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On the existence, uniqueness and stability of spatial equilibrium in an open city with externalities
JOURNAL OF URBAN ECONOMICS 8, 13% 149 (1980)
On the Existence, Uniqueness and Stability of Spatial Equilibrium in an Open City with Externalities TAKAHIRO MIYAO, PERRY SHAPIRO, AND DAVID KNAPP Unicersi~ of Southern Califoornia.Las Angeles, Calijbmia 9GiM7,and Unicersi@ of Calflomnia,Santa Barbara, Califomii 93106
Received May 5, 1978; revised October 16, 1978 This paper deals with the existence, uniqueness and stability of a spatial equilibrium in an open city with external diseconomles like air pollution generated by manufacturing activities. First, assuming that both production functions and utility functions are Cobb-Douglas, we prove the existence of a spatial equilibrium under some reasonable assumptions. It is shown, however, that the uniqueness and stability of the equilibrium may not be obtained, unless the degree of externality is sufficiently small. In fact, none of the equilibria will be stable for a certain set of speeds of adjustment in the labor market and the land market, if the degree of externality is relatively large. Finally, some implications of our results are derived in regard to the application of the hedonic price concept within the context of a spatial equilibrium model.
1. INTRODUCTION In the recent literature on urban economics, spatial equilibrium models with externalities like air pollution have been studied in relation to zoning of hedonic and taxation policies,’ as well as the proper interpretation prices. * Those studies have implicitly assumed the existence, uniqueness and stability of a spatial equilibrium and have failed to show under what conditions such desirable properties of the equilibrium are obtained. As suggested by Starrett (1972) there is reason to believe that in the presence of external diseconomies like air pollution, we may not ensure the existence, uniqueness or stability of an equilibrium under usual assumptions. In this paper, we set up an open city model with external diseconomies, assuming that both production functions and utility functions are CobbDouglas and that the utility level of households is adversely affected by certain by-products of manufacturing activities, e.g., noise, air pollution, etc. It is shown that although a spatial equilibrium always exists under reasonable assumptions, its uniqueness and stability may not be ensured unless the degree of externality is relatively small. In fact, none of the ‘See Henderson [l] and Stull [7] for example. %ce Polinsky and Shavell[3] for example. 139 0094-l 190/80/050139-11So2.00/0 copyl+ght015%0by-cRgh1ac. Au Ii&U
of mpmduction
ill any fcmm -d.
140
MIYAO, SHAPIRO, AND KNAPP
equilibria will be stable for a certain set of speedsof dynamic adjustment, if the degree of externality is relatively large. 2. THE MODEL Let us assumea linear city with a market place at one end. Manufacturing activities take place in the “inner” segment between the market and the “inner” boundary at distance s* from the market. On the other hand, households are located in the “outer” segment between the inner boundary and the “outer” boundary at distance s** from the market. All products must be transported to the market where they are sold at an exogenously given price, and also households must travel to the market in order to sell their labor services and to buy consumption goods. Transport cost per unit of product and commuting cost per household are assumed to be sole functions of distance. The city considered here is open in that both the utility level and product price are given exogenously from outside of the city, whereas the wage rate is determined endogenously within the city. All firms have identical Cobb-Douglas production functions Y(s) = H(s)aL(S)‘-a,
O
for a typical firm at distance s from the market, where Y(s), H(s) and L(S) are output, land and labor, respectively, at distance s. Under perfect competition we have the zero profit condition, i.e., product price net of transport cost be equal to unit production cost P - V(s) = AR(S)aW’-a,
(1)
where P and w are the product price and the wage rate, respectively, given exogenously, V(S) and R(s) are transport cost and manufacturing bid rent, respectively, at distance S, and A is some positive constant. From (1) the manufacturing bid rent function R(s) can be obtained as R(s) = B(P - v(/(s))“aw-(‘-a’)/~,
(2)
where B is a positive constant. All households have identical utility functions, depending on residential land h, a consumption good q, and the amount of pollution 2, which they receive at their location S.
