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The effects of output and factor subsidies or taxes on an urban area
Regional
Science
and Urban
THE EFFECTS
Economics
14 (1984)
533-546.
North-Holland
OF OUTPUT AND FACTOR SUBSIDIES TAXES ON AN URBAN AREA
OR
Donald R. HAURIN” The Ohio State University, Columbus, OH 43210, USA Received
July
1982, final
version
received
April
1984
This study investigates the effects of five types of factor and output subsidies upon an urban area. It represents an improvement upon prior regional tax or subsidy analyses since it includes a non-traded good in the urban model. Also, the supply elasticities of the three inputs to production; land, labor, and capital are parameterized, thus increasing the generality of the results. A suggested application of the model is to the study of urban decentralization policies in countries with a low or mid-level of development.
1. Introduction
The economic stimulation of specific urban areas has frequently been a goal of national governments. Among the policy tools that have been utilized are factor and output subsidies for goods produced or consumed in the selected area. The model presented in this paper analyzes the effects of these subsidies on a small or mid-size urban area. The emphasis is upon an area of this particular size since it is a likely target for subsidies, especially in less developed countries pursuing decentralization policies. Another reason for urban subsidies, mentioned by Renaud (1979), is that they are one method of securing claims to disputed international borders. The growth of selected cities is seen as one means of strengthening national claims to the territory. This paper will discuss responses to both factor and output subsidies or taxes. Of particular concern is the identification of the direction of flows of factors into or from the urban area as well as the change in the level of outputs. Also, changes in the rate of return to factors and the price of outputs are discussed. It is likely that no single public policy has the desired effects in all environments (countries). Thus, one goal of this study is to derive a fairly complete set of reactions to the various subsidies so that a comparison among them is possible. The literature in this area has concentrated on the analysis of regional, not urban, subsidies or taxes. Many of the models are variations of the twosector, general equilibrium model of Harberger (1962) which was used to *I wish
to thank
01660462/84/$3.00
the anonymous 0
1984, Elsevier
referee
for comments
Science
Publishers
on the earlier
draft.
B.V. (North-Holland)
534
D.R. Ham-in,
Effects
of subsidies
or taxes
on an urban
area
analyze the incidence of a factor tax, on capital. Harberger’s model was extended by Mieszkowski (1967) and later regionalized by McLure (1970). In McLure’s model, the two sectors are renamed as two regions, each region producing only one good. One attractive feature of McLure’s analysis is that factor supply elasticities are parameterized, contrasted with a number of later articles that restrict certain factors to be immobile. The application of Harberger’s model to a regional economy is particularly appropriate since, in theory, the taxes subject to analysis should be ‘small’, a reasonable assumption for the level of regional subsidies or taxes. The incidence of interregional taxes has been analyzed by McLure (1969) and Homma (1977). Both models are limited by an assumption that labor is immobile. Also, each region specializes in the production of a single good which limits the application of the analysis in the urban framework. One of the characteristics of cities that urban economists have found interesting is the presence of non-traded goods, housing in particular. If regions are assumed to specialize in the production of a single good, the possibility of intersectoral shifts in production within a region are eliminated. The analysis of subsidies in an urban context requires a different framework. An extension in this direction was made by Gerking and Mutti (1981), where they develop a model which allows for incomplete specialization in production in one region of a two-region economy. They further assume that capital is perfectly mobile between regions, labor is immobile, and that production exhibits a linear homogeneous technology. An output tax is analyzed, and they find the existence of the second sector in a region is important in that the tax incidence also depends on whether the taxed good is capital intensive within the two-good region. Limitations include the labor immobility assumption, and the requirement that all goods are traded. The analysis indicates that the existence of a second sector modifies the incidence results found in the earlier literature. Another study was by Dewees and Kotowitz (1980), again analyzing regional subsidies. Their contributions include allowing for the existence of non-constant returns to scale in production and the possibility of less than full employment. Both of these extensions are relevant for urban areas, thus they appear to be fruitful areas for further research. Their analysis is limited, similar to the earlier studies, by the assumptions that labor is immobile and regions specialize in production. The final study that is relevant for the analysis of factor or output tax effects on urban areas is by Jones (1982). He includes a non-traded good in a general equilibrium urban model, thus begins to directly address the problem of interest. However, .when performing the comparative statics, specific numerical values were assumed for all parameters in the model, thus generalizations are severely limited. The results derived from the model indicate that the addition of a non-traded good to an urban model importantly modify the conclusions drawn from the earlier models.
D.R. Haurin,
Effects
of subsidies
or taxes
on an urban
area
535
The most noticeable omission in the literature examining the effect of subsidies or taxes, is the development of models relevant for urban areas. Of primary importance is the inclusion of a non-traded good representing housing and other local services. Other models have found that certain policies encourage the growth of a particular region. For example, assume that growth occurs in the capital stock of the region. A conclusion that could ,be derived in the framework of a regional model is that this policy is desirable, given that capital growth was the target. However, if most of the increase in capital occurs in the local sector (housing), the policy verdict may be less favorable compared to the situation where the capital is employed in the export good sector. In any case, it is important that the effects of taxes or subsidies be fully identified, especially differentiating local good production from export good production. The model developed in this paper includes both a traded and a nontraded good. Factor’s supply elasticities are parameterized, avoiding an unrealistic a priori assumption and increasing applicability. Another facet of the model is that since land is a particularly important factor in urban analysis, the local sector is modeled as a three-factor sector (land, labor, capital). Thus, specific assumptions about land’s supply elasticity can be incorporated directly, rather than inappropriately assuming land and capital are equivalent factors. Limitations of the model include the assumption of constant returns to scale in production, restrictions upon consumers’ demand elasticities, and assuming the full employment of factors. The model is developed to analyze taxes or subsidies in a small or mid-size urban area. Thus a ‘small city’ assumption is made, fixing the price of the exported good. Factors flow to or from the rest of the country, but the other region or regions are not explicitly modeled. The next section of the paper presents the model of the urban area, introduces factor and output subsidies, and derives the equations of change. (Throughout the analysis, taxes can be analyzed rather than subsidies by making the appropriate sign changes.) The derivation of the responses of the endogenous variables to the various subsidies is presented in the third section. The concluding section of the paper discusses some applications of the analysis and investigates some special cases. 2. The urban model
Following a general outline of the urban model, the specifications for each sector are presented. The equations describing the reactions of the endogenous variables to factor or output subsidies are then derived. Two outputs are produced in the locality, one is only consumed locally, the other is both exported and consumed locally. Producers of the export good consider its price to be fixed, determined in the national or international economy. The
536
D.R. Haurin,
Effects
of subsidies
OY taxes
on an urban
area
price of the non-traded good is endogenous, determined by local aggregate demand and supply, and generally this price will vary in response to the imposition of subsidies. Inputs to the export good are capital and labor while local good production also utilizes land. The supply elasticities of all of the inputs are parameterized. Labor is assumed mobile between sectors in the urban area. Throughout the analysis, the markets are assumed to be competitive. Consumers are utility maximizers, selecting the consumption quantities of two goods, and earning income through their ownership of labor. Subsidies (taxes) can be given to any of the three factors or either of the outputs. Since the model is of a smaller urban area and the application of the model is .the analysis of national or international development of certain areas through subsidies, it is most appropriate to consider the revenue source for the subsidy to be external to the urban area. 2.1. The output sectors The export good (X) is produced with a linear homogeneous utilizing capital (K,) and labor (L,),
technology
x =fK, Lx).
