Urban growth, externality and welfare

ELSEVIER Regional Science and Urban Economics 24 (1994) 565-576

Urban growth, externality and welfare

Chi-Chur Chaos,*, Eden S.H. YubsC

“Department of Economics, Oregon State University, Cowallis, OR 97331, USA hDepartment of Decision Sciences and Managerial Economics, The Chinese University of

Hong Kong, N.T., Hong Kong ‘Department of Economics, Louisiana State University, Baton Rouge, LA 70803, USA

Received August 1992, final version received September 1993

Abstract

This paper examines the short-run and long-run welfare implications of policies for promoting urban growth for less-developed countries. It is shown that the short-run effects of tariffs, wage and production subsidies on the urban unemployment ratio do not carry over to the long run. Along with agglomeration economies and inter- sectoral externality, the urban unemployment ratio plays a key role in determining social welfare.

JEL classifications: R13; R23

1. Introduction

A 1983 survey conducted by the United Nations of 236 less-developed countries (LDCs) reveals that more than three-quarters of these countries are pursuing policies to slow down urban growth.’ Why do LDCs want to

curtail urban growth? The present paper attempts to develop a model to provide a plausible answer to this question.

* Corresponding author.

’ See Shukla and Stark (1986) for a detailed discussion.

0166-0462194/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDZ 0166-0462(94)02062-L

566 C.C. Chao, E.S.H. Yu I Reg. Sci. Urban Econ. 24 (1994) 565-576

The concentration of firms and agglomeration of economic activities in urban areas are instrumental to urban growth.’ However, rapid urban growth may generate a substantial agglomeration of diseconomies as well. The diseconomies, in the forms of congestion and pollution, are external to the firms but internal to the city. In addition, urban growth has also exerted various favorable and/or detrimental externalities to other sectors of an economy. The inter-sectoral spillover effects could arise from human and technological transfers to various types of pollution. Thus, all these internal and external externalities should be taken into account in determining the socially optimal size of a city.

Several features of our model may be noted at the outset. First, since urban unemployment is a pervasive problem in LDCs which accompanies urban growth, we introduce urban unemployment d la Harris and Todaro (1970), except that our model focuses on the externalities and on ag- glomerative scale economies and diseconomies, a distinct feature of urban growth. Second, we consider the normative implications of several urban developmental policies, e.g., tariffs, urban wage subsidies and urban production subsidies, to shed light on the optimal level of urbanization. The policy effects are examined both for the short run and the long run.

Several interesting results are found. In particular, it is demonstrated that the urban unemployment ratio, along with agglomeration economies and inter-sectoral externalities, plays a key role in identifying the optimal urban size and social welfare. In the short run, specific-factors version of the Harris-Todaro model, the policies to encourage urban growth considered in this paper enhance national welfare. However, in the long run, where capital is perfectly mobile between urban and rural sectors, the welfare effects of these policies are ambiguous and are likely to be negative.3 The policy implications of our results for the developing world are straight- forward. Government interventions in the presence of market imperfections can be useful for fostering healthy urban growth when factors are intersectorally immobile. However, when factors are intersectorally mobile, government interventions are likely to cause harm to the society.

The paper is organized as follows. Section 2 develops the model and provides the transformation relation between urban and rural outputs. Section 3 examines the welfare effects of urban growth policies in the short run, as well as in the long run. Section 4 offers conclusions.

* Mulligan (1984) provides a comprehensive review on this issue.

3 In the original Harris-Todaro (1970) model, capital is treated as sector-specific. Corden and Findlay (1975) extended it by incorporating perfect capital mobility between sectors.

C.C. Chao, E.S.H. Yu I Reg. Sci. Urban Econ. 24 (1994) 565-576 567

2. The model

The economy consists of two sectors: urban manufacturing, X, and rural agriculture, Y. Good Y is produced by labor services L, and capital K, and also influenced by output in urban sector, X:

Y= gGWW,, KY) = Y(L,, K,; X) > G,>O,Gii<O,G,j>O, (1)

where the subscript i and j (i, j = L, K) denote the partial derivatives. G is the internal technology for sector Y which is linearly homogeneous in factors. The role of inter-sectoral externality is described by g, where marginal external economies (diseconomies) exist if dg/dX > (c)O. Hence, output elasticity of inter-sectoral externality can be defined as: e = (dg/ dX)(X/g), where e > (<)O reflects the intersectoral economies (diseconomies) .4

