumption, goods). Each production is assumed to satisfy a neoclassical productton function with constant returns to scale, positive marginal products and a diminishin: rate of substitution. The production functions are
where Yi, K, and L, stand for the output, capital and labor in the region i (i = I, 2). Thus they may be written in the intensive form as ~1 =
f,(kiYt
9
i= J,2,
where
y1
=
;,
k, +,
1, =
L
$
1
and
and where L is inelastic labor suppfy in the economy as a whole.”
Thetax system in the region i is composed of the tax (as a percentage of cost) on the commodity denoted by Ti and the ad-valorem tax rate on capital (labor) denoted by T&I’,,).’ Under this circumstance, the equality sf marginal products and gross factor prices implies PJ;I = p.+?, - I prffi-kJ;]
=:
i=
s-
(2)
1,2,
IV,, i = I, 2, I
where P 1, R and W, are the price of the output, the net rental rate of capital and the net wage+rate in the region i, respectively. A mention may be needed that ween digerent regions, and labor (capital) is perfectly immobile (mobile) en regions, while the rental therefore the wage-rate is not getxrally same: rate of capital service is the same. From the conditions (2) and (3), we obtain the following relations :
h
2-k’
= ?@ir
(4)
i = 1,2,
and
(5) where
Full employment of capital service requires K1 + K~ = K, so
where K is total quantity of ~v~li~~b~~capital, and where k zz Kit. ern~~~~ that the ratio of labor ern~il~y in the region i, whit by Ii, is positive eonsumt by the assumption of
rfe@tlyh3-mmki’e Mmr-
380
hf. Hmnma, 7% rheory
of interregiond
fax incidence
On the demand side of the economy, we distinguish separate consumptionsaving properties of capital incomes, labor incomes and the tax proceeds enoted by
Cp(Sp), CL(SL)and CG(SG)are the propensities to consume (save) out of net capital incomes, labor incomes and the tax proceeds. The equilibrium condition for the consumption-goods market is expressible by the following equation:’ Y, = CPR(X’, -t-X,) + CL(IV&, + W2L2)i-
ing both sides in (7) by Z, and substituting
e Q.
(3)
(2) through (6) yields, after
3. Causality of tbe system fn this seciion we are concerned with causality of the system relative to a
given tax structure. Tne entire system of simultaneous equations which is dated in the previous section may be summarized as
j-5
5
I-6
3:
Ilk-, +i,k,-k
= 0,
where A, =
A, =
A, x= CLSCGT,~ A4 = CIC;T,K---,
(9)
71
These six equations composed of P to P describe a complete model for the six variables, i.e., y, , y2, cot , 02, k, , and k, . As is shown by Burmeister t,1968), the two-sector model is causal if and only if the Jacobian determinant IJ[ of the above system does not change sign, and thus the variable vector, i.e..,
is uniquely detecz\ined by the parameter vector, i.e.,
In other words, we can assert the existence of a fun&n function theorem such that
$ by tlx implicit
We want to show that the Jacobian determinant jJ! of the above system is is causal. The Jacobian detersitive, slnd tkus our unambitiously ~~i~~~t (bl caff
~~~~~~~far F’ to I ’ as follows:
Ai. Homma, The theory of interregiond tax incidence
382
0
-1 0 =
0 0 0 0
-1
0
0
llf;
0
4fi 0 -21 0 -fij;'lm2 0 -yx.f~~2 0 0 -52 0 r, 0 0 0 0
0
0
0
0
0
r;
&
Wkr&
and
f;
=
I2 Al+A2-Aa
fzfi" CM*I*
Direct calcuJaticn verifies that VI = Substituting
(12)
7,7;![lJ$-rl*r~~.