(1)
The elasticity of substitution in production is S,, defined to be a positive number. Both factors are paid their values of marginal product. The price of the output is symbolized as px and is assumed exogenous in this model. The non-traded good (H) is produced with inputs of capital (KJ, labor (Lh) and land (Z),
The Cobb-Douglas production function was chosen to avoid the complications that arise if the elasticities of substitution in the three-factor production function are parameterized. The local market is competitive, with factors paid their values of marginal product; the output price for the local good is ph. The prices of both the local and export goods are determined by marginal costs.
2.2. Consumers The welfare level of a household depends upon its consumption traded and non-traded goods, specifically,
u = xalhaz.
of the
(3)
D.R. Haurin,
Effects
of subsidies
or taxes
on an urban
area
531
This form is chosen so that the market equilibrium for the non-traded good can be determined in a tractable fashion. A household’s take home income is w and is spent on both goods, either of which may be subsidized at per unit rates s, and s,, where s,, s,,> 0. The budget constraint for a household is1
w=(Ph-s,)h+(Pl--s,)4
(4)
where pf; is the local production cost of the traded good (p, =pL -s,). The demand equations are h = a,w(p, - sJ ‘, and x = a,wp; I. Substituting the demand equations back into the utility function yields the indirect utility function u = a”,‘ayWp,al(ph -sJ -a2.
(5)
2.3. Inputs
The total quantities of capital (K), labor (L), and land (2) employed in the urban area are determined within the model based on each factor’s compensation rate. The ratio of labor used in production of the local good to total labor is derived from the equilibrium conditions in the local goods market. Aggregate demand is H = hL= ~w(p, -sJl L. The value of marginal product equation for labor employed in the local sector is w-s, =62ph(H/Lh), where sL is the per unit subsidy for labor. Solving for H yields H =&‘(w -sJp; ‘Lh and equating H to the aggregate demand for the non-traded good implies Lt&=
~2~2(W/(W-sJ)(PhMPh -&I)).
(6)
2.4. Equations of change
The subsidies analyzed include per unit subsidies for labor (Q), land (s,), capital (s,J, and both outputs. Originally, no local subsidies (or taxes) exist and the changes are assumed small (which is not unrealistic since we are considering the implementation of local subsidies or taxes). Also, units of inputs and outputs are defined such that all pre-tax prices are equal to one.’ A positive value for ds represents an increase in the subsidy. Differentiating eq. (5) yields ti=dw-a,dp,+u,ds,, ‘The returns to local capital and land of the effects of including the return mechanism, see Haurin (1980). Assuming reasonable given the likely applications land rentals as part of local income are ‘These assumptions are typical of the
(7) are exported, earned by non-residents. For a discussion to local land in local income through a land bank that the return to capital accrues to non-residents is of the model. The modifications resulting from including discussed further in footnote 6. Harberger-Mieszkowski-McLure analysis.
D.R. Haurin,
538
Effects
of subsidies
or taxes
on an urban
area
where the dot above a variable represents a percentage change. The export sector is represented by a substitution equation and a cost equation, K,-L,,=S,(dw-ds,-dr+ds,), dp;=o,(dr-ds,)+a,(dw-ds,).
(9)
Derivations of these equations are presented in Harberger (1962), Mieszkowski (1967), or Tresch (1981). In eq. (9), g1 is the share of capital costs in total costs for X, and c2 is the share of labor costs. Since dp: =dp,+ds, and dp, = 0, eq. (9) can be rewritten as 0 = o,(dr - ds,) -f a,(dw - dsJ - ds,.
(9’)
Changes in the local sector are presented in eqs. (10)-(12), including substitution possibilities and a cost equation.
the
K,-i=dp,-ds,-dr+ds,, i-
t,=dw-ds,-dp,+ds,,
(11)
dp,=&(dv-ds,)+&(dw-ds,)+&(dp,-ds,).
(12)
In (12), the ai represent the cost shares of factor i in the production of the local good. Note that in this formulation, dp, is the change in the net of subsidy price of the local good, consistent with the notation for dp,, dw, and dr. [The Mieszkowski analysis defines the change in the output price variable to be the change in the gross of subsidy (tax) price.] Turning to the input market, the differentiated version of eq. (6), which allocates labor among sectors is3 t,-L=ds,+dsh.
(13)
The changes in the total quantities of labor and capital are dL,+dL,=dL,
(14)
dK,+dK,=dK.
(15)
Finally, in eqs. (16)-(18), the mobility of the factors to this urban area are ‘In the derivation critical.
of eq. (13),
the assumption
relationships that
initially
are specified. The supplies no subsidies
exist
and
w = 1 are
D.R.
Haurin,
Effects
of subsidies
or taxes
on an urban
area
539
it=ekdr,
(16)
(’ \
i=ezdpz,
(17)
L=e,O.
(18)
The elasticities of supply of capital, land, and labor are symbolized as ek, e,, and eL. Laborers respond to changes in the level of utility which is related to the real wage rate through eq. (7). The supply elasticity of land is e,, assumed to be 0 5 e, 5 co. If the city is of fixed size then e, = 0, but this assumption is too extreme to impose initially. (The fact that a particular plot of land is immobile does not imply e, =O.) Frequently, urban models assume that at the area’s border, agricultural land is available to the city and the price is fixed which implies e, = co. However, in a spatial urban model, the cost of locating at the edge of a city increases with distance from city center due to increased transport costs. Thus, the ‘full cost’ of marginal land increases as the city expands. This fact can be captured by allowing 0
-1
1 1
KIL
i B 1[I [I
a,&
- allo2 *1
dr
-(1+4 -(l+4KhlK,
dp,
B
=ds,+a2ds,+a,6,ds,+(l-a,6,)o,lds,+(K/L)ds,,
C
= -ds,-ds,-ds,/a,-(o,/oJds,-ds,,
4The pre-subsidy model can be solved for certain ratios that are subsidy solution. Noting Lh + L, = L and utilizing eq. (6), Lb/Lx = and LJX=o,, then X/H=(6,/o,)(L,/L,,). Also K,/X =(rl, K,/H= Combining the above expressions, K,/K, =(~,/o,)a,S,/(l -G~~SJ. (fL+K,YX
M+WKJ)=a
YE= (L,+L,)/x
K
= o,(l+(L,/L,))
In the matrix
IEl, the IElI,
element
c D
6
Z 1
is xZSI +(uI/oz)(l
+2(1-u
2 2
-q6,),
that
(19)
of some importance in the postc(,6,/( 1 -a&. Since L,/H = 6, 6,, thus X/H =(BJcT~)(K~/KJ. Finally,
6 ),
62
2 where
is, K/L.
D.R. Haurin,
+(oJaJ(l
K/L=c&
Effects
of subsidies
or taxes
on an urban
area
-a,6,).
Computing the determinant of the 3 x 3 matrix (IEl) reveals that IEl >O. The solutions for i, dr, and dp, can be found directly by applying Cramer’s Rule; the solutions for the other endogenous variables of interest can then be determined from the equations of the original system.5 Since the change in the level of household welfare, the capital stock, and the area1 size of the city are directly related to i, dr, and dp, ,through the mobility equations, their solutions are not presented. It is sufficient to note that capital, land, and labor flow into the urban area if dr, dp,, and ri are positive. In the next section the discussion of the effects of giving local subsidies is in the following order: subsidies for land, capital, labor, the local good, and the traded good. A general summary of the directions of change in all endogenous variables resulting from a subsidy is presented in table L6 Summary
ds, ds, ds, dsc ds,
of directions of change subsidized. (Many
Table 1 in the endogenous variables if either of the changes can also be 0 in specific
factors cases.)
or outputs
are
i
dr
dpz
dp,
dw
1
ri
XjH
wk
qk,
ft,
K
+ + + ? +
+ + ? ? +
+ + + + +
? ? ? +
+ ? ? +
+ ++? ? ? +??
+
-
? +-
?
0 0 +
? ? ? ? ?