U(s) = U(h(s), q(s), Z(s)), where the partial derivatives of U with respect to h, q and Z are
u, > 0,
uq> 0,
u, < 0.
(3)
141
OPEN CITY WITH EXTERNALJTIES
Note that distance does not directly enter the utility function, as we assume away the disutility (or utility) of travel for the sake of simplicity. It may be reasonable to suppose that the amount of pollution received by a household is increasing with total manufacturing output X, but is decreasing with the distance from the inner boundary s* to the location of the household s.
z, > 0,
Z(s)= z(x,t),
z, < 0,
where t E s - s*. Combining this with (3), we can write U(s) = Wh(s),q(s),X,
t),
where U, < 0 and U, > 0.3 Let us take up the Cobb-Douglas case with U(s) = h(s)“q(s)‘-“(X0
+ x)-“(to
+ t)‘,
O
l,b>O,c>O, (4
where X0 and to are positive constants.4 With the production function being also Cobb- Douglas, we find total manufacturing output as
xc
s*-Y(s) * J0 H(s)
=[‘( !$)‘-‘.
Since profit maximization leads to Lw/(HR) obtain xc
,( 0
= (1 - a)/a for all s, we
S* 1 - a -R(s) 1lmadr = Cw-(~-a)/a I S’cp _ v(s)yl-4/a~ a
W
0
9 (5)
in view of (2), where C is a positive constant. Considering the budget constraint’ r(s)h(s)
+ Pq(s)
= W - u(s),
‘This treatment of externality is slightly more general than Stull[7] which neglects the total amount of pollution produced by manufacturing activities, but is slightly more special than Henderson [ 11which allows substitution between polluting and nonpolluting activities. ‘This assumption is ncctssary to keep the utility at a finite positive level when X - 0 or t = 0. It is reasonable to have a finite utility level even if there is no pollution, and also to have a positive utility level at the worst location, i.e., at the inner boundary. ‘The definition of net income as w - u(s) may be justified by assuming that each household takes a trip to the central market to sell labor services and to buy consumption goods. For the same assumption in a similar model, see Solow (51.
142
MIYAO, SHAPIRO, AND KNAPP
we can write the indirect utility function as u = Dr(s)-(w
- u(s))(X,
+ x)-“(to
+ t)‘,
(6)
where D is a positive constant and u is an exogenously given utility level which is common for all households. From (6) we can determine the residential bid rent function as
r(s) = Eu+yw - u(s))““(X, + x)-“‘yt,
+ p,
(7)
where E is a positive constant. In the labor market, the excess demand for labor should be zero in equilibrium
For the land market to be in equilibrium, the overall rent function must be continuous at both the inner boundary and the outer boundary R(s*) - r(P) = 0,
(9)
r(s**)
(10)
= 0.
Note that for the sake of simplicity the opportunity cost of land is assumed to be zero at the outer boundary, as seen in (1O).6 Furthermore, in equilibrium the manufacturing bid rent should be higher (lower, resp.) than the residential bid rent at every point in the inner (outer, resp.) segment of the city: R(s) > r(s) R(s) < 4s)
for alls < s* foralls*
(11) (14
These conditions ensure that manufacturing activities will take place in the inner segment while residential activities in the outer segment, as assumed at the outset.’ ‘jIt can be asked whether r’(s) might be positive around s** so that (10) could not determine the outer boundary. It turns out, however, that if u’(s) is greater than some positive constant for all s (which will be assumed later), we have r’(s) = 0 at s - s** and r’(s) > 0 - u(s))“-“‘/“(x, + for s(> s**) sufficiently close to s**, because r’(s) = (l/a)Eu--““(W X)-*/yr, + f)(-qC(W - u(s)) - u’(s)(to + r)]. ‘For a more general formulation of this type of model and further details concerning derivations using indirect utility functions, see Polinsky and Rubinfeld [4].