(11)
into1(12) and rearranging yields
IJI =
A,-A,
f2f2”
hfi”
@JP43
(-jYp
1 ’
(13)
Since each production function satisfies the neoclassical postulates, and A3 is shown to ‘be positive directly from (9), the Jacobian determinant i.?l is positive if Al > 0 and x;=i A j > 0. The desired muIts are easily obtained a!5follows:
383
=
(1-c7T,)(i+T;,)-(~-T,)~T,K+~c(1+T’K)+c,T I -
=
1
TJ
(*-~T,)+SGTz_r+CV
+T’g)+CGT 1 -T,
I-T,
-T,
IK
1R’
This completes the proof for iJ/ > 0, and thus causality of the system is guaranteed since A, > 0 and xf_, A, > 0 for 0 5 T 5 1 and 0 6 C’, CL, Co S 1. 4. A
csmpamtive static
aadyds
of reghd
incidence
In this &t&n we wish to establish some propositions relating to the current problem by using a method of conducting a comparative static analysis. Following the usual way to measure the incidence, we will investigate the effects of the once and for all changes in each tax not only on the wage/rental ratio in each region, but also on the labor incomes relative to capital incomes in the economy as a whole. Totally differentiating the erltire system of equations for fr to r6 yields Jd.2 = HdT, where J is the Jacobian of the entire system, and where
dz = ‘Ldy,, dy2, dw, , dcu,, dT = [d?‘, * dT2, dI& and
d&,
a,
9 &21’, dT,,
, c&,1’,
M. Hommu, The theory of interregiond tax inciderrce
1 -.
Here
37.: foliowing relations 14 be derived by using Cramer’s rule to solve for changes in the wage/rentai ratio resulting from those of each tax: (al Effects of the commodity
the
tax in the region I !,
(21) ‘A final
formof
P TZIC can be obtained as follows:
+
L-~T, I-T,
M. Hurrtma, 7&e themy
ufinterregiud tax incidence
385
(b) Efkcts of the commodity tax in the region 2,
(c) Effects of the factor tax on capital in the region 1,
do2 -= dT, K -;,
rrxK41;
(d) Effects of the factor tax on capital in the region 2,
dw2 -s-B
d&x
1
;
(e) Effects of the factor tax on labor in the region 1,
(f) EffWs of the factor tax on labor in the region 2,
where AfIis the (ga n) cofactor of the Jacobian J. It will Fx convenient for the analysis that follows to calculate the cofactors used as follows:8
M. Horn,
l%e theoryof interregionaltax incidence
It is now time to examine the e ts on the wage/rental ratio. We begin with th: case of commodity taxes. From (Iti), (21), (22) and (27) together with /Ji r 0, we can observe the following results :
rvation leads to: ~~~~~~~tio~f . A rise in the commodity tax in we region results in (Jdecline in the wage/rental ratio in rhe tax-changed region, but a rise $1 the wage/rental ratio in another region.
We turn TVthe case of factor taxes on capital. With regard to the effects of e region on the wage~r~nta~ratio in another region, it immediately (17), (Iti), (23) and (24) that do2
->O
d7;K
and
d%->O.
387
It can be derived from substituting (4) and (13) into (28) and factoring out r,~1,1,/1+ T,, that
(29)
The reIation (29) can be simplified by using &A, and Pl =- -
7lWl k
u3’ ff
1
1
1
as
where p1 is the elasticity of factor substitution in the region 1.’ A careful exa-nination of (30) reveals that
w
> 0 if pI 2 I
E-R
or
CG = 0
The similar derivation procedure gives us
Then we can state
:
i
(SC = 1).
the wagelrerital ratio in the region either if the elasticity offactor substitution in the tax-changedregion is greater or equal to unity, or ty the propensity to save (consume)out of the tax proceeds is unity. We are now concerned with the case of factor tax on labor. The following result is easy to derive from (2§)-(27) along with (19) and (20) :
do2 d&i
-$go
as c%$#,
dot $g0 as CL gj CC.
dT,,
e result can be summarized as :
Proposition 4. A rise in the factor tax on la&oremployed in a region, where the imestment go& (the conmmption goodr) are produced, results in a rise or decline (a decline or rise) in another region according as the propensity to consume out ofthe tax praceeds is greater or less than that ou,tof the labor incomes. T&c derivation of dq/dT,, (d:do2/dT21jis nol so straightforward. Substituting (41, (191, and cofactors di and dz in (27) into dw,fdT,, in (25) yields
(31) Again substituting (13) into (31) and factoring out ~2~1~1~ gives
q!dT$,, < 0 follows directly from (32). O.$ldT~L< 0 throu the similar procedure. We now have :
And,
we are able to
show
Pr~~po~~ti~~ 5. A rise in the factor tax on labor in a region results in a decline in the wage/rental ratio in the tab-changed
M. fkmma,
The theory qf interregimal tax incidence
389
the labor incomes relative to the capital incomes in the economy as a whole. The ratio is expressed as
on
(33)
Differentiating (33) with respect to some tax in question denoted by T,, taking into account that k and !I are positive constants, gives
In order to show the effect on the labor incomes relative to the capital incomes, it suflices fron (34) to determine the sign of the weighted average of changes of the wage/rental ratio in each region caused by the once and for ail change in each tax. Consider first the effect of the commodity tax. Taking the weighted average of da+/dTt leads us to (35) From (4), we obtain i3Wj
1 .f&
ak, = _;;(JJ”
i==
1,2.