? + ? ? ?
+ + ? ? +
0’
‘The solutions for L, dr, dp,, and dw are contained in an appendix available from the author. 6An alternative specification of the model would allow for the return of land rents to the local population through a ‘land bank’ mechanism. If an individual’s receipts from the land bank are A,, the modified model is described as follows assuming the additional restriction that no land subsidies are given, ds, =O. In eqs. (4), (5), and (6), replace w with w +A,. In eq. (7), replace i with (wiA,). Eq. (13) is replaced by &i=(A-G)()(A/(w+A))+ds,,+ds,. Total land bank payments are AiL=p,Z=G,p,(H/Z)Z and substituting for H yields AiL=G3(w-s,)L,G;‘, thus k =3-ds,+& -i. The model now consists of 13 equations and 13 unknowns. Following the same solution technique reveals the interesting result that the inclusion of a land bank mechanism for returning land rentals to the local population changes the conclusions only for a subsidy to the local good. Eq. (19) remains the same except that the ds, terms in B, C, D are Thus, only a minor’ replaced with (CQ -(A/w)); ((A + w)/w); -((L,/L,)(A + w)/w) respectively. modification of the model results if payments to land are retained by the local population. The equations for ds,, ds,, and ds, remain unchanged (ds, =0 by assumption).
D.R. Ham-in,
Effects
of subsidies
or taxes
on an urban
area
541
3. Analysis of the effects of local taxation upon an urban area 3.1. Land subsidies
A subsidy for land raises the net rental of land and lowers the gross of subsidy rental (unless e, =O).’ In local production, land is substituted for the other factors, tending to lower their prices. Also, the price of the local good declines. With local goods now less expensive, the urban area becomes more attractive to households. Wage rates decline, but by less than a factor of a,; thus, the level of household welfare rises and laborers immigrate into the area. (If e; 1 = 0, labor is fully mobile, then dp, = dwfa,, as expected, since ri=O in this case.) In the traded good sector, eq. (9’) implies that wage rates and the return to capital must move in opposite directions; thus, dr>O. Therefore, the subsidy for land results in an inflow of capital and labor to the city and a spatially larger city. The change in the aggregate demand for the local good occurs as a result of changes in population and in each household’s level of demand. Both increase in response to the land subsidy, and production of the local good increases. The change in the output of the traded good depends on the changes in the inputs of labor and capital.8 Previously, ‘we found that, in aggregate, capital and labor immigrate into the urban area. Also, there is no change in the ratio L,/L,, thus, the labor input to the export good sector increases. The change in the ratio of capital in the local sector to the traded sector is (Kh/Kx)=(K/K,)((l+e,)dp,-ds,-(l+e,)dr:+ds,). In the case of a land subsidy, l&,/K, falls if S,< 1. Also, although unlikely, K, may decline (it requires S, > l/cr, +e,K/K,.) However, the output of the traded good unambiguously increases if land is subsidized. It is also of interest to compare the changes for the traded to those of the local good (X/H can be derived from the equations presented in footnote 8). For a land subsidy, the increase in the production of the traded good is smaller in percentage terms than the increase in the production of the local good. If e, = 0, the city is of fixed size. The only change that results from a land subsidy is in the net price of land, dp, = ds,. Landowners fully capture the benefits of the subsidy if land is inelastic in supply. If the city can expand in size, the benefits are shared by the three inputs, the benefits of labor (capital) increasing as labor’s (capital’s) mobility declines. ‘Usually, the extreme cases where a supply elasticity is 0 or cc are not discussed in the text. *Given the production function for the traded good in eq. (I), then 2 =cr,xi, +g,&. An alternative form that indicates the importance of the elasticity of substitution is x=(1 -s,)((cr,/crz)(dr-dsk))-((1-~,S,)/cTz)ds,-ds,-(L/L,)ds,+(l+e,)dp,. A general expression for the change in output level of the local good is Ei=(6,+6,+e,)dp,-(6,+6,)ds,+(6,(a,/a,) -d,)(dr-ds,)-(b,/c,)ds,.
542
D.R. Haurin,
Effects
of subsidies
or taxes
on an urban
area
3.2. Capital subsidies If capital inputs in this urban area are subsidized, then the gross rental falls (the net rental rate rises) inducing factor substitution in both the traded and non-traded sectors. Since the price of the export good is fixed, the wage rate must rise. The general expectation is that the price of the local good will also increase but, if labor is relatively immobile and the local sector is capital intensive, the price may fall.’ The increase in the wage rate dominates the change in the price of the local good resulting in an increase in household welfare levels and an inflow of labor to this city. The rental of land must rise even though the direction of change in the price of housing is ambiguous. This can be seen by inspecting eq. (11). Since L,/L, = 0, the quantity of labor employed in the local sector must increase as’ the total labor force rises. Also, d w > 0, and i and dp, must be of the same sign since e,zO. The only way to satisfy the equation is for land’s rental to increase and for the area to become spatially larger. Increased levels of both inputs (L,,K,) for the traded good result in increased output. (Even though the direction of change in K,/K, is ambiguous, K, > 0.) lo Similarly, the output of the local good increases. Comparing the change in output levels between the traded and local good, the percentage increase in X is greater if labor is fully mobile (e,’ =O), but in general, the sign is ambiguous. 3.3. A subsidy for labor Labor subsidies create a series of changes that are ambiguous in direction. The only observations that can be stated precisely are that the population of the area increases, the ratio of local sector to export sector labor increases, and the return to land rises. The additional intersectoral shift in labor’s allocation creates the ambiguities (it is only present in this model if labor is subsidized). If the traded good sector is labor intensive, then the return to capital rises and capital flows into the urban area in response to the labor subsidy. Similarly, the net of subsidy wage and the price of the local good rise if the traded good sector is labor intensive. It is possible for the gross of subsidy wage to increase, a sufficient condition for it to decline is that the traded good sector be labor intensive (dr and dw -ds, must move in opposite directions). ‘For dp, 20, either ei 1 = 0 or 6,~~ i ~?,a, is sufficient. The latter inequality can be related to factor intensities as follows. Since 6,/o, =(K,/H)/(K,/X) and &a, =(L,/H)/(L,/X), substituting into the inequality yields K,/K, < L,,/L, or K,/L, < K,/L,. Thus, if the local sector is labor intensive, the price of the local good increases. A suffkient condition for dp, < 0 is: eL = 0, e, = 0, the local sector be capital intensive, and 6, > (rl. “‘The general expression for K, is K, = ((K,, + e,K)/K,)dr - (Kh/Kx)dsk -(K,JKJ( 1 + e,)dp, + W,lKNz.
D.R.