Turning to the urban sectors, its production function is specified as

x =f(L,)w‘,~ Kx) = XL, Kx) 7 F, >O,Fii < O,Fij > 0, (2)

where F is the internal technology for sector X. The role of agglomeration economies or diseconomies, f, is determined by urban size, which can be approximated by employment in the urban sector, L,.5 To simplify the analysis, it is assumed that the economies of agglomeration remain invariant over city size.6 Cobb-Douglas specifications are used for the urban production as follows:

F= A(L,)"(K,)'-" ,

and

f = (L,lY

where y > (<)O denotes agglomeration economies (diseconomies). Since economies of agglomeration are external to firms in the urban

sector, individual firms do not see themselves influenced by city size and perceive themselves to have constant returns to scale technologies;7 the urban production function displays increasing (decreasing) returns to scale through the agglomeration effect.

4 See Herberg et al. (1982) and Yu (1987) for analysis on the implications of intersectoral

externality. ’ This specification is adapted from Shukla and Stark (1990). ’ This assumption implies that ez (i = 1, 2) is given in the model.

’ For related discussions, see, for example, Choi and Yu (1985, 1987) among others.


Let MP, denote the private marginal product of labor and MP: denote the social marginal product of labor in the urban sector. From (3) and (4), we obtain:

MP,(=fF,) = (YA(L,)“+~+‘(K,)~-~ ,

MP;(=X, = fFL + fLF) = (a + y)A(L,)“+Y+(Kx)‘-” ,

(5)

(6)

where FL = dF(L,, K,)ldL,, etc. For a positive and decreasing social marginal product of labor, we require that 0 < (Y + y < 1. Dividing (5) by (6) we have

MP,IMP; = a/((~ + y) , (7)

which is less (greater) than one if y is positive (negative). The private marginal product of labor in urban sector is less (greater) than the social marginal product of labor, which is consistent with agglomeration economies (diseconomies). The discrepancy between the private and social marginal products of labor in X implies that a competitive equilibrium will not be characterized as a point of tangency between the marginal rate of transformation (MRT) and the goods price ratio (shown below). Consequently, this distortion plays a crucial role for determining the implications of various urban policies.

We now turn to consider a labor market structure that allows us to capture the pervasive problem of urban unemployment for LDCs. In many LDCs the urban wages (wx) are set institutionally at levels higher than the market clearing wages, and hence urban unemployment emerges. Following Harris-Todaro, it is hypothesized that in the labor market equilibrium, the rural wage, wy, is equal to the expected urban wage, which is wx times the probability of employment in the urban sector:8

w,/(l + A) = WY ) (8)

where A = L, /Lx is the urban unemployment ratio and l/(1 + A) serves as an index for the probability of finding a job. Here, L, represents the level of urban unemployment.

Under perfect competition, labor is paid according to the value of its private marginal product:

wx = PDF, > (9)

WY = YL 9 (10)

where p is the price ratio of good X in terms of good Y.

‘For empirical studies of internal migration with expected real wage rates as the central

economic variables, see Salvatore (1977, 1980) for an example. For a recent theoretical analysis

of a version of the Harris-Todaro model, see Hazari and Sgro (1987).


Denoting the total endowment of labor by L, we can write

(l+A)L,+L,=L. (II)

With regard to capital, in the short run, it is sector-specific; that is, capital in each sector is a fixed factor, and so rentals, paid according to the marginal value products of capitals, differ between the two sectors:

pXK=rx#ry=YK, (12)

where ri (i =X, Y) is the rental rate. Denoting the short-run endowment of capital in the ith sector by K,, that is,

K; = I?, , i=X,Y. (13)

However, in the long run, capital will flow between the two sectors in response to these rental differentials, and eventually the rentals will be equalized:

pXK=rK=ry=YK,

and sector’s capital employment will satisfy

(14)

K,+K,=I&+I?,=l?, (15)

where K denotes the economy’s total capital endowment.’ This completes the production side of the model.

It is essential to derive the equilibrium condition concerning transformation between outputs and the goods price ratio. Differentiations of (l), (2), (11) and (13) or (15), and using (5)-(10) and (14) yield

p dX + dY = p[yI(a + y)]X, dL, + pue dX - wyL, dh , (16)

where u = YIpX is the relative value of good Y. The right-hand side (RHS) of (16) captures three types of distortion: an agglomeration effect (y), an inter-sectoral externality (e), and an unemployment effect (A). Consider the special case of no distortions, i.e., y =e =A =O. It is clear that dY/dX reduces to -p which is the traditional, well-known result regarding the tangency between the goods price line and the transformation frontier at production equilibrium. However, the RHS of (16) in general differs from zero. These distortions will be shown to be crucial in determining the welfare implications of urban growth policies.