(36)
It then follows from (10, (34) and (35) with IJi > 0 that (37) Applying the similar procedure, we obtain
ofinterregiunal tar kidence
M. Hornma,The theory
Proposition 6. A rise in the commodity tax in one regiun rest&s in a rise or decline in the labor incomes relative to the capita1incomes in the economy as Q whale accorditzgas the wage/rentalratio in the tax-changed region is less or more sensitive to the capitalflaborratio rhanthat in another region.
We turn our attention to the eff&ztof factor tax on capital, Combining Propositions 2 and 3 with attention to the relation (M), we obtain with ease dM dT;;;
dM
> 0 if p1 2 1 or SG = 1,
>O
if ~~21
or cG=l.
dT,, us we can state our result as : Proposition 7. A r&e in the factor tax on capital employed in a region, where the investmentgo& (the consumptiongoodss)are produced, results in a rise in the l&or incc,mcsrelative to the capital incomes in the economy as a whole either if the elasticity of $ac tar substitutionin the tax-changed region is greater than or equal to unity, or if the propensity to save (c@nsumej out ofthe tax proceeds is unity.
We can derive another result on the efTectof factor tax on capital. ‘Taking the weighted averageof d(ti,/dT,, and dw,/dT,, and using (36) by arrangement, yields
Hence, it foHows from Ai c 0 and r&, < 0 that
e following is then i
te:
Proposition 8. A rise in the fuctor tax on capital employed in one region results in a rise in the lubor incomes relative to the capital incomes in the economy as a whole if the wage/rental ratio in the tax-changed region is less sensitive to the capital]lubor ratio than or equal to that in another region.
Finally, we are interested in the effect of factor tax on labor. To obtain the definite result, we need following derivation procedure. Taking the weighted average of dw,/d‘I’,, and doz/d7’,, and dividing by k by arrangement, leads to
Substituting (19) along with dt and LIZin (27) into (393, we obtain
It can be derived from substituting (4) and (13) into (40) and factoring out tlfJI i2 that
-Co
(41) E
This can be simplaed by using A, and A3 in (9). A quick inspection of (41) reveals that dM/dT,L is unambi show that dM/dT,, < 0. We nate then the foilawing:
tive. Simi1a-W we will k abEe to
392
h4. Hommo, The theory of
ir;terregbnaltax
iirclaknce
5. Concluding remarks IKIthis paper, the attempt has been made to follow a new method of conducting comparative static analysis, md to establish some new propositions on in*zrregional incidence of taxes. A few fin& remarks are appropriate. First, the present paper has limited the analysis to the economy in which each region is perfectly specialized in the production of one good, and hence has not pursued the investigation of differentia! taxes on only some products or some uses of a given factoF’in a region. This limitation, which is common to the usual model for interregional incidence, suggests the important area for further research. To drop the sumption o!’ perfect specialization is to represent a sign&ant step forward from the traditional analysis of interregional incidence* and to introduce fairly interesting results into the field of the current issue. Second, our analysis has opted constant separate consumption-saving properties of capital incomes, labor incomes and tax proceeds. While the distinction of taste patterns betweeu each agent is an extension of the usual model for interregional incidence, it rnzf be of great advance to introduce more general taste patterns into our model. It is clear that much remains to be done beyond the limitations mentioned above to develop more general theoretical models with which to study the interaction between local commuities. References Balkntine,J.G. and i Eris, 1975, On the general equilibrium analysis of tax incidence, Journal of Political Econol?y 83,633-644. Ekrmekkr, E., 1968, ;%e role of the Jacobian determimmt in the two-sector model, International Economic 1,wiew 9, 195-203. Harkger, AC,, 1962, ‘he incidence of the wrporatilon income tax, Journal of Political Economy 70,215_24O Homma, hi., 1977, A corn] urative static analysis of tax incidence,Journalof PublicFxonomics,
