Otemon Economic Studies, 29（1996）

A Variation on General Equilibrium Analysis

　　　　　　ofProperty Tax Incidence

Masazo　KiSHI　and　Shigeaki　Watanabe

1

Introduction

1

　　　After Mieszkowski's well-known article (1972)regarding property tax incidence,

there have been many related works. Lin (1986)is one of the most important

contributions to this field. He analysed property tax with a two sector model

incorporating both local and national goods. However, he treated property taχ as an

excise taχon a local good (housing), though it is equivalent to a taχ on capita] and

land in the housing sector. In fact it is a partial rather than a general tax on capital

and land. He left unmentioned a general type of property taχ as imposed on capital

and land in all sectors.!)The purpose of the present paper is to deal with such a

property taχ，using Lin's two sector model.

　　　We follow all the standard assumptions of Lin's model except for modelling

property taχ and treating its revenue. We assume the taχ proceeds are returned in

a lump-sum to the landlords and the capitalists.

2. The model

　　　We assume two communities, 1 and 2. Starting from a zero-taχinitialsituation、

we analyse the effectsof an introduction of property taχby community 1｡

　　　We begin with the supply side of the model. Capital is assumed to be fixed in

supply for the whole economy and perfectly mobile across communities. Assuming

equal initialallocations between the two communities, Ki （i＝l, 2）,we have the

following equation of change

1 ）Lin just suggested that to consider this type of property taχ would be more realistic

　（Lin（1986, p.115, footnote 7）.

　　　　　　　　　　　　　　　　　　（1）

2　　　　　　　　　　MASAZO　KISHI　and　SHIGEAKI　WATANABE

（1）　　　　　　　　　　　　　　　　　　君x十君2＝0,

where the symbol （＾）represents the rate of change.

　　Each community is assumed to have the same fiχedamount ofland, Li（i＝l,2）,

which is transferred across sectors for rents to be equalized within each jurisdiction.

Denoting each allocation of land between the local good sector,H, and the national

good sector, X, by L汀and L'x repectively, we have

（2）　　　　　　　　　　　　　　　出L?十び舒＝O，　i=l, 2.

where l″=Lダ/Li and 戸＝L句Li｡

　　The totalnumber of workers is assumed to be fiχedfor the whole economy and

an individual worker is taken to possess one unit oflabor. The workers are assumed

to be perfectly mobile across communities, responding to intercommunity differ-

ences in utilitylevels. Letting the number of workers residingin each community be

Ni （i－1,2）,and assuming they are initiallyequal, we have

（3）　　　　　　　　　　　　　　　　　譚,十iV2＝0.

　　Lin assumes that local goods are produced by capital and land, and national

goods by capitaland labor. This is a nice simplification,which allows us to avoid the

compleχity of the three factor model. Production functions are assumed to be

identical across communities and homogeneous of degree one.　In a competitive

market equilibrium, we have the following relations among the output oflocal goods,

Hi, capitalinput, 瓦，and labor input, Lダ

（4）　　　　　　　　　　　島＝几島十几£ﾀﾞ，　1 = 1, 2

where几and ルare the initial factor shares of capital and land in local goods sector.

We have similar relations for national goods sector, which are not needed here by

Walras' law. As we have already assumed, the economy starts in equilibrium with a

zero property taχΓate in every community, and community l introduces the taχ.

Comparatively, we then have the following relations of local good prices, pi, to

capital service net prices, s,land service net prices, ri and the property taχΓate, t,

(5a)

(5b)

(6a)

p＼＝几(s十dt)十几(r,十直)＝ｱ八十几八十叱

　　　　　　　　　　　　　　　　　　　　　列＝/μ十几几.

We have similar relations for national goods as follows,

O＝gN倒l十g衣剔十dt），

　　　　（2）

A VARIATION ON GENERAL EQUILIBRIUM ANALYSIS OF PROPERTY TAX INCIDENCE3

(6b)　　　　　　　　　　　　　　　　　　0＝乱功2十gift.

where national goods are taken to serve as the numeraire and are set equal to one，

and 匹･denotes the wage rate in each region, ga:＝耀iNi/X，and 乱＝r,£芦/X. The

property taχΓaisesgross rent in the national goods sector of the community 1.