Haurin,
Effects
of subsidies
or taxes
on an urban
area
543
Generally, the output of the local good will rise in response to the labor subsidy. A necessary condition for it to fall is that the export sector be labor intensive and even if true, a reduction is unlikely. The direction of change in the output of the traded good is ambiguous, stemming from the uncertainty over whether capital inputs rise or fall. It is also possible that the labor input to the traded good declines due to the intersectoral shift in labor’s allocation. Finally, the direction of change in X/H is uncertain. 3.4. A subsidy for the local good
A subsidy for the local good increases production in the local sector and raises the proportion of labor devoted to local production. There is also. a shift in production towards the local sector from the export sector (X/H declines) even though the output level in the export sector may rise or fall. Land values in the area increase as does the spatial size of the urban area. In all but the most unusual cases, the population of the area will increase. The local sector being capital intensive is sufficient to guarantee an increased return to capital and an inflow of capital to the area. For this subsidy, the direction of wage change is opposite of the direction of change in the return to capital [6= -(c~~/gJ+]. A capital intensive local sector also yields a higher production price for the local good. If both capital and land are perfectly elastic in supply, the subsidy for the local good is passed fully on to the consumers. An interesting and unusual result occurs if both land and laborers are perfectly elastic in supply. Here dpZ = ri=O, therefore dw = a, (dp, -ds,). The production cost of the local good falls if 6,0, <6,cr1, that is, the local sector is labor intensive. The net price paid by consumers falls further resulting in a decline in local wages and increase in the return to capital. However, the increased cost of capital does not offset the wage decline and because land prices are fixed, the local sector’s production costs dec1ine.r l 3.5. A subsidy for the traded good
The last policy tool analyzed is a subsidy for the good that is exported from this urban area. The effects of the output subsidy upon factors are generally as expected. Since the gross price of the good is fixed in the national market, the subsidy allows for an increase in the net price. The result is that both the wage rate and the return to capital rise. All three factors flow into the urban area. With the increase in input prices, the price of the local good also rises. I’This case -(~I/~Z)(K/Kx)lEI-ld~h dr= -(o,/g,)dw, lEla,](6,cr,-6,a,)ds,.
requires
dp, = 0 = 0, therefore, dw=cc,(dp,-ds,). Note that dp,-ds,= ~0. Substituting into the expression for dw reveals dw0. Using (12), dp,=61dr+6,dw, or dp,=[a,(K/K,)/ If the local sector is labor intensive then 6,(r,>6,~~~ and dp,
544
D.R.
Ha&n,
Effects
of subsidies
or taxes
on an urban
area
The increased wage more than compensates for the higher cost of the local good, thus the population of the city grows. Since the ratio of L,/L, is unchanged, the labor inputs to both the traded good and the non-traded good rise. The land input to the non-traded good also increases and since, in aggregate, capital flows into the urban area, the production of both outputs tends to rise (both can not fall). However, the direction of change in K,/K, is ambiguous, depending upon the elasticity of substitution of production in the traded good sector and the supply elasticities of capital and land. Either &, or r(;r, may be negative. In the local good sector, if S, 2 1, then the capital input rises and since the quantity of other inputs’also increases, the output of H must rise in this case. Also, even though the capital input to the export sector may fall, g>O, the quantity of the traded good that is produced must increase in response to the subsidy. 4. Special cases and conclusions A brief inspection of table 1 reveals that unless some of the parametric values are specified, the general reactions to urban subsidies are frequently of uncertain direction. Three special cases are now analyzed yielding better insights into how an urban area changes in response to various subsidies. In the first case, capital is assumed fully mobile and marginal land is available at a fixed price for urban expansion. No particular values are assumed for the elasticity of substitution in export good production or laborers’ supply elasticity. The results are reported in table 2. In the second special case, laborers are assumed fully mobile, the city is of fixed spatial size, and a unitary elasticity of substitution is assumed (table 3). The final case assumes capital and labor are fully mobile (table 4). Which type of subsidy is favored depends on the policy goal of the national government. A few comments can be made without further specifying the social welfare function. If land is in fixed supply (table 3), a subsidy to land is clearly ineffective while a subsidy for capital increases output of the traded good both absolutely and relative to the output of the local good. If marginal land is available at a fixed price and capital is mobile (table 2), then subsidies for labor or the local good are associated with an uncertain change
Summary
ds, ds, ds, ds, ds,
+ + + + +
0 0 0 0 0
of directions
0 0 0 0 0
? 0 0 +
of change
0 + 0“ +
Table 2 if e, = co, ek = co, 0
++++? 7 i ++?
++-
< co, 0
0 0 + 0’
0 + + ?
1 < co.
+ + ? ? ?
+ + + ? +
D.R. Haurin,
Summary
ds, ds, ds, ds, ds,
Effects
of directions
oj”subsidies
or taxes
dr dpz
dp,
dw
Tt
0 + + + +
0 f ? + +
0 + ? ? +
0 + ? +
000 +++ ? ? ++?
Summary
ds, ds, ds, ds, ds,
of directions
area
545
Table 3 of change if e, = 0, eL 1 = 0,C3x = 1,O
i
+ + + + +
on an urban
of change
fi
X;H
+? +-
LdL
K,;K,
f?,
k
0 0 +
0 0 + + +
0 + ? ? ?
0 + ? + +
:
Table 4 if ek = co, e; 1 = 0, 0 < S, < co, 0 < e, < co.
i
dr
dpz
dp,
dw
8
+ + + + +
0 0 0 0 0
+ + + + +
0 + + + +
0 + + 0 +
++o +++ ++? +++++
fi
X;H
LIL 0 0 + 0’
K,;K,
&,
0 ? + + +
+ + + + ?
r( + + + + +
in export good production, which may be undesirable. Also, in this case, subsidies for land, the local good, or labor lead to a relative decline in traded good output compared to local good output. The assumptions behind the results presented in table 4 make the example applicable to many of the more developed countries and the less developed countries that exhibit a high degree of mobility of capital and labor. All subsidies raise the output of the traded good and result in an inflow of capital and population to the subsidized city. The ratio X/H increases for subsidies to capital, the export good, and possibly for labor. The magnitudes can not be orderedI except that the increase resulting from a subsidy for the traded good is larger than that for a subsidy for the capital good if ds, = ds,. The general conclusion reached is that a labor subsidy or subsidy for the local good frequently results in uncertain effects. A subsidy for land is only effective to the extent that land is elastic in supply, and it increases production of the local good relative to the export good. Subsidies for the exported good or capital always result in a capital inflqw and increased production of the export good. The general ordering of X/H resulting from subsidies for capital, labor, or the export good can not be specified without further information concerning the parameters’ values. For specific applications, the model can be easily implemented since most parameters’ values 12The coefficients of X;H are (if ek= co,e-I-L -0) +u201Sx)/a2; for dsL:63(Cc,-G126Z)/(1-C(Z82). RSUE-
C
for
ds,:6,(cT,/~,)(cc,+GIZS,);
for
ds,:6,(cc,
546
D.R.
Haurin,
Effects
of subsidies
or taxes
on an urban
area
can be obtained by estimating factor or consumption shares. Once the national goals are detailed, the subsidy that is most effective can be selected. The special cases help to illuminate the importance of assumptions concerning factor mobility. Those models that restrict mobility possibilities with a priori assumptions limit the generality of their applications. Also, the importance of including a local (non-traded) good in the urban model is demonstrated by the possibility of intersectoral shifts in capital and labor. Laborers locational choice is guided not only by the wage rate, but also by the cost of living in the area which depends upon i. the (endogenous) price of the local good. References Dewees, Donald and Y. Kotowitz, 1980, A general equilibrium analysis of regional subsidies, Public Finance 36, no. 3, 344-361. Gerking, Shelby and J. Mutti, 1981, Possibilities for the exportation of production taxes, Journal of Public Economics 16, 233-252. Harberger, Arnold, 1962, The incidence of the corporation income tax, Journal of Political Economy 70,215-240. Haurin, Donald, 1980, The regional distribution of population, migration, and climate, Quarterly Journal of Economics 95,293-308. Homma, Masaaki, 1977, On the theory of interregional tax incidence, Regional Science and Urban Economics 7, 377-392. Jones, Donald, 1982, The interregional incidence of local taxes with migration and a nontradable good, Geographical Analysis 14, 109-123. McLure, Charles, 1969, The inter-regional incidence of general regional taxes, Public Finance 24, 457-483. McLure, Charles, 1970, Taxation, substitution, and industry iocation, Journal of Foiiticai Economy 78, 112-132. Mieszkowski, Peter, 1967, On the theory of tax incidence, Journal of Political Economy 75, 251& 262. Renaud, Bertrand, 1979, National urbanization policies in developing countries, Working paper no. 347 (The World Bank, Washington, DC). Tresch, Richard, 1981, Public finance, A normative theory (Business Publications, Plano, TX).