f43rthcotig. McClure,L’ i”., Jr., 1969, Interregional incident of general regional taxes, Public Finance 24, 457-483. !&Lure, C.E, Jr., 197Oa,Tax incidence, macroeconomic poky and absolute prices, Quarterly Journal of Ekonomics 84,254-267. .McLue, C.E., Jr., 197Ob,Taxation, substitution, and industrial location, Journal of Politicai Economy 78,121-132. McLure, C.E., Jr., 1971, The theory of tax incidence with @K&A factor mobiliry,Finanzarchiv 30, 27-48. B&Lure, C.E., Jr., 1974, A diagrammatic exposition of the Warberger model, Journal of Palirical Economy 82, 5&82. Jr., 1975, General equilibrium incidence analysis; The Har tlnodel afur ‘&~~a1 of Public Economics 4,125161. P.M., 1967, On the theory Of tax incidence, Journal Of Political Economy 75, 1972, A om capi
al equilibrium calctition 0 e U.S., Journal of Public
ON TWE THEORY OF
EGIONAP, TAX INCIDENCE’
Masaaki HOMMA Departmenr of Economies, Osaka UniversiZy,Topnuka, Osaka 560, Japan Received June 1976 The state of factor income distribution prevailing in a local jurisdiction may be greatly influenced by the change in tax policies carried out by some other local jurisdictions throlggh the market mechanism. The theory of interregional tax incidence is concerned with the fb otlem of interactive distributional effects among local jurisdictions when each local govgrnment executes independently tax policies. This paper develops a new metho.d of con@cting a comparative static analysis on the current issue by making use of a general equilibrium model of interregional tax incidence characterized by the assumption that labor is perfsctly immobile between different jurisdictions.
More than a decade has passed since Harberger (1962) established the theory of tax incidence in a two-sector g/:neral equihbrium setting. In rhe intervening years, a number of other authorci have developed the logical ,jtruoture of the incidence model under the assumption t!zat factors of production are perfectly mobile between different sectors [Mieszkowski (1967), MeLure (1970a), Shaven and Whalley (1972)&l An interrelated extension of the general equilibrium a.pproach is to the' fieid of interregional incidence characterized by the assumption that Iabd.rr is perfwtly immobile between different regions. There has been a rapid gawth of interest in interregional incidence as one of the problems of inter-governmental finance. Despite the importance of the problem, there has been virtually no cumparative static analysis done on the current issue except McLure’s theoretical investigations (1959, 1970b, 1971, 1974). While the McLure’s works are excellent, his direct appiicaticm of the Harberger model to this problem would seem to limit tbc scope and generaiity of his results because of the concomitant shortcomings with the Harberger model.’
M. Homma, Y?tetheory of rTnterregkmaI tax hi&me
3?8
In view of the present state of the method of conducting a comparative comprehensive results of interregional the sta&lard procedure as described
study, this paqer aims to follow a new static analysis, and then to obtain more incidence. In so doing, we shall follow
below. Section 2 formulates the interregional model associated with a set of tax policy instruments in two different regions. !&zction3 is devoted to the study of causality of the system relative to a given tax structure in wh region. It is an indispensable step to conduct a comparative static analysis that causality of the system is shown to be guaranteed. Section 4 establishes some new propositions of inter-regional tax incidence. Not only the wage/rental ratio in each region but the labor incomes share relative to capital incomes share in the economy as a whole will be examined as measures of tax incidence resulting from the once and for all changes of various taxes imposed by one local government. Section 5 briefly mentions the limitations of our analysis and concluding remarks for further study.
The economy consists of two regions hbelled by 1 and 2, respectivtily. Without losing of its generality, we assume that the region I (2) is ‘lerfectly specialized in the production of the investment goods (the consumption, goods). Each production is assumed to satisfy a neoclassical productton function with constant returns to scale, positive marginal products and a diminishin: rate of substitution. The production functions are
where Yi, K, and L, stand for the output, capital and labor in the region i (i = I, 2). Thus they may be written in the intensive form as ~1 =
f,(kiYt
9
i= J,2,
where
y1
=
;,
k, +,
1, =
L
$
1
and
and where L is inelastic labor suppfy in the economy as a whole.”