　　　Byassuming linear homogeneous production functions, we have the following

equations relating factor ratios and factor prices for the local goods sector of each

region:

(7)　　　　　　　　　　　　　　抑/一瓦i＝o/(s一fi)，　1＝1, 2

where a" denotes the elasticityof substitution in thissector. In community l gross

prices of all factors are increased by the taχ，but the gross price ratio remains

unchanged, so factor substitution in that community is not affected｡

　　　Similarly,we have the following equations for the national goods sector,

(8a)　　　　　　　　　　　　　　乙ぞ一良,＝ダ(利一fl一dt)，

(8b)　　　　　　　　　　　　　　　f X一良2 ＝ダ(功2一几)，

where ff^ denotes the elasticityof substitution in this sector. Eq. (8a)for the first

community includes the rate of change in the tax rate,as land is used in the national

goods sector as well.

　　　We now turn to the demand side of the model. The demand for local goods in

each region,亙,,is assumed to depend on their price,拓,and on the disposable income

of each region, Yi.So we have the following demand function of local goods for the

firstregion

(9)　　　　　　　　　　　　　　　　H、＝D(j)u Fi),

where r, is defined as the sum of wage income, gross capitalincome and rent income

(by assuming the tax revenues are returned in a lump sum to capitalists and

landlords).That is,

(10)　　　　　　　　　Yi＝叫N,十s(l十t)A十r,(l十t)(Lぐ十ム1 )，

where A is the totalamount of capitalowned by the residents (capitalists)of the first

region.

　　　Differentiatingeq. (9), we get

皿） d亙^＝(∂w∂凱)dp 1十(dD/∂Y,)dYy．

　　　　　　　(3)

4

　　　　　　　　MASAZO　KISHI　and　SHIGEAKI　WATANABE

And differentiatingeq.(10),we have

dYχ＝Nidiv[十切idNχ十(1十)Ads十sAdt十(1十t)(Lf十£ﾀﾞ)心,十r,£f十£ｸﾞ)血

And transforming this,we have

dYi＝Xi[(wiTVi/Xi)ど匹/wi)十r,(l十)£f/X,){面,/r,十dt/{l十t))]

　　　十拓耳,[(s(l十£)A/凱H、){dsﾉs十dび{1十t)}

　　　十{r,(l十)£,//),亙,}{dri/r,十dt/(l十雨)

　　　十widNi.

Here we assume the capitalistsown theircapitalwithin the region they residein, and

the initialallocations and ownerships of capital are equal across regions. ＼.e.,A＝Ki.

Applying the definitions(4)and (6)of gN,gい八andル, to the above equation, we

have

dYi＝Xi[iTw勧十gdfl十dt)]

　　　十凱Hχ[八(S十dt)十//,(八十dt)]

　　　十UハdNし

And, substituting（6a）and （5a）into the above equation gives

（12）　　　　　　　　　　　　　　　　dYχ＝♪i拓pi十切idNi.

And, substituting the above into (11)gives

　　dHχ＝(dD/∂/,)dpi十(屁)/∂Yy)φ,ど^か十四dNi).

Dividing both sides of the above equation by耳, and rearranging gives

dHﾉH、＝(凱ﾉH,)(∂Dﾉ∂凱)(d凱ﾉ釦)＋Pi(∂oﾉ∂Y,)p,

　　　　十加(∂£)/∂Y,)(心気か^気)気/気．

Here we use the following definitions,

　　　㈲/瓦)(dD/∂ipi)＝－ε，

　　m勺）i∂'D/∂Yu

　　M＝凱(∂£)/∂Y,)(叫皿か刀,)＝mw、N、ﾉ凱H、，

where £is the price elasticityof demand forlocal goods, m is the marginal propensity

to consume local goods, and M is the ratio oflocal goods eχpenditures by the workers

to community aggregate eχpenditures for the same goods. Applying these defini-

　　　　　　　　　　　　　　　　　　　　　　　(4)

A VARIATION ON GENERAL EQUILIBRIUM ANALYSIS OF PROPERTY TAX INCIDENCE

tions to the preceding equation gives

　　dHi/H I ＝(－a十剛pi十肛斤x.

5

The eχpression一ε十min the above equation is the elasticityofincome-compensated

demand for local goods, because m＝pWD/∂71＝夕1 (∂£)/∂YD)(∂oﾉ∂71)(Yl/D).

hence m is equal to the income effectterm of the Slutsky equation in elasticityform.