143
OPEN CITY WITH EXTERNALITIES
3. EXISTENCE In order to prove the existence of a spatial equilibrium, we shall assume that V(0) 2 0,
u(0) 2 0;
V’(s) 2 e > 0,
u’(s) 2 e > 0, (13)
where e is a positive constant. First, we find from (7) and (10) that w - u(s**) = 0, or
s** = u-‘(w)
(14
with dr**/dw = l/u’(s**)
> 0,
(15)
and s**+Oasw+u(O)
and
2 0;
s**+
cc asw+u(cc)
= co. (16)
Next, denoting the excess demand for labor in (8) by F and using (2), (5), (7) and (14), we obtain
F(w,s*) =
"'$2 s0
Ai+
= Jw-yy'(P
- 1:":
&
- V(s))""dS
1 u-yw -u(S))(‘-wyfo +s-S*)c/%, Is*
_ K x0 + cw-(l-aVa [ X
w ""1,,
$*(p - qs))('-")/"& s0
-b'a
(17)
where J and K are positive constants. By partially differentiating (17) with respect to w and s*, it is easy to see that
aF/aw< 0
and
aF/W > 0,
(18)
and therefore ~/~*l,dJ
= - (aF/as*)/ (aFlaw) > 0,
(19)
which means that in the (s*, w) plane the curve representing F(w, s*) = 0 is continuous and upward-sloping as illustrated in Fig. 1. We can also
144
MIYAO, SHAF’IRO, AND IWAFT
FIOURE1
show the following: As s* + 0, we must have s**( = o-‘(w)) + 0 to satisfy
F = 0, because it would follow from (17) that F < 0 if s** > 0 = s*. Therefore, u-r(w) + 0, or w + u(0) 2 0
as
P-0.
(20)
Assuming that
p > W),
(21)
we can find a positive number, say S > 0, such that
P - V(S)= 0,
(22)
in view of (13). Then, in order to satisfy F = 0, we should have s** > s* as s* +5, because (17) would lead to F > 0 if s** = s* = 5 > 0. That is, s** = u-‘(w) > s* = 5, or w > u(S)
as
s*+s.
(23)
Considering (20) and (23) in addition to (19), we can illustrate the curve representing F(w, s*) = 0 as in Fig. 1. Turning to (9) and denoting the rent differential R(s*) - r(P) by G, we obtain from (2), (5) and (7)
G(w,s*) = B(p - ~(I(s*))‘/~w-(‘-)/~ - Eu-‘/“(w - u(s*))“”
_@))(‘-“)/a& )(1x0+c,-U-)/a ss*(p 1-b’ut;,.* 0
(24)
145
OPEN CITY WITH EXTERNALITIES
We can readily see that the curve representing G(w, 9) = 0 is continuous for 0 < s* < 5 and w > u(O), since hv/dr*Ic,e = - (aG/%*)/(aG/Elw) and ClG/aw # 0. Furthermore, when s* = 0, we have G(w, 0) = 0, yielding W(‘-+yW
- u(()))‘/a = BE-'u""(P
- V(0))""X,f~"r~c~o > 0. (25)
This implies that as
w > u(0)
s* + 0.
(26)
On the other hand, when s* = S; we have G(w, S) = 0, giving &yw
- u(j))‘/a
j(p - v(s))(‘-4/y& /0
x0 + c,-U-)/a [
1
-b’atp
= 0. (27)
That is, w+u(S)
as
s*+s.
(28)
Thus, in the (s*, w) plane the curve representing G(w, s*) = 0 is located above the curve representing F(w, s*) = 0 when s* = 0, and the former is below the latter when s* = S. As a result, the two curves must intersect at least once, as illustrated in Fig. 1. This proves the existence of a set of values, w > u(0) and 0 < s* < S, satisfying 8’(w, s*) = 0 and G(w, s*) = 0 simultaneously. Finally, conditions (11) and (12) will be ensured by assuming that the manufacturing bid rent curve is more elastic than the residential bid rent curve at every location s s -~-r’(s) -~-R’(s) s s R(s) > s 4s)
for all s .