’ The adjustment process from short-run equilibrium to long-run equilibrium is demonstrated

in Neary (1981). Chao and Yu (1990) utilize this concept to examine the welfare effects of the

terms of trade.


3. Urban growth policies and welfare

A duality approach is used to examine the welfare impacts of several urban growth policies. Let the demand side of the economy be represented by an expenditure function: E(p, U) = minimizing (PC, + C,) with respect to the demand for the two commodities, C, and C,, subject to a strictly quasi-concave aggregate utility function u(C,, C,) 3 u. We assume that the developing country exports the agricultural product and imports the manufacturing product.” The equilibrium condition of the commodity markets is

&(P, u) =X + M , (17)

where E, = aEl( p, u) I ap = C, and M denotes the import of the manufacturing goods.

For simplicity, assume that this is a small open economy and a tariff is imposed on the imports of X initially. The budget constraint may then be written as

Wp,u)=pX+Y+(p-p*)M, (18)

where p* is the foreign price ratio of good X. The first two terms on the RHS of (18) represent the real income from domestic production, and the third term denotes the tariff revenue, which is assumed to be redistributed to the private sectors in a non-distortionary manner, e.g., a lump-sum transfer.

Differentiating (18) and denoting the change in welfare by dW= E, du, we have

dW=p[y/(a+y)]X,dL,+pvedX-w,L,dh+(p-p*)dM, (19)

which is the key equation for our welfare analysis. The first term on the RHS of (19) captures the agglomeration effect, the second term denotes the inter-sectoral externality effect, the third term is the unemployment effect, and the four term represents the volume-of-trade effect. Note that since this is a small economy, i.e., p* is fixed in the world market, the terms of trade effect vanishes.

3.1. Short-run analysis

We examine first the economy’s response to various urban developmental policies in the short run, in which the amount of capital allocated to each of two sectors is assumed to be fixed. To accomplish this task, we carry out some comparative statics exercises. The three equations of immediate

‘“This assumption describes nations in the early stages of development. Some NICs export

mainly manufacturing goods, however.


relevance are (S), (9) and (11)) where (9) can be rewritten to accommodate the policy variable of urban wage subsidy sx as (1 - sx)wx = pfF,. Totally differentiating them, and solving the equations yields

d~,id~>o,d~,idp~o, aMap< 0, aL,ias,~o,aL,las,<o,

aA/as,co. (20)

The signs for dL,ldp, dL,ldp, dL,lds, and dL,lds, should be intuitive- ly clear. Given the neoclassical production function, the output responses to the changes in the goods price ratio and the subsidy are normal. The economic explanation for aA\/ap < 0 is as follows. An increase in p results in an expansion in the urban product and more labor employment in the presence of fixed amount of urban capital. Consequently, the urban unemployment ratio falls. Hence, urban growth through tariff protection would lead to a fall in urban unemployment in the short run. aAIds, < 0 can be similarly explained.

Now, we can use (19) with the aid of (20) to deduce the welfare implications of urban developmental policies. To begin with, consider the issue of optimality without policy interventions (laissez faire), defined as a situation where the domestic and world prices of all traded goods are equal, i.e., p = p*. The necessary condition for laissez faire to be an optimal policy is dW = 0, whereas the sufficient condition requires d*W <Cl. Hence under Zaissez faire, from (19), we have

dW = p[yI(cx + y)]X, dL, + pve dX - wyL, dh , (21)

which is generally not equal to zero. Thus, laissez faire in this expanded version of Harris-Todaro economy is not optimal. So let us consider government interventions in the form of various urban developmental policies. Specifically, we will study the welfare impacts of such policies as tariff, wage subsidy and production subsidy to the urban sector.”

3.1 .l. Tariff (t) The imposition of a tariff at rate t creates a wedge between the domestic

and foreign price ratios. The domestic price ratio for producers and consumers is: p =p*(l + t). Using this, (19) can be expressed as

I’ The welfare effects of these policies have been examined recently for the standard

Harris-Todaro model without agglomeration effect and inter-sectoral externalities by Batra and Naqvi (1987) and Beladi (1988). While their analyses provide a useful insight into the desirability of certain trade policies for LDCs, the distinctive features of urban growth, are not

modelled at all.