Defining －??=－£十m, 7]gives the same elasticityin positive sign. Using this,we get

　　H＼ ＝一顔i十MN,.

　　Carrying on in the same way as above for the demand function of the second

region, we arrive at a similar equation. So we have

(13)　　　　　　　　　　　拓＝－がi十MNi,　1 ＝1, 2.

　　Finally, residents mobility remains to be modelled. Workers are assumed to

move across regions to equalize their utility. Each individual worker's utilityis

assumed to be a function of local goods and national goods. But, we assume that

capital owners and landlords are immobile. By assumption, the price of national

goods is unity and each worker owns one unit of labor. The indirect utilityof an

individual worker as a resident is a function of the local goods prices and the wage

rates in each community, i.e.,V(j)i,耀丿.This function is assumed to be identical

across individuals. In equilibrium we have

　　ド,凱，切i)＝ゾφ2，W2)

Differentiating the above equation gives

　　(∂v/∂Pi)dpi十(∂V/∂匹)dw 1＝(∂1ク砂2)㈲2十(∂lう/∂wi)dwo.

Substituting Roy's Identity for the present model,

　　　づ∂万作)バ∂口∂叫)べ

into the preceding equation yields

（14）　　　　　　　　　　　　　　　　功1－砂I ＝功2－顛2，

where b is the initialconsumption of local goods by an individual worker.

（5）

6

MASAZO　KISHI　and　SHIGEAKI　WATANABE

3. Comparative static results

　　　Solvingfor endogenous variables of the equations of change in the preceding

section yields

(15)

(16a)

(16b)

(17a)

(17b)

(18a)

(18b)

　　　　　　S＝－dt/2,

　わ＝－[bfμtﾉ2(gi./gN十ね几)一dtl

　　　た＝b八dﾉ2(gﾑﾉgN十bfム)，

功l＝(gむ/gN)b八dt/2(g t/g N十町ム

飢＝－(刄ソ宮N)b八dtﾉ2(gソポN十占几)

　み＝(召ソgw)几dt/2(si./gN十げム

加＝－(gL/gN)fKdt/2(gl/gN十町ハ

（See Appendix for derivations.)

　　Eq.（15）supports Mieszkowski's argument that the property tax reduces the

after-taχreturn on capital by the average taχΓatein the economy｡

　　Eq.（16a）and （16b）show that the rent change in the taxing community is,in

absolute terms, by dt,larger than the rent change in the no-taχ community. This

means that the burden of the land portion of the property tax, as imposed on both

capital and land, is borne by the landowners. We have shown elsewhere that when

we examine a tax imposed on capital only with the present model, the following

resultis obtained,

f,十几＝0.≫

In our present model, property taχ and a tax levied on capital only make no

difference in effect on variables other than rent.

　　　Eqs.（17a）and （17b）predict that the wage rate rises in the taxing community

and falls an equal percentage in the no-taχ community.

　　　Eqs.（18a）and （18b）indicate that the price of local goods goes up in the taxing

community and goes down in the no-taχ community.

2）See Kishi and Watanabe （1996）.

（6）

　　　　A VARIATION ON GENERAL EQUILIBRIUM ANALYSIS OF PROPERTY TAX INCIDENCE　　　7

　　Let us make a comparison between our preceding results and Lin's, whose

resultsanalogous to ours (15)to(18b)are as follows :

　　j＝－[(1十B(戸/円/丿)/2{l十B(び/戸(f丿丿十几/)}]励

　　　＝－[ri(a"十B(び/門/2{♂v十方(びﾉ岬(がK十丿几)}]励

　　た＝－[1/2 {1十B(戸/戸(/K/♂十几洵)}]冶

　　　　＝－[♂刀ﾉ2{♂η十駅ぴ/戸(v八十♂几)}]dt

　　功i＝－(£Lﾉ八)ri

　　則＝[1－(1十組Z/の几/a")/2{l十B(びﾉ円(l丿丿十几力)}]励

　　　　＝[l－v(丿十召(戸/戸几/u佞％十方(Zケ戸(がK十丿几)}]励

which are presented with our notation. Qualitative effects are the same between

Lin's model and ours, but quantitative ones are different.