(29)
In view of (3) and (7), condition (29) is equivalent to the following: 1
V(s)
a P - V(s)
-- 1
u’(s)
a w - u(s)
+2a
1 t, + s -
s*
>o
for all s. (30)
This is a modified version of von Thiinen’s well-known condition, stating that the ratio of marginal transport cost to land for the manufacturing sector should be greater than that for the residential sector minus a certain number which is associated with externalities.8 In summary, we have 8Herc we implicitly assume that t0 + s - s* > 0 for 0 < s < s**, so that the third term in (30) is always positive. Note that (30) is weaker than the usual von Thbn condition which involves only the first two terms in (30).
146
MIYAO, SHAPIRO, AND KNAPP
proved the following proposition: PROPOSITION 1. Under the assumptions (13), (21) and (30) in the present Cobb-Douglas case with zero opportunity cost of land, there exists a spatial equilibrium, i.e., a set of values w > v(O), 0 < s* < S ands** > s*, satisfuing all the equilibrium conditions (8)- (12).
4. UNIQUENESS AND STABILITY We are now in a position to examine the uniqueness property of the equilibrium in our model. As will be shown below, the assumptions which have been made so far are not enough to ensure the uniqueness, and we need an additional assumption whose validity depends on the degree of external diseconomy. First, it is straightforward to see from (24) aG/i3w < 0.
(31)
However, from (24) we have aG as*=
-- R(P) a
V(s*) + r(s*) P - V(s*) a
+ br(s*) a
(P - V(S*))(‘-~)‘~ x, + x
v’(s*) w - v(s*) 2 o >
9
(32)
according as
v’(s*)
MEL
-- 1
a P - v(s*)
v’(s*)
a w - v(s*)
-- b (P - V(s*))(‘-a)‘u x0 + x a
1 o < *
If the expression M in (33) is non-negative, then aG/as*
I 0,
(34)
and therefore in view of (31)
h/dr*l,,,
= - (aG/as*)/
(aG/aw)
I 0,
(35)
meaning that in the (s*, w) plane the curve representing G( w, s*) = 0 has a non-positive slope everywhere. This ensures the uniqueness of the equilibrium as seen in Fig. 2. PROPOSITION 2.
The equilibrium whose existence is ensured in Proposition
1 is unique, .if the expression A4 in (33) is non-negative for al/possible values ofw>v(O)andO
OPEN CITY
WITH
147
EXTERNALITIES
0
> 5’
-s
FIGURE 2
Note that the additional assumption M 2 0 may be violated if the degree of external diseconomy, b, is sufficiently large relative to the elasticity of the utility function with respect to residential land, a.’ Next, we shall investigate the stability property of the equilibrium by assuming a dynamic adjustment process of the wage rate and the inner boundary. It turns out that we need exactly the same additional assumption as in Proposition 2 in order to ensure the dynamic stability of the equilibrium for any positive speedsof adjustment. In particular, we assume that the wage rate is adjusted in response to the excessdemand for labor as I+ = +[ F(w,s*)]
with
+‘( ) > 0
and
$48 = 0, (36)
where the dot denotes differentiation with respect to time. On the other hand, the inner boundary is adjusted in response to the rent differential at that boundary asi’ S* = I//[ G(w,s*)]
with
a 1> 0
and
l)(O) = 0. (37)
Here we shall focus on local stability in a small neighborhood of the equilibrium (G, i*) by linearizing the system (36) and (37): r#J’(O)aF/aw c#a’(o)aF/as* $‘(O)aG/tIw 1+5’(0)tIG/b*
I[ I W-8
s*
-
s^*
’
(38)
Considering (18) (3 l), (32) and (33), we can show that the equilibrium is locally stable for any positive speeds of adjustment r+‘(O)and 4’(O), if the 91t should also be noted that (30) implies M > 0, leading to uniqueness, if there is no externality, i.e., b = c = 0. ‘OFor this kind of boundary adjustment, see Miyao [2].