572 C.C. Chao, E.S.H. Yu I Reg. Sci. Urban Econ. 24 (1994) 56.5-576

dW/dt =p[y/(cz + r)]X,(aL,l+)(dp/dt) +pue(aXl@)(dp/dt)

- wYL,(&Uap)(dpldt) + tp*(dM/dt) . (22)

Recall that aL,lap >Q, axlap > 0, ahlap < 0, dpldt > 0 and dM/dt 2 0.” To see the welfare effects of a ‘small’ tariff, we may set t = 0 initially in (22). It immediately reveals that an introduction of a tariff as a tool to promote the urban sector may enhance social welfare, if there exist agglomeration and inter-sectoral economies to accompany the favorable short-run employment effects.

3.1.2. Wage tax-cum-subsidy to manufacturing (s,) Alternatively, the government may introduce a wage subsidy in the

manufacturing sector. The wage subsidy causes a fall in the labor cost to urban producers, i.e., ~~(1 - sx). However, the domestic price ratio faced by both consumers and producers remains at p, which is equal to the world price ratio, p*. Hence (19) is modified as

dW/ds, = [pX,y/(a + r) - SxWx](aLx/aSx) +pue(aX/as,)

- w,L,(aA/as,) , (23)

where aL,las, >O, ax/as, > 0 and ah/as, CO. Setting sX = 0 initially, dW/ds, is positive, if agglomeration and inter-sectoral economies prevail. Thus, the wage subsidy given to urban workers raises welfare in the short run.

3.1.3. Production tax-cum-subsidy to manufacturing (e,) When a production tax-cum-subsidy to urban manufacturing at the rate e,

in lieu of a tariff or wage subsidy is introduced, the goods price ratio facing consumers remains at p*, but the price ratio facing producers is p,. Note that p, = p*(l + ex). Incorporating the production subsidy (exp*X), the social budget constraint (18) changes to: E(p, u) = p,X + Y - e,p*X. Differentiation of this budget constraint, and some manipulations yields

dW/% = P,X,[Y/(~ + r)l(aL,lap,)(dp,/de,) + (we - e,P*)(axlap,)(dp,/de,) - w,L,(ahlap,)(dp,/de,) y (24)

I2 dMldr can be solved from (17) and (18) as

dMldt = T{-[(aXlap) - EP,]

+ Wp)[~(rla + r)x,(wap) +pue(axlap) - w,ua~lap)l} ,

where T = l/[l - Iml(l + r)] is the tariff multiplier and m = pE,,IE, is the marginal propensity

to consume for good X. The first part of dM/dt denotes the change in supply of imports and the second part is the change in demand for imports due to the change of tariffs. The sign of dMldt is generally ambiguous.


where dp,/de, > 0. Equation (24) reveals that if initially e, = 0, then dW/de, > 0 in the presence of agglomeration and inter-sectoral economies. Hence, an introduction of production subsidy to urban producers improves the short-run welfare through the operation of the agglomeration economies, inter-sectoral economies and the reduction of the urban unemployment ratio.

3.2. Long-run analysis

In the foregoing analysis, we have analyzed the short-run consequences of urban developmental policies. For the long run, capital becomes perfectly mobile between the sectors. Two additional equations, (14) and (15), along with (S), (9), and (11) are needed for our comparative statics analysis. The results are derived as follows (see Appendix for derivations):

We can use (19) with the aid of (25) to study the long-run welfare implications of the urban developmental policies. It turns out that (22), (23) and (24) are still applicable for analyzing the effects of tariff, wage subsidy and production subsidy to the urban sector, but with a reversal in the sign of ah/ ap in the long run. The reason for Map > 0 now is as follows. An increase in p results in a rise in the rental rate in the urban sector and thus capital moves out of the rural sector into the urban sector. This leads to a fall in the marginal product of labor and hence a fall in w,. Consequently, workers migrate from the rural to the urban sector, and the urban unemployment ratio rises. Similarly, other urban developmental policies, such as the wage subsidy to urban workers, result in a rise in the urban unemployment ratio in the long run (ah/as, > 0).

As for the long-run welfare impact, it suffices to note that, from (22), (23) and (24), the policies for promoting urban growth may harm the economy as the inter-sectoral diseconomies and the unfavorable unemployment effect would be likely to dominate the agglomeration economies.

4. Concluding remarks

This paper has examined the short-run and long-run welfare implications of policies for promoting urban growth for LDCs. We have shown that the effects of tariffs, urban wage subsidies and urban production subsidies in the short run, in which capital is immobile, do not carry over to the long run with capital mobility. Policies for promoting urban growth reduce the urban


unemployment ratio in the short run, but, by contrast, raise the urban unemployment ratio in the long run. Along with agglomeration economies and inter-sectoral externality, the urban unemployment ratio plays a key role in determining social welfare.