Science
and Urban
THE EFFECTS
Economics
14 (1984)
533-546.
North-Holland
OF OUTPUT AND FACTOR SUBSIDIES TAXES ON AN URBAN AREA
OR
Donald R. HAURIN” The Ohio State University, Columbus, OH 43210, USA Received
July
1982, final
version
received
April
1984
This study investigates the effects of five types of factor and output subsidies upon an urban area. It represents an improvement upon prior regional tax or subsidy analyses since it includes a non-traded good in the urban model. Also, the supply elasticities of the three inputs to production; land, labor, and capital are parameterized, thus increasing the generality of the results. A suggested application of the model is to the study of urban decentralization policies in countries with a low or mid-level of development.
1. Introduction
The economic stimulation of specific urban areas has frequently been a goal of national governments. Among the policy tools that have been utilized are factor and output subsidies for goods produced or consumed in the selected area. The model presented in this paper analyzes the effects of these subsidies on a small or mid-size urban area. The emphasis is upon an area of this particular size since it is a likely target for subsidies, especially in less developed countries pursuing decentralization policies. Another reason for urban subsidies, mentioned by Renaud (1979), is that they are one method of securing claims to disputed international borders. The growth of selected cities is seen as one means of strengthening national claims to the territory. This paper will discuss responses to both factor and output subsidies or taxes. Of particular concern is the identification of the direction of flows of factors into or from the urban area as well as the change in the level of outputs. Also, changes in the rate of return to factors and the price of outputs are discussed. It is likely that no single public policy has the desired effects in all environments (countries). Thus, one goal of this study is to derive a fairly complete set of reactions to the various subsidies so that a comparison among them is possible. The literature in this area has concentrated on the analysis of regional, not urban, subsidies or taxes. Many of the models are variations of the twosector, general equilibrium model of Harberger (1962) which was used to *I wish
to thank
01660462/84/$3.00
the anonymous 0
1984, Elsevier
referee
for comments
Science
Publishers
on the earlier
draft.
B.V. (North-Holland)
534
D.R. Ham-in,
Effects
of subsidies
or taxes
on an urban
area
analyze the incidence of a factor tax, on capital. Harberger’s model was extended by Mieszkowski (1967) and later regionalized by McLure (1970). In McLure’s model, the two sectors are renamed as two regions, each region producing only one good. One attractive feature of McLure’s analysis is that factor supply elasticities are parameterized, contrasted with a number of later articles that restrict certain factors to be immobile. The application of Harberger’s model to a regional economy is particularly appropriate since, in theory, the taxes subject to analysis should be ‘small’, a reasonable assumption for the level of regional subsidies or taxes. The incidence of interregional taxes has been analyzed by McLure (1969) and Homma (1977). Both models are limited by an assumption that labor is immobile. Also, each region specializes in the production of a single good which limits the application of the analysis in the urban framework. One of the characteristics of cities that urban economists have found interesting is the presence of non-traded goods, housing in particular. If regions are assumed to specialize in the production of a single good, the possibility of intersectoral shifts in production within a region are eliminated. The analysis of subsidies in an urban context requires a different framework. An extension in this direction was made by Gerking and Mutti (1981), where they develop a model which allows for incomplete specialization in production in one region of a two-region economy. They further assume that capital is perfectly mobile between regions, labor is immobile, and that production exhibits a linear homogeneous technology. An output tax is analyzed, and they find the existence of the second sector in a region is important in that the tax incidence also depends on whether the taxed good is capital intensive within the two-good region. Limitations include the labor immobility assumption, and the requirement that all goods are traded. The analysis indicates that the existence of a second sector modifies the incidence results found in the earlier literature. Another study was by Dewees and Kotowitz (1980), again analyzing regional subsidies. Their contributions include allowing for the existence of non-constant returns to scale in production and the possibility of less than full employment. Both of these extensions are relevant for urban areas, thus they appear to be fruitful areas for further research. Their analysis is limited, similar to the earlier studies, by the assumptions that labor is immobile and regions specialize in production. The final study that is relevant for the analysis of factor or output tax effects on urban areas is by Jones (1982). He includes a non-traded good in a general equilibrium urban model, thus begins to directly address the problem of interest. However, .when performing the comparative statics, specific numerical values were assumed for all parameters in the model, thus generalizations are severely limited. The results derived from the model indicate that the addition of a non-traded good to an urban model importantly modify the conclusions drawn from the earlier models.
D.R. Haurin,
Effects
of subsidies
or taxes
on an urban
area
535
The most noticeable omission in the literature examining the effect of subsidies or taxes, is the development of models relevant for urban areas. Of primary importance is the inclusion of a non-traded good representing housing and other local services. Other models have found that certain policies encourage the growth of a particular region. For example, assume that growth occurs in the capital stock of the region. A conclusion that could ,be derived in the framework of a regional model is that this policy is desirable, given that capital growth was the target. However, if most of the increase in capital occurs in the local sector (housing), the policy verdict may be less favorable compared to the situation where the capital is employed in the export good sector. In any case, it is important that the effects of taxes or subsidies be fully identified, especially differentiating local good production from export good production. The model developed in this paper includes both a traded and a nontraded good. Factor’s supply elasticities are parameterized, avoiding an unrealistic a priori assumption and increasing applicability. Another facet of the model is that since land is a particularly important factor in urban analysis, the local sector is modeled as a three-factor sector (land, labor, capital). Thus, specific assumptions about land’s supply elasticity can be incorporated directly, rather than inappropriately assuming land and capital are equivalent factors. Limitations of the model include the assumption of constant returns to scale in production, restrictions upon consumers’ demand elasticities, and assuming the full employment of factors. The model is developed to analyze taxes or subsidies in a small or mid-size urban area. Thus a ‘small city’ assumption is made, fixing the price of the exported good. Factors flow to or from the rest of the country, but the other region or regions are not explicitly modeled. The next section of the paper presents the model of the urban area, introduces factor and output subsidies, and derives the equations of change. (Throughout the analysis, taxes can be analyzed rather than subsidies by making the appropriate sign changes.) The derivation of the responses of the endogenous variables to the various subsidies is presented in the third section. The concluding section of the paper discusses some applications of the analysis and investigates some special cases. 2. The urban model
Following a general outline of the urban model, the specifications for each sector are presented. The equations describing the reactions of the endogenous variables to factor or output subsidies are then derived. Two outputs are produced in the locality, one is only consumed locally, the other is both exported and consumed locally. Producers of the export good consider its price to be fixed, determined in the national or international economy. The
536
D.R. Haurin,
Effects
of subsidies
OY taxes
on an urban
area
price of the non-traded good is endogenous, determined by local aggregate demand and supply, and generally this price will vary in response to the imposition of subsidies. Inputs to the export good are capital and labor while local good production also utilizes land. The supply elasticities of all of the inputs are parameterized. Labor is assumed mobile between sectors in the urban area. Throughout the analysis, the markets are assumed to be competitive. Consumers are utility maximizers, selecting the consumption quantities of two goods, and earning income through their ownership of labor. Subsidies (taxes) can be given to any of the three factors or either of the outputs. Since the model is of a smaller urban area and the application of the model is .the analysis of national or international development of certain areas through subsidies, it is most appropriate to consider the revenue source for the subsidy to be external to the urban area. 2.1. The output sectors The export good (X) is produced with a linear homogeneous utilizing capital (K,) and labor (L,),
technology
x =fK, Lx).