　　In addition, we find the following properties in our model :

(19)

(20)

(21)

(22)

尹^十八＝一心(た＜Q,わ＞0)，

飢十iX>2=0(iiJi ＞0, 恥＜0)，

Px十^2＝0(5i＞0,瓦＜0)，

　　　　　ibi＝bp i

(See Appendix for derivations.)

Eq. (19)is simply rewritten asら＝－(几十dt). As we have already noted about (16a)

and (16b), the rent change in the firstcommunity is,in proportion to the taχΓate,

larger than the rent change in the second community. That is explained by the fact

that the burden of the land tax component of property tax fallson landowners in the

taxing community. Eqs. (20)and (21)indicate that wage rates and local goods prices

respectively change in opposite directions between communities. Eq. (22)says that

wage rates and local goods prices in the same community change in the same

direction｡

　　　Meanwhile, the properties of Lin's model corresponding to the above are as

follows:

剔＝た(＜O)，

功1＝功2(＞0)

Pl＝か十励(β2＜0,づ2－＜叱則＞O).

　　　　　　　　　　　　　　　　　　　(7)

8　　　　　　　　　　MASAZO　KISHI　and　SHIGEAKI　WATANABE

Rents change in the same direction across communities, and so do wage rates. But

local goods prices change in opposite directions.In Lin's model, as readily seen from

the above eχpressions,wage rates and local goods prices go up in the taχed commu-

nity, but they go in opposite directions in the no-tax community. One remarkable

difference between Lin's model and ours is that an asymmetry appears in local goods

price changes in Lin's,and in rent changes in ours. This reflects a difference in

treating property tax between these models.

4. Concluding remarks

　　　Inthis paper we have considered a conventional type of property taχ as levied on

both capital and land in all sectors. The results of our model differ in quantitative

effects from those of Lin's original model｡

　　　Our results support Mieszkowski's view that property taχ will depress the return

to capital through the economy and his proposition that the eχcise tax effects of a

property taχ will be offset among communities

Append!χ

　　　　　　　　　　　　　　　　　｡-―s

f

^

｡-'―＼

／　＾

/""N

j　j　j　j　　a　　b　　a　　b　　j　　a

1　　2　　3　　4　　5　　5　　6　　6　　7　　oo

ぐ　ぐ　ぐ　ぐ　ぐ　ぐ　ぐ　ぐ　ぐ　ぐ

In what follows, the derivation of equations（15）to（22）is shown.

At the outset, the key equations in the teχtare enumerated.

　　　　　　　　　　　　　　　　　　　君,十君2＝0，

　　　　　　　　　　　　　　　tH聊十びび＝0,　1 = 1,2

　　　　　　　　　　　　　　　　　　　刀,十iV2=0,

　　　　　　　　　　　　　　彦i＝几K,十几Lf, i＝l，2

　　　　P1＝fK§十几八十叱

　　　　　加＝ﾁﾞ八十几八,

　　　0=i

　　　　　0＝gN切2十g乙几，

LダーKi＝a"{s一fi)，　1 ＝1,2

　　　Lぎ一譚i＝が(功i一戸l),

（8）

司　㈲　蜀

印　Ｇ　Ｇ

　　AVARIATION ON GENERAL EQUILIBRIUM ANALYSIS OF PROPERTY TAX INCIDENCE　　　9

　　　　　　　　　　　　　　　　£ぞ一譚2＝ダ(功2一fl).

　　　　　　　　　　　　　　耳i＝－Vか十MNi，　i＝l, 2

　　　　　　　　　　　　　　　　　泌.一如l＝功2－如2.

We now proceed to solve the above set of equations for endogenous variables.

We begin to work with the equations of community 1. We rewrite (6a)as

(Ala)　　　　　　　　　　勧= －(gL/gN)(八十dt).

Substituting the above (Ala)into(8)to eliminate wl yields for the firstregion

　　　　　　　　　　　　　　　　び=乱－£(ﾀ,十dt).

where i?＝<7'^(l十乱/gN)，which is positive. And substituting the above into (2)to

eliminate Lf gives

(A2a)　　　　　　　　　　剔( ＝－(戸/戸)譚,十召(び/戸)(八十dt).