Thetax system in the region i is composed of the tax (as a percentage of cost) on the commodity denoted by Ti and the ad-valorem tax rate on capital (labor) denoted by T&I’,,).’ Under this circumstance, the equality sf marginal products and gross factor prices implies PJ;I = p.+?, - I prffi-kJ;]
=:
i=
s-
(2)
1,2,
IV,, i = I, 2, I
where P 1, R and W, are the price of the output, the net rental rate of capital and the net wage+rate in the region i, respectively. A mention may be needed that ween digerent regions, and labor (capital) is perfectly immobile (mobile) en regions, while the rental therefore the wage-rate is not getxrally same: rate of capital service is the same. From the conditions (2) and (3), we obtain the following relations :
h
2-k’
= ?@ir
(4)
i = 1,2,
and
(5) where
Full employment of capital service requires K1 + K~ = K, so
where K is total quantity of ~v~li~~b~~capital, and where k zz Kit. ern~~~~ that the ratio of labor ern~il~y in the region i, whit by Ii, is positive eonsumt by the assumption of
rfe@tlyh3-mmki’e Mmr-
380
hf. Hmnma, 7% rheory
of interregiond
fax incidence
On the demand side of the economy, we distinguish separate consumptionsaving properties of capital incomes, labor incomes and the tax proceeds enoted by
Cp(Sp), CL(SL)and CG(SG)are the propensities to consume (save) out of net capital incomes, labor incomes and the tax proceeds. The equilibrium condition for the consumption-goods market is expressible by the following equation:’ Y, = CPR(X’, -t-X,) + CL(IV&, + W2L2)i-
ing both sides in (7) by Z, and substituting
e Q.
(3)
(2) through (6) yields, after
3. Causality of tbe system fn this seciion we are concerned with causality of the system relative to a
given tax structure. Tne entire system of simultaneous equations which is dated in the previous section may be summarized as
j-5
5
I-6
3:
Ilk-, +i,k,-k
= 0,
where A, =
A, =
A, x= CLSCGT,~ A4 = CIC;T,K---,
(9)
71
These six equations composed of P to P describe a complete model for the six variables, i.e., y, , y2, cot , 02, k, , and k, . As is shown by Burmeister t,1968), the two-sector model is causal if and only if the Jacobian determinant IJ[ of the above system does not change sign, and thus the variable vector, i.e..,
is uniquely detecz\ined by the parameter vector, i.e.,
In other words, we can assert the existence of a fun&n function theorem such that
$ by tlx implicit
We want to show that the Jacobian determinant jJ! of the above system is is causal. The Jacobian detersitive, slnd tkus our unambitiously ~~i~~~t (bl caff
~~~~~~~far F’ to I ’ as follows:
Ai. Homma, The theory of interregiond tax incidence
382
0
-1 0 =
0 0 0 0
-1
0
0
llf;
0
4fi 0 -21 0 -fij;'lm2 0 -yx.f~~2 0 0 -52 0 r, 0 0 0 0
0
0
0
0
0
r;
&
Wkr&
and
f;
=
I2 Al+A2-Aa
fzfi" CM*I*
Direct calcuJaticn verifies that VI = Substituting
(12)
7,7;![lJ$-rl*r~~.
(11)
into1(12) and rearranging yields
IJI =
A,-A,
f2f2”
hfi”
@JP43
(-jYp
1 ’
(13)
Since each production function satisfies the neoclassical postulates, and A3 is shown to ‘be positive directly from (9), the Jacobian determinant i.?l is positive if Al > 0 and x;=i A j > 0. The desired muIts are easily obtained a!5follows:
383
=
(1-c7T,)(i+T;,)-(~-T,)~T,K+~c(1+T’K)+c,T I -
=
1
TJ
(*-~T,)+SGTz_r+CV
+T’g)+CGT 1 -T,
I-T,
-T,
IK
1R’
This completes the proof for iJ/ > 0, and thus causality of the system is guaranteed since A, > 0 and xf_, A, > 0 for 0 5 T 5 1 and 0 6 C’, CL, Co S 1. 4. A
csmpamtive static
aadyds
of reghd
incidence
In this &t&n we wish to establish some propositions relating to the current problem by using a method of conducting a comparative static analysis. Following the usual way to measure the incidence, we will investigate the effects of the once and for all changes in each tax not only on the wage/rental ratio in each region, but also on the labor incomes relative to capital incomes in the economy as a whole. Totally differentiating the erltire system of equations for fr to r6 yields Jd.2 = HdT, where J is the Jacobian of the entire system, and where
dz = ‘Ldy,, dy2, dw, , dcu,, dT = [d?‘, * dT2, dI& and
d&,
a,
9 &21’, dT,,
, c&,1’,
M. Hommu, The theory of interregiond tax inciderrce
1 -.