(1)
The elasticity of substitution in production is S,, defined to be a positive number. Both factors are paid their values of marginal product. The price of the output is symbolized as px and is assumed exogenous in this model. The non-traded good (H) is produced with inputs of capital (KJ, labor (Lh) and land (Z),
The Cobb-Douglas production function was chosen to avoid the complications that arise if the elasticities of substitution in the three-factor production function are parameterized. The local market is competitive, with factors paid their values of marginal product; the output price for the local good is ph. The prices of both the local and export goods are determined by marginal costs.
2.2. Consumers The welfare level of a household depends upon its consumption traded and non-traded goods, specifically,
u = xalhaz.
of the
(3)
D.R. Haurin,
Effects
of subsidies
or taxes
on an urban
area
531
This form is chosen so that the market equilibrium for the non-traded good can be determined in a tractable fashion. A household’s take home income is w and is spent on both goods, either of which may be subsidized at per unit rates s, and s,, where s,, s,,> 0. The budget constraint for a household is1
w=(Ph-s,)h+(Pl--s,)4
(4)
where pf; is the local production cost of the traded good (p, =pL -s,). The demand equations are h = a,w(p, - sJ ‘, and x = a,wp; I. Substituting the demand equations back into the utility function yields the indirect utility function u = a”,‘ayWp,al(ph -sJ -a2.
(5)
2.3. Inputs
The total quantities of capital (K), labor (L), and land (2) employed in the urban area are determined within the model based on each factor’s compensation rate. The ratio of labor used in production of the local good to total labor is derived from the equilibrium conditions in the local goods market. Aggregate demand is H = hL= ~w(p, -sJl L. The value of marginal product equation for labor employed in the local sector is w-s, =62ph(H/Lh), where sL is the per unit subsidy for labor. Solving for H yields H =&‘(w -sJp; ‘Lh and equating H to the aggregate demand for the non-traded good implies Lt&=
~2~2(W/(W-sJ)(PhMPh -&I)).
(6)
2.4. Equations of change
The subsidies analyzed include per unit subsidies for labor (Q), land (s,), capital (s,J, and both outputs. Originally, no local subsidies (or taxes) exist and the changes are assumed small (which is not unrealistic since we are considering the implementation of local subsidies or taxes). Also, units of inputs and outputs are defined such that all pre-tax prices are equal to one.’ A positive value for ds represents an increase in the subsidy. Differentiating eq. (5) yields ti=dw-a,dp,+u,ds,, ‘The returns to local capital and land of the effects of including the return mechanism, see Haurin (1980). Assuming reasonable given the likely applications land rentals as part of local income are ‘These assumptions are typical of the
(7) are exported, earned by non-residents. For a discussion to local land in local income through a land bank that the return to capital accrues to non-residents is of the model. The modifications resulting from including discussed further in footnote 6. Harberger-Mieszkowski-McLure analysis.
D.R. Haurin,
538
Effects
of subsidies
or taxes
on an urban
area
where the dot above a variable represents a percentage change. The export sector is represented by a substitution equation and a cost equation, K,-L,,=S,(dw-ds,-dr+ds,), dp;=o,(dr-ds,)+a,(dw-ds,).
(9)
Derivations of these equations are presented in Harberger (1962), Mieszkowski (1967), or Tresch (1981). In eq. (9), g1 is the share of capital costs in total costs for X, and c2 is the share of labor costs. Since dp: =dp,+ds, and dp, = 0, eq. (9) can be rewritten as 0 = o,(dr - ds,) -f a,(dw - dsJ - ds,.
(9’)
Changes in the local sector are presented in eqs. (10)-(12), including substitution possibilities and a cost equation.
the
K,-i=dp,-ds,-dr+ds,, i-
t,=dw-ds,-dp,+ds,,
(11)
dp,=&(dv-ds,)+&(dw-ds,)+&(dp,-ds,).
(12)
In (12), the ai represent the cost shares of factor i in the production of the local good. Note that in this formulation, dp, is the change in the net of subsidy price of the local good, consistent with the notation for dp,, dw, and dr. [The Mieszkowski analysis defines the change in the output price variable to be the change in the gross of subsidy (tax) price.] Turning to the input market, the differentiated version of eq. (6), which allocates labor among sectors is3 t,-L=ds,+dsh.
(13)
The changes in the total quantities of labor and capital are dL,+dL,=dL,
(14)
dK,+dK,=dK.
(15)
Finally, in eqs. (16)-(18), the mobility of the factors to this urban area are ‘In the derivation critical.
of eq. (13),
the assumption
relationships that
initially
are specified. The supplies no subsidies
exist
and
w = 1 are
D.R.
Haurin,
Effects
of subsidies
or taxes
on an urban
area
539
it=ekdr,
(16)
(’ \
i=ezdpz,
(17)
L=e,O.
(18)
The elasticities of supply of capital, land, and labor are symbolized as ek, e,, and eL. Laborers respond to changes in the level of utility which is related to the real wage rate through eq. (7). The supply elasticity of land is e,, assumed to be 0 5 e, 5 co. If the city is of fixed size then e, = 0, but this assumption is too extreme to impose initially. (The fact that a particular plot of land is immobile does not imply e, =O.) Frequently, urban models assume that at the area’s border, agricultural land is available to the city and the price is fixed which implies e, = co. However, in a spatial urban model, the cost of locating at the edge of a city increases with distance from city center due to increased transport costs. Thus, the ‘full cost’ of marginal land increases as the city expands. This fact can be captured by allowing 0
-1
1 1
KIL
i B 1[I [I
a,&
- allo2 *1
dr
-(1+4 -(l+4KhlK,
dp,
B
=ds,+a2ds,+a,6,ds,+(l-a,6,)o,lds,+(K/L)ds,,
C
= -ds,-ds,-ds,/a,-(o,/oJds,-ds,,
4The pre-subsidy model can be solved for certain ratios that are subsidy solution. Noting Lh + L, = L and utilizing eq. (6), Lb/Lx = and LJX=o,, then X/H=(6,/o,)(L,/L,,). Also K,/X =(rl, K,/H= Combining the above expressions, K,/K, =(~,/o,)a,S,/(l -G~~SJ. (fL+K,YX
M+WKJ)=a
YE= (L,+L,)/x
K
= o,(l+(L,/L,))
In the matrix
IEl, the IElI,
element
c D
6
Z 1
is xZSI +(uI/oz)(l
+2(1-u
2 2
-q6,),
that
(19)
of some importance in the postc(,6,/( 1 -a&. Since L,/H = 6, 6,, thus X/H =(BJcT~)(K~/KJ. Finally,
6 ),
62
2 where
is, K/L.
D.R. Haurin,
+(oJaJ(l
K/L=c&
Effects
of subsidies
or taxes
on an urban
area
-a,6,).
Computing the determinant of the 3 x 3 matrix (IEl) reveals that IEl >O. The solutions for i, dr, and dp, can be found directly by applying Cramer’s Rule; the solutions for the other endogenous variables of interest can then be determined from the equations of the original system.5 Since the change in the level of household welfare, the capital stock, and the area1 size of the city are directly related to i, dr, and dp, ,through the mobility equations, their solutions are not presented. It is sufficient to note that capital, land, and labor flow into the urban area if dr, dp,, and ri are positive. In the next section the discussion of the effects of giving local subsidies is in the following order: subsidies for land, capital, labor, the local good, and the traded good. A general summary of the directions of change in all endogenous variables resulting from a subsidy is presented in table L6 Summary
ds, ds, ds, dsc ds,
of directions of change subsidized. (Many
Table 1 in the endogenous variables if either of the changes can also be 0 in specific
factors cases.)
or outputs
are
i
dr
dpz
dp,
dw
1
ri
XjH
wk
qk,
ft,
K
+ + + ? +
+ + ? ? +
+ + + + +
? ? ? +
+ ? ? +
+ ++? ? ? +??