148
MIYAO, SHAPIRO, AND KNAPP
expression M in (33) is non-negative; because in that case the trace of the matrix in (38) is negative, i.e., c#‘(O)i3F/~w + pY(O)tIG/b* < 0,
(39)
and the determinant is positive, i.e., +‘(O)$‘(O)[ (tW/aw)(
aG/as*)
- (i3G/aw)( W/b*)]
> 0.
(40)
PROPOSITION 3. The equilibrium whoseexistence is proved in Proposition 1 is locally stable for any positive speerisof adjustment +‘(O) and q’(O), if the expressionM in (33) is non-negative at the equilibrium.
Finally, it should be noted that if the degree of external diseconomy, 6, is so large as to give M < 0, for any possible values of w and s*, then there will exist a set of positive speedsof adjustment cp’(0)and #‘(O) (the latter is sufficiently larger than the former) such that the trace becomes positive, i.e., #(O)W/8w
+ \Cl’(O)aG/ih* > 0,
(41)
for all possible equilibria. This means the following: PROPOSITION 4. Zf the expressionM is negativefor allpossible values of w and s*, then there exist a set of positive speed of adjustmentfor which none of the equilibria will be loca& stable.
The results obtained above have particular significance for the applied work on hedonic prices. As is well known, the hedonic price approach to evaluating environmental quality is to discover the functional relationship between housing prices and externalities. It has been shown by Polinsky and Shave11(1976) that hedonic prices should properly be interpreted only within the context of a spatial equilibrium model. The disturbing feature of our results is the implication that the hedonic price may not be unique and may not even be observed, as the uniqueness and stability of the underlying spatial equilibrium may not be ensured without some prior restrictions on the degree of externality. Although this fact does not deny the applicability of the hedonic price approach, one must be careful in using it for making a policy argument in the context of spatial equilibrium with external diseconomies. ACKNOWLEDGMENTS The authors wish to thank the editor and a referee for helpful comments and suggestions.
OPEN CITY WITH EXTERNALITIES
149
REFERENCES 1. J. V. Henderson, Externalities in a spatial context, J. Public Econ., 7, 89-110 (1977). 2. T. Miyao, Dynamics and comparative statics in the theory of residential location, J. Icon Tbeofy, 11, 133- 146 (1975). 3. A. M. Polinsky and S. Shavell, Amenities and property values in a model of an urban area, J. Public Econ, 5, 119- 129 (1976). 4. A. M. Polinsky and D. L. Rubinfeld, The long-run effects of a residential property tax and local public services, J. Urban &on., 5, 241-262 (1978). 5. R. M. Solow, On equilibrium models of urban location, in “Essays in Modem Economics” (M. Parkin, Ed.), Longman, London (1973). 6. D. A. Starrett, Fundamental nonconvexities in the theory of externalities, J. Econ Zheoty, 4, 180- 199 (1972). 7. W. J. Stull, Land use and zoning in an urban economy, Amer. Econ Rev., 64, 337-347 (1974).
On the Existence, Uniqueness and Stability of Spatial Equilibrium in an Open City with Externalities TAKAHIRO MIYAO, PERRY SHAPIRO, AND DAVID KNAPP Unicersi~ of Southern Califoornia.Las Angeles, Calijbmia 9GiM7,and Unicersi@ of Calflomnia,Santa Barbara, Califomii 93106
Received May 5, 1978; revised October 16, 1978 This paper deals with the existence, uniqueness and stability of a spatial equilibrium in an open city with external diseconomles like air pollution generated by manufacturing activities. First, assuming that both production functions and utility functions are Cobb-Douglas, we prove the existence of a spatial equilibrium under some reasonable assumptions. It is shown, however, that the uniqueness and stability of the equilibrium may not be obtained, unless the degree of externality is sufficiently small. In fact, none of the equilibria will be stable for a certain set of speeds of adjustment in the labor market and the land market, if the degree of externality is relatively large. Finally, some implications of our results are derived in regard to the application of the hedonic price concept within the context of a spatial equilibrium model.
1. INTRODUCTION In the recent literature on urban economics, spatial equilibrium models with externalities like air pollution have been studied in relation to zoning of hedonic and taxation policies,’ as well as the proper interpretation prices. * Those studies have implicitly assumed the existence, uniqueness and stability of a spatial equilibrium and have failed to show under what conditions such desirable properties of the equilibrium are obtained. As suggested by Starrett (1972) there is reason to believe that in the presence of external diseconomies like air pollution, we may not ensure the existence, uniqueness or stability of an equilibrium under usual assumptions. In this paper, we set up an open city model with external diseconomies, assuming that both production functions and utility functions are CobbDouglas and that the utility level of households is adversely affected by certain by-products of manufacturing activities, e.g., noise, air pollution, etc. It is shown that although a spatial equilibrium always exists under reasonable assumptions, its uniqueness and stability may not be ensured unless the degree of externality is relatively small. In fact, none of the ‘See Henderson [l] and Stull [7] for example. %ce Polinsky and Shavell[3] for example. 139 0094-l 190/80/050139-11So2.00/0 copyl+ght015%0by-cRgh1ac. Au Ii&U
of mpmduction
ill any fcmm -d.
140
MIYAO, SHAPIRO, AND KNAPP
equilibria will be stable for a certain set of speedsof dynamic adjustment, if the degree of externality is relatively large. 2. THE MODEL Let us assumea linear city with a market place at one end. Manufacturing activities take place in the “inner” segment between the market and the “inner” boundary at distance s* from the market. On the other hand, households are located in the “outer” segment between the inner boundary and the “outer” boundary at distance s** from the market. All products must be transported to the market where they are sold at an exogenously given price, and also households must travel to the market in order to sell their labor services and to buy consumption goods. Transport cost per unit of product and commuting cost per household are assumed to be sole functions of distance. The city considered here is open in that both the utility level and product price are given exogenously from outside of the city, whereas the wage rate is determined endogenously within the city. All firms have identical Cobb-Douglas production functions Y(s) = H(s)aL(S)‘-a,
O
for a typical firm at distance s from the market, where Y(s), H(s) and L(S) are output, land and labor, respectively, at distance s. Under perfect competition we have the zero profit condition, i.e., product price net of transport cost be equal to unit production cost P - V(s) = AR(S)aW’-a,
(1)
where P and w are the product price and the wage rate, respectively, given exogenously, V(S) and R(s) are transport cost and manufacturing bid rent, respectively, at distance S, and A is some positive constant. From (1) the manufacturing bid rent function R(s) can be obtained as R(s) = B(P - v(/(s))“aw-(‘-a’)/~,
(2)
where B is a positive constant. All households have identical utility functions, depending on residential land h, a consumption good q, and the amount of pollution 2, which they receive at their location S.
U(s) = U(h(s), q(s), Z(s)), where the partial derivatives of U with respect to h, q and Z are
u, > 0,
uq> 0,
u, < 0.
(3)
141
OPEN CITY WITH EXTERNALJTIES
Note that distance does not directly enter the utility function, as we assume away the disutility (or utility) of travel for the sake of simplicity. It may be reasonable to suppose that the amount of pollution received by a household is increasing with total manufacturing output X, but is decreasing with the distance from the inner boundary s* to the location of the household s.
z, > 0,
Z(s)= z(x,t),
z, < 0,
where t E s - s*. Combining this with (3), we can write U(s) = Wh(s),q(s),X,
t),
where U, < 0 and U, > 0.3 Let us take up the Cobb-Douglas case with U(s) = h(s)“q(s)‘-“(X0
+ x)-“(to
+ t)‘,
O
l,b>O,c>O, (4
where X0 and to are positive constants.4 With the production function being also Cobb- Douglas, we find total manufacturing output as
xc
s*-Y(s) * J0 H(s)
=[‘( !$)‘-‘.