Here
37.: foliowing relations 14 be derived by using Cramer’s rule to solve for changes in the wage/rentai ratio resulting from those of each tax: (al Effects of the commodity
the
tax in the region I !,
(21) ‘A final
formof
P TZIC can be obtained as follows:
+
L-~T, I-T,
M. Hurrtma, 7&e themy
ufinterregiud tax incidence
385
(b) Efkcts of the commodity tax in the region 2,
(c) Effects of the factor tax on capital in the region 1,
do2 -= dT, K -;,
rrxK41;
(d) Effects of the factor tax on capital in the region 2,
dw2 -s-B
d&x
1
;
(e) Effects of the factor tax on labor in the region 1,
(f) EffWs of the factor tax on labor in the region 2,
where AfIis the (ga n) cofactor of the Jacobian J. It will Fx convenient for the analysis that follows to calculate the cofactors used as follows:8
M. Horn,
l%e theoryof interregionaltax incidence
It is now time to examine the e ts on the wage/rental ratio. We begin with th: case of commodity taxes. From (Iti), (21), (22) and (27) together with /Ji r 0, we can observe the following results :
rvation leads to: ~~~~~~~tio~f . A rise in the commodity tax in we region results in (Jdecline in the wage/rental ratio in rhe tax-changed region, but a rise $1 the wage/rental ratio in another region.
We turn TVthe case of factor taxes on capital. With regard to the effects of e region on the wage~r~nta~ratio in another region, it immediately (17), (Iti), (23) and (24) that do2
->O
d7;K
and
d%->O.
387
It can be derived from substituting (4) and (13) into (28) and factoring out r,~1,1,/1+ T,, that
(29)
The reIation (29) can be simplified by using &A, and Pl =- -
7lWl k
u3’ ff
1
1
1
as
where p1 is the elasticity of factor substitution in the region 1.’ A careful exa-nination of (30) reveals that
w
> 0 if pI 2 I
E-R
or
CG = 0
The similar derivation procedure gives us
Then we can state
:
i
(SC = 1).
the wagelrerital ratio in the region either if the elasticity offactor substitution in the tax-changedregion is greater or equal to unity, or ty the propensity to save (consume)out of the tax proceeds is unity. We are now concerned with the case of factor tax on labor. The following result is easy to derive from (2§)-(27) along with (19) and (20) :
do2 d&i
-$go
as c%$#,
dot $g0 as CL gj CC.
dT,,
e result can be summarized as :
Proposition 4. A rise in the factor tax on la&oremployed in a region, where the imestment go& (the conmmption goodr) are produced, results in a rise or decline (a decline or rise) in another region according as the propensity to consume out ofthe tax praceeds is greater or less than that ou,tof the labor incomes. T&c derivation of dq/dT,, (d:do2/dT21jis nol so straightforward. Substituting (41, (191, and cofactors di and dz in (27) into dw,fdT,, in (25) yields
(31) Again substituting (13) into (31) and factoring out ~2~1~1~ gives
q!dT$,, < 0 follows directly from (32). O.$ldT~L< 0 throu the similar procedure. We now have :
And,
we are able to
show
Pr~~po~~ti~~ 5. A rise in the factor tax on labor in a region results in a decline in the wage/rental ratio in the tab-changed
M. fkmma,
The theory qf interregimal tax incidence
389
the labor incomes relative to the capital incomes in the economy as a whole. The ratio is expressed as
on
(33)
Differentiating (33) with respect to some tax in question denoted by T,, taking into account that k and !I are positive constants, gives
In order to show the effect on the labor incomes relative to the capital incomes, it suflices fron (34) to determine the sign of the weighted average of changes of the wage/rental ratio in each region caused by the once and for ail change in each tax. Consider first the effect of the commodity tax. Taking the weighted average of da+/dTt leads us to (35) From (4), we obtain i3Wj
1 .f&
ak, = _;;(JJ”
i==
1,2.
(36)
It then follows from (10, (34) and (35) with IJi > 0 that (37) Applying the similar procedure, we obtain
ofinterregiunal tar kidence
M. Hornma,The theory
Proposition 6. A rise in the commodity tax in one regiun rest&s in a rise or decline in the labor incomes relative to the capita1incomes in the economy as Q whale accorditzgas the wage/rentalratio in the tax-changed region is less or more sensitive to the capitalflaborratio rhanthat in another region.