+
-
? +-
?
0 0 +
? ? ? ? ?
? + ? ? ?
+ + ? ? +
0’
‘The solutions for L, dr, dp,, and dw are contained in an appendix available from the author. 6An alternative specification of the model would allow for the return of land rents to the local population through a ‘land bank’ mechanism. If an individual’s receipts from the land bank are A,, the modified model is described as follows assuming the additional restriction that no land subsidies are given, ds, =O. In eqs. (4), (5), and (6), replace w with w +A,. In eq. (7), replace i with (wiA,). Eq. (13) is replaced by &i=(A-G)()(A/(w+A))+ds,,+ds,. Total land bank payments are AiL=p,Z=G,p,(H/Z)Z and substituting for H yields AiL=G3(w-s,)L,G;‘, thus k =3-ds,+& -i. The model now consists of 13 equations and 13 unknowns. Following the same solution technique reveals the interesting result that the inclusion of a land bank mechanism for returning land rentals to the local population changes the conclusions only for a subsidy to the local good. Eq. (19) remains the same except that the ds, terms in B, C, D are Thus, only a minor’ replaced with (CQ -(A/w)); ((A + w)/w); -((L,/L,)(A + w)/w) respectively. modification of the model results if payments to land are retained by the local population. The equations for ds,, ds,, and ds, remain unchanged (ds, =0 by assumption).
D.R. Ham-in,
Effects
of subsidies
or taxes
on an urban
area
541
3. Analysis of the effects of local taxation upon an urban area 3.1. Land subsidies
A subsidy for land raises the net rental of land and lowers the gross of subsidy rental (unless e, =O).’ In local production, land is substituted for the other factors, tending to lower their prices. Also, the price of the local good declines. With local goods now less expensive, the urban area becomes more attractive to households. Wage rates decline, but by less than a factor of a,; thus, the level of household welfare rises and laborers immigrate into the area. (If e; 1 = 0, labor is fully mobile, then dp, = dwfa,, as expected, since ri=O in this case.) In the traded good sector, eq. (9’) implies that wage rates and the return to capital must move in opposite directions; thus, dr>O. Therefore, the subsidy for land results in an inflow of capital and labor to the city and a spatially larger city. The change in the aggregate demand for the local good occurs as a result of changes in population and in each household’s level of demand. Both increase in response to the land subsidy, and production of the local good increases. The change in the output of the traded good depends on the changes in the inputs of labor and capital.8 Previously, ‘we found that, in aggregate, capital and labor immigrate into the urban area. Also, there is no change in the ratio L,/L,, thus, the labor input to the export good sector increases. The change in the ratio of capital in the local sector to the traded sector is (Kh/Kx)=(K/K,)((l+e,)dp,-ds,-(l+e,)dr:+ds,). In the case of a land subsidy, l&,/K, falls if S,< 1. Also, although unlikely, K, may decline (it requires S, > l/cr, +e,K/K,.) However, the output of the traded good unambiguously increases if land is subsidized. It is also of interest to compare the changes for the traded to those of the local good (X/H can be derived from the equations presented in footnote 8). For a land subsidy, the increase in the production of the traded good is smaller in percentage terms than the increase in the production of the local good. If e, = 0, the city is of fixed size. The only change that results from a land subsidy is in the net price of land, dp, = ds,. Landowners fully capture the benefits of the subsidy if land is inelastic in supply. If the city can expand in size, the benefits are shared by the three inputs, the benefits of labor (capital) increasing as labor’s (capital’s) mobility declines. ‘Usually, the extreme cases where a supply elasticity is 0 or cc are not discussed in the text. *Given the production function for the traded good in eq. (I), then 2 =cr,xi, +g,&. An alternative form that indicates the importance of the elasticity of substitution is x=(1 -s,)((cr,/crz)(dr-dsk))-((1-~,S,)/cTz)ds,-ds,-(L/L,)ds,+(l+e,)dp,. A general expression for the change in output level of the local good is Ei=(6,+6,+e,)dp,-(6,+6,)ds,+(6,(a,/a,) -d,)(dr-ds,)-(b,/c,)ds,.
542
D.R. Haurin,
Effects
of subsidies
or taxes
on an urban
area
3.2. Capital subsidies If capital inputs in this urban area are subsidized, then the gross rental falls (the net rental rate rises) inducing factor substitution in both the traded and non-traded sectors. Since the price of the export good is fixed, the wage rate must rise. The general expectation is that the price of the local good will also increase but, if labor is relatively immobile and the local sector is capital intensive, the price may fall.’ The increase in the wage rate dominates the change in the price of the local good resulting in an increase in household welfare levels and an inflow of labor to this city. The rental of land must rise even though the direction of change in the price of housing is ambiguous. This can be seen by inspecting eq. (11). Since L,/L, = 0, the quantity of labor employed in the local sector must increase as’ the total labor force rises. Also, d w > 0, and i and dp, must be of the same sign since e,zO. The only way to satisfy the equation is for land’s rental to increase and for the area to become spatially larger. Increased levels of both inputs (L,,K,) for the traded good result in increased output. (Even though the direction of change in K,/K, is ambiguous, K, > 0.) lo Similarly, the output of the local good increases. Comparing the change in output levels between the traded and local good, the percentage increase in X is greater if labor is fully mobile (e,’ =O), but in general, the sign is ambiguous. 3.3. A subsidy for labor Labor subsidies create a series of changes that are ambiguous in direction. The only observations that can be stated precisely are that the population of the area increases, the ratio of local sector to export sector labor increases, and the return to land rises. The additional intersectoral shift in labor’s allocation creates the ambiguities (it is only present in this model if labor is subsidized). If the traded good sector is labor intensive, then the return to capital rises and capital flows into the urban area in response to the labor subsidy. Similarly, the net of subsidy wage and the price of the local good rise if the traded good sector is labor intensive. It is possible for the gross of subsidy wage to increase, a sufficient condition for it to decline is that the traded good sector be labor intensive (dr and dw -ds, must move in opposite directions). ‘For dp, 20, either ei 1 = 0 or 6,~~ i ~?,a, is sufficient. The latter inequality can be related to factor intensities as follows. Since 6,/o, =(K,/H)/(K,/X) and &a, =(L,/H)/(L,/X), substituting into the inequality yields K,/K, < L,,/L, or K,/L, < K,/L,. Thus, if the local sector is labor intensive, the price of the local good increases. A suffkient condition for dp, < 0 is: eL = 0, e, = 0, the local sector be capital intensive, and 6, > (rl. “‘The general expression for K, is K, = ((K,, + e,K)/K,)dr - (Kh/Kx)dsk -(K,JKJ( 1 + e,)dp, + W,lKNz.
D.R.