For the firstregion, equating (4)and (13)to eliminate Hi, substituting the above

(A2a)to eliminate L＼/and substituting (5a)to eliminate p i gives

(A3a)　　ﾉ八十几(1十B(lケ戸力)r,＝(び/の[(M十几)/77]剣－け丿v)瓦

　　　　　　　　　　　　　　　　　　　　　－(1十几Biび/l")/v)dt.

Substituting(A2a)into (7)to eliminateLぴyields

(A4a)　－s十(1十B(戸/l")/a")剔＝(l/a″)(ぴ/l")N,十(/戸)瓦,－(β(びﾉn/戸血

　　We now turn to the equations of community 2. Beginning with (6b)and

followingthe same procedures for the equations of community 2,we obtain

(Alb)

(A2b)

and

(A3b)

(A4b)

　　　功2 ＝－gム/gNf-l，

舅＝－(び/戸)譚2十B(門/l≫)f2.

ｱ八十几(1十召(ﾉ円)/?7)た＝(/戸[(M十几)力]刄t－(ブソv)君i，

　　一丿十(1十召(び//″/丿わ＝(＼/a")(戸/l")譚2十(l/丿君2.

　　The above eqs.(A3a), (A4a), (A3b)and (A4b)are fundamentally important to

solve the present equation system. Adding both sides of (A3a)and (A3b)and using

(1)and (3)gives

　　　　　　　　　　　　　　　　　　　(9)

10 MASAZO　KISHI　and　SHIGEAKI　WATANABE

(A5)　　　　　写μ十几(1十B(､びﾉぴ")/)(f.十尹2)＝－(l十几B(び/n/ri)dt.

And, adding both sides of (A4a)and (A4b)and using (1)and (3)gives

(A6)　　　　　　－2s十(1十B(戸/門/♂)(fl十几)＝一iB(び/n/a")dt.

Eliminating g from the above (A5)and (A6)yields

(A7) 剔十fi ＝一血

By substituting the above into （A6）we get

（A8）　　　　　　　　　　　　　　　　　　s＝－（l/2）血

We now go on to find f, and f2. Substituting（A8）into （5a）and （5b）yields

(A9)

(A10)

加＝一几八十(1一几/2)dt,

　　か=几たー(fK/2)dt.

Substituting these into (14)to eliminateβi andβ2 and substituting (Ala)and(Alb)

into (14)to eliminate 副and 功2 gives

（AI1）　　　　　－fe/ノgw十わル）剔十（Sむ/gN十わ几）た

Solving for八and わfrom （A7）and （All）we obtain

(A12a)

(A12b)

＝(乱ﾉ肘午b)dt.

ri＝－[bfKdt/2(g,./gN十か几)]一dl，

　　た＝町バび2fe/./g-w十bfむ).

By adding both sides of（Ala）and（Alb）and using （A7）, we get

(A13) 功l十功2 ＝0.

　　　Byadding both sides of（5a）and （5b）and using（A8）and （A7）, we obtain

（A14）　　　　　　　　　　　　　　　　　　　痢十夕2＝O.

　　　Sincewe have already found the solutions of s,f1 and た，we can obtain those of

the other variables by appropriate substitutions,as follows

(A15a)

(A15b)

and

til1＝bf ic(g ム/g N)dt/2(g i/g N十わ几)

　功2 ＝一b八dtﾉ2(gム/gN十町ハ

（10）

A VARIATION ON GENERAL EQUILIBRIUM ANALYSIS OF PROPERTY TAX INCIDENCE 11

(A16a)　　　　　　　　　　　釦こ八dt/2(sソ匈N十bfi)，

(A16b)　　　　　　　　　　　　彭＝－ノバび2(Bソ政N十町八

　　Lastly, from the above equations, (A15a), (A15b), (A16a)and (16b), it follows

that

(A17)　　　　　　　　　　　　　　　　　　　飢＝bpi.

　　　　　　　　　　　　　　　　　　　　　　References

Kishi, M. and s. Watanabe （1996）,Property tax incidence, Otemon Keizai Ronsyu （Otemon

　　　Economic Review）31 （in Japanese）.

Lin, C.（1986）,A general equilibrium analysis of property tax incidence, Journal of Public

　　　Economics 29,113－32.

Mieszlowski, P. (1972），The property tax : an excise lax or a profits taχ?，Journal of Public

　　　Economics 1,73 －96.

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