We have made for analytical convenience a number of assumptions in this paper. The assumption of homogeneity of labor can be relaxed by recogniz- ing heterogeneity in the labor market. It may be fruitful to draw a distinction between skilled and unskilled labor in view of the fact that the unskilled are usually the target group of minimum wage laws. Legislation is more likely to be binding in smaller cities than it is in larger cities, because the market clearing wages in the smaller cities tend to be lower due to a lower cost of living. Consequently, the number of unemployed unskilled workers in the smaller cities will tend to grow. This will induce migration of unskilled labor to the larger cities.

Another assumption involves constant economies of agglomeration over city size. It is notable that the growth of larger cities in developing nations has been catastrophic. A detailed structural model of variable economies of agglomeration may be useful to explain the observed catastrophic and discontinuous growth pattern, though the mathematics for the analysis will be more complex.

Finally, there are three traditional policy options of a small open economy discussed in the present paper. Other policy options designed to either internalize or minimize the effects of externalities may also be considered. Further, the issues of industry location sought to reduce urban unemployment can be explored in a somewhat modified framework.

Acknowledgements

The authors are indebted to two anonymous referees for many useful comments. The authors, however, are solely responsible for any remaining shortcomings.

Appendix

(A) The short-run case

The comparative statics results can be obtained by totally differentiating and solving (8), (9), and (11) as:

dL,lap = -FLIpFLL > 0,

dL,ldp = -FLYL(l + h)iD ~0,

ahlap = FLYLL(l + h)‘lD ~0,

C.C. Chao, E.S.H. Yu I Reg. Sci. Urban Econ. 24 (1994) ~565-576 575

aL,l as, = -wJpfF& > 0,

iIL,las, = -(wJf)Y,(l + h)lD <O ,

dhlds, = (wX/f)Y,,(l + h)*/D < 0,

where

D =pF,,[L,Y,,(l + A) - Y,] > 0.

(B) The long-run case

Totally differentiating (8), (9), (ll), (14) and (15), we have

PF~ 0 0 (1 + A)Yu :I + A)YLK PfF,, - Y,, - YKK

(1+/i) 0 1 0

1 0 1 -FL dp - (wxlf) b.x

= :fF,dp I 1 0 0

The determinant of the coefficient of matrix is given by

A = -fp2AE +pL,F,,(l + A)B -pY,C ,

where A = FLLFKK - (FLK)‘, B = YLLYKK - (YLK)2, C = FLLYKK - (1 +

A)FLKYLK and E = Y, - L,Y,,( 1 + A) > 0. Since the production function F

and Y are linearly homogeneous in Li and K,, we obtain that A = B = 0 and C = - FLKYKK[kX - (1-t A)k,], w h ere ki = K, lL, (i = X, Y). Since the Neary (1981) stability condition requires that k, - (1 + A)k, > 0, the determinant of the coefficient is therefore negative.

By solving the system, we obtain:

aL,Jap = (pf(FLFm -FKFLK)E-LXFL(l+A)B+FLYLYKK}/A>O,

aK,Jap = {pf(FKFu -FLFKL)E-FLYLYKL(l+A)}lA>O,

aL,lap = (1 + A){pfL,Y,,(F,F,, - FKFu) - F,Y,(pfFm + Ym)

+pfF,yY,F,,)fA <O,

sm = -(pf(FKFLL - FLFKL)E - FLYLYKL(l + A)}/A < 0,

aA\lap = (1 + A){F,(l + A)B

- Pfy,,(F,F,, - FLFKK)tkx - (I+ A)k,lIlA >O , dL,/as, = (w,/f){psy,F,, + YLYKK - L,(l + A)B

576 C.C. Chao, E.S.H. Yu I Reg. Sci. Urban Econ. 24 (1994) 565576

aK,/as,=-(wx/f){pfF,,E+Y,Y,,(l+A)}/A>O,

aWas, = (w,V)(l + A)bW,~,,Y,, -~.f’Y~h - Y,Y,,>/A<O >

aK,/as, = (WX/f){pfFKLE + YLYKL(l + A)}/A < 0 )

ah/as, = (~,if)(l + ~){(l + h)B +ppKKGLK[kX - (1 + h)k,])ld > 0.

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Urban growth, externality and welfare