We turn our attention to the eff&ztof factor tax on capital, Combining Propositions 2 and 3 with attention to the relation (M), we obtain with ease dM dT;;;
dM
> 0 if p1 2 1 or SG = 1,
>O
if ~~21
or cG=l.
dT,, us we can state our result as : Proposition 7. A r&e in the factor tax on capital employed in a region, where the investmentgo& (the consumptiongoodss)are produced, results in a rise in the l&or incc,mcsrelative to the capital incomes in the economy as a whole either if the elasticity of $ac tar substitutionin the tax-changed region is greater than or equal to unity, or if the propensity to save (c@nsumej out ofthe tax proceeds is unity.
We can derive another result on the efTectof factor tax on capital. ‘Taking the weighted averageof d(ti,/dT,, and dw,/dT,, and using (36) by arrangement, yields
Hence, it foHows from Ai c 0 and r&, < 0 that
e following is then i
te:
Proposition 8. A rise in the fuctor tax on capital employed in one region results in a rise in the lubor incomes relative to the capital incomes in the economy as a whole if the wage/rental ratio in the tax-changed region is less sensitive to the capital]lubor ratio than or equal to that in another region.
Finally, we are interested in the effect of factor tax on labor. To obtain the definite result, we need following derivation procedure. Taking the weighted average of dw,/d‘I’,, and doz/d7’,, and dividing by k by arrangement, leads to
Substituting (19) along with dt and LIZin (27) into (393, we obtain
It can be derived from substituting (4) and (13) into (40) and factoring out tlfJI i2 that
-Co
(41) E
This can be simplaed by using A, and A3 in (9). A quick inspection of (41) reveals that dM/dT,L is unambi show that dM/dT,, < 0. We nate then the foilawing:
tive. Simi1a-W we will k abEe to
392
h4. Hommo, The theory of
ir;terregbnaltax
iirclaknce
5. Concluding remarks IKIthis paper, the attempt has been made to follow a new method of conducting comparative static analysis, md to establish some new propositions on in*zrregional incidence of taxes. A few fin& remarks are appropriate. First, the present paper has limited the analysis to the economy in which each region is perfectly specialized in the production of one good, and hence has not pursued the investigation of differentia! taxes on only some products or some uses of a given factoF’in a region. This limitation, which is common to the usual model for interregional incidence, suggests the important area for further research. To drop the sumption o!’ perfect specialization is to represent a sign&ant step forward from the traditional analysis of interregional incidence* and to introduce fairly interesting results into the field of the current issue. Second, our analysis has opted constant separate consumption-saving properties of capital incomes, labor incomes and tax proceeds. While the distinction of taste patterns betweeu each agent is an extension of the usual model for interregional incidence, it rnzf be of great advance to introduce more general taste patterns into our model. It is clear that much remains to be done beyond the limitations mentioned above to develop more general theoretical models with which to study the interaction between local commuities. References Balkntine,J.G. and i Eris, 1975, On the general equilibrium analysis of tax incidence, Journal of Political Econol?y 83,633-644. Ekrmekkr, E., 1968, ;%e role of the Jacobian determimmt in the two-sector model, International Economic 1,wiew 9, 195-203. Harkger, AC,, 1962, ‘he incidence of the wrporatilon income tax, Journal of Political Economy 70,215_24O Homma, hi., 1977, A corn] urative static analysis of tax incidence,Journalof PublicFxonomics,
f43rthcotig. McClure,L’ i”., Jr., 1969, Interregional incident of general regional taxes, Public Finance 24, 457-483. !&Lure, C.E, Jr., 197Oa,Tax incidence, macroeconomic poky and absolute prices, Quarterly Journal of Ekonomics 84,254-267. .McLue, C.E., Jr., 197Ob,Taxation, substitution, and industrial location, Journal of Politicai Economy 78,121-132. McLure, C.E., Jr., 1971, The theory of tax incidence with @K&A factor mobiliry,Finanzarchiv 30, 27-48. B&Lure, C.E., Jr., 1974, A diagrammatic exposition of the Warberger model, Journal of Palirical Economy 82, 5&82. Jr., 1975, General equilibrium incidence analysis; The Har tlnodel afur ‘&~~a1 of Public Economics 4,125161. P.M., 1967, On the theory Of tax incidence, Journal Of Political Economy 75, 1972, A om capi
al equilibrium calctition 0 e U.S., Journal of Public