Since profit maximization leads to Lw/(HR) obtain xc
,( 0
= (1 - a)/a for all s, we
S* 1 - a -R(s) 1lmadr = Cw-(~-a)/a I S’cp _ v(s)yl-4/a~ a
W
0
9 (5)
in view of (2), where C is a positive constant. Considering the budget constraint’ r(s)h(s)
+ Pq(s)
= W - u(s),
‘This treatment of externality is slightly more general than Stull[7] which neglects the total amount of pollution produced by manufacturing activities, but is slightly more special than Henderson [ 11which allows substitution between polluting and nonpolluting activities. ‘This assumption is ncctssary to keep the utility at a finite positive level when X - 0 or t = 0. It is reasonable to have a finite utility level even if there is no pollution, and also to have a positive utility level at the worst location, i.e., at the inner boundary. ‘The definition of net income as w - u(s) may be justified by assuming that each household takes a trip to the central market to sell labor services and to buy consumption goods. For the same assumption in a similar model, see Solow (51.
142
MIYAO, SHAPIRO, AND KNAPP
we can write the indirect utility function as u = Dr(s)-(w
- u(s))(X,
+ x)-“(to
+ t)‘,
(6)
where D is a positive constant and u is an exogenously given utility level which is common for all households. From (6) we can determine the residential bid rent function as
r(s) = Eu+yw - u(s))““(X, + x)-“‘yt,
+ p,
(7)
where E is a positive constant. In the labor market, the excess demand for labor should be zero in equilibrium
For the land market to be in equilibrium, the overall rent function must be continuous at both the inner boundary and the outer boundary R(s*) - r(P) = 0,
(9)
r(s**)
(10)
= 0.
Note that for the sake of simplicity the opportunity cost of land is assumed to be zero at the outer boundary, as seen in (1O).6 Furthermore, in equilibrium the manufacturing bid rent should be higher (lower, resp.) than the residential bid rent at every point in the inner (outer, resp.) segment of the city: R(s) > r(s) R(s) < 4s)
for alls < s* foralls*
(11) (14
These conditions ensure that manufacturing activities will take place in the inner segment while residential activities in the outer segment, as assumed at the outset.’ ‘jIt can be asked whether r’(s) might be positive around s** so that (10) could not determine the outer boundary. It turns out, however, that if u’(s) is greater than some positive constant for all s (which will be assumed later), we have r’(s) = 0 at s - s** and r’(s) > 0 - u(s))“-“‘/“(x, + for s(> s**) sufficiently close to s**, because r’(s) = (l/a)Eu--““(W X)-*/yr, + f)(-qC(W - u(s)) - u’(s)(to + r)]. ‘For a more general formulation of this type of model and further details concerning derivations using indirect utility functions, see Polinsky and Rubinfeld [4].
143
OPEN CITY WITH EXTERNALITIES
3. EXISTENCE In order to prove the existence of a spatial equilibrium, we shall assume that V(0) 2 0,
u(0) 2 0;
V’(s) 2 e > 0,
u’(s) 2 e > 0, (13)
where e is a positive constant. First, we find from (7) and (10) that w - u(s**) = 0, or
s** = u-‘(w)
(14
with dr**/dw = l/u’(s**)
> 0,
(15)
and s**+Oasw+u(O)
and
2 0;
s**+
cc asw+u(cc)
= co. (16)
Next, denoting the excess demand for labor in (8) by F and using (2), (5), (7) and (14), we obtain
F(w,s*) =
"'$2 s0
Ai+
= Jw-yy'(P
- 1:":
&
- V(s))""dS
1 u-yw -u(S))(‘-wyfo +s-S*)c/%, Is*
_ K x0 + cw-(l-aVa [ X
w ""1,,
$*(p - qs))('-")/"& s0
-b'a
(17)
where J and K are positive constants. By partially differentiating (17) with respect to w and s*, it is easy to see that
aF/aw< 0
and
aF/W > 0,
(18)
and therefore ~/~*l,dJ
= - (aF/as*)/ (aFlaw) > 0,
(19)
which means that in the (s*, w) plane the curve representing F(w, s*) = 0 is continuous and upward-sloping as illustrated in Fig. 1. We can also
144
MIYAO, SHAF’IRO, AND IWAFT
FIOURE1
show the following: As s* + 0, we must have s**( = o-‘(w)