Haurin,
Effects
of subsidies
or taxes
on an urban
area
543
Generally, the output of the local good will rise in response to the labor subsidy. A necessary condition for it to fall is that the export sector be labor intensive and even if true, a reduction is unlikely. The direction of change in the output of the traded good is ambiguous, stemming from the uncertainty over whether capital inputs rise or fall. It is also possible that the labor input to the traded good declines due to the intersectoral shift in labor’s allocation. Finally, the direction of change in X/H is uncertain. 3.4. A subsidy for the local good
A subsidy for the local good increases production in the local sector and raises the proportion of labor devoted to local production. There is also. a shift in production towards the local sector from the export sector (X/H declines) even though the output level in the export sector may rise or fall. Land values in the area increase as does the spatial size of the urban area. In all but the most unusual cases, the population of the area will increase. The local sector being capital intensive is sufficient to guarantee an increased return to capital and an inflow of capital to the area. For this subsidy, the direction of wage change is opposite of the direction of change in the return to capital [6= -(c~~/gJ+]. A capital intensive local sector also yields a higher production price for the local good. If both capital and land are perfectly elastic in supply, the subsidy for the local good is passed fully on to the consumers. An interesting and unusual result occurs if both land and laborers are perfectly elastic in supply. Here dpZ = ri=O, therefore dw = a, (dp, -ds,). The production cost of the local good falls if 6,0, <6,cr1, that is, the local sector is labor intensive. The net price paid by consumers falls further resulting in a decline in local wages and increase in the return to capital. However, the increased cost of capital does not offset the wage decline and because land prices are fixed, the local sector’s production costs dec1ine.r l 3.5. A subsidy for the traded good
The last policy tool analyzed is a subsidy for the good that is exported from this urban area. The effects of the output subsidy upon factors are generally as expected. Since the gross price of the good is fixed in the national market, the subsidy allows for an increase in the net price. The result is that both the wage rate and the return to capital rise. All three factors flow into the urban area. With the increase in input prices, the price of the local good also rises. I’This case -(~I/~Z)(K/Kx)lEI-ld~h dr= -(o,/g,)dw, lEla,](6,cr,-6,a,)ds,.
requires
dp, = 0 = 0, therefore, dw=cc,(dp,-ds,). Note that dp,-ds,= ~0. Substituting into the expression for dw reveals dw
544
D.R.
Ha&n,
Effects
of subsidies
or taxes
on an urban
area
The increased wage more than compensates for the higher cost of the local good, thus the population of the city grows. Since the ratio of L,/L, is unchanged, the labor inputs to both the traded good and the non-traded good rise. The land input to the non-traded good also increases and since, in aggregate, capital flows into the urban area, the production of both outputs tends to rise (both can not fall). However, the direction of change in K,/K, is ambiguous, depending upon the elasticity of substitution of production in the traded good sector and the supply elasticities of capital and land. Either &, or r(;r, may be negative. In the local good sector, if S, 2 1, then the capital input rises and since the quantity of other inputs’also increases, the output of H must rise in this case. Also, even though the capital input to the export sector may fall, g>O, the quantity of the traded good that is produced must increase in response to the subsidy. 4. Special cases and conclusions A brief inspection of table 1 reveals that unless some of the parametric values are specified, the general reactions to urban subsidies are frequently of uncertain direction. Three special cases are now analyzed yielding better insights into how an urban area changes in response to various subsidies. In the first case, capital is assumed fully mobile and marginal land is available at a fixed price for urban expansion. No particular values are assumed for the elasticity of substitution in export good production or laborers’ supply elasticity. The results are reported in table 2. In the second special case, laborers are assumed fully mobile, the city is of fixed spatial size, and a unitary elasticity of substitution is assumed (table 3). The final case assumes capital and labor are fully mobile (table 4). Which type of subsidy is favored depends on the policy goal of the national government. A few comments can be made without further specifying the social welfare function. If land is in fixed supply (table 3), a subsidy to land is clearly ineffective while a subsidy for capital increases output of the traded good both absolutely and relative to the output of the local good. If marginal land is available at a fixed price and capital is mobile (table 2), then subsidies for labor or the local good are associated with an uncertain change
Summary
ds, ds, ds, ds, ds,
+ + + + +
0 0 0 0 0
of directions
0 0 0 0 0
? 0 0 +
of change
0 + 0“ +
Table 2 if e, = co, ek = co, 0
++++? 7 i ++?
++-
< co, 0
0 0 + 0’
0 + + ?
1 < co.
+ + ? ? ?
+ + + ? +
D.R. Haurin,
Summary
ds, ds, ds, ds, ds,
Effects
of directions
oj”subsidies
or taxes
dr dpz
dp,
dw
Tt
0 + + + +
0 f ? + +
0 + ? ? +
0 + ? +
000 +++ ? ? ++?
Summary
ds, ds, ds, ds, ds,
of directions
area
545
Table 3 of change if e, = 0, eL 1 = 0,C3x = 1,O
i
+ + + + +
on an urban
of change
fi
X;H
+? +-
LdL
K,;K,
f?,
k
0 0 +
0 0 + + +
0 + ? ? ?
0 + ? + +
:
Table 4 if ek = co, e; 1 = 0, 0 < S, < co, 0 < e, < co.
i
dr
dpz
dp,
dw
8
+ + + + +
0 0 0 0 0
+ + + + +
0 + + + +
0 + + 0 +
++o +++ ++? +++++
fi
X;H
LIL 0 0 + 0’
K,;K,
&,
0 ? + + +
+ + + + ?
r( + + + + +
in export good production, which may be undesirable. Also, in this case, subsidies for land, the local good, or labor lead to a relative decline in traded good output compared to local good output. The assumptions behind the results presented in table 4 make the example applicable to many of the more developed countries and the less developed countries that exhibit a high degree of mobility of capital and labor. All subsidies raise the output of the traded good and result in an inflow of capital and population to the subsidized city. The ratio X/H increases for subsidies to capital, the export good, and possibly for labor. The magnitudes can not be orderedI except that the increase resulting from a subsidy for the traded good is larger than that for a subsidy for the capital good if ds, = ds,. The general conclusion reached is that a labor subsidy or subsidy for the local good frequently results in uncertain effects. A subsidy for land is only effective to the extent that land is elastic in supply, and it increases production of the local good relative to the export good. Subsidies for the exported good or capital always result in a capital inflqw and increased production of the export good. The general ordering of X/H resulting from subsidies for capital, labor, or the export good can not be specified without further information concerning the parameters’ values. For specific applications, the model can be easily implemented since most parameters’ values 12The coefficients of X;H are (if ek= co,e-I-L -0) +u201Sx)/a2; for dsL:63(Cc,-G126Z)/(1-C(Z82). RSUE-
C
for
ds,:6,(cT,/~,)(cc,+GIZS,);
for
ds,:6,(cc,
546
D.R.
Haurin,
Effects
of subsidies
or taxes
on an urban
area
can be obtained by estimating factor or consumption shares. Once the national goals are detailed, the subsidy that is most effective can be selected. The special cases help to illuminate the importance of assumptions concerning factor mobility. Those models that restrict mobility possibilities with a priori assumptions limit the generality of their applications. Also, the importance of including a local (non-traded) good in the urban model is demonstrated by the possibility of intersectoral shifts in capital and labor. Laborers locational choice is guided not only by the wage rate, but also by the cost of living in the area which depends upon i. the (endogenous) price of the local good. References Dewees, Donald and Y. Kotowitz, 1980, A general equilibrium analysis of regional subsidies, Public Finance 36, no. 3, 344-361. Gerking, Shelby and J. Mutti, 1981, Possibilities for the exportation of production taxes, Journal of Public Economics 16, 233-252. Harberger, Arnold, 1962, The incidence of the corporation income tax, Journal of Political Economy 70,215-240. Haurin, Donald, 1980, The regional distribution of population, migration, and climate, Quarterly Journal of Economics 95,293-308. Homma, Masaaki, 1977, On the theory of interregional tax incidence, Regional Science and Urban Economics 7, 377-392. Jones, Donald, 1982, The interregional incidence of local taxes with migration and a nontradable good, Geographical Analysis 14, 109-123. McLure, Charles, 1969, The inter-regional incidence of general regional taxes, Public Finance 24, 457-483. McLure, Charles, 1970, Taxation, substitution, and industry iocation, Journal of Foiiticai Economy 78, 112-132. Mieszkowski, Peter, 1967, On the theory of tax incidence, Journal of Political Economy 75, 251& 262. Renaud, Bertrand, 1979, National urbanization policies in developing countries, Working paper no. 347 (The World Bank, Washington, DC). Tresch, Richard, 1981, Public finance, A normative theory (Business Publications, Plano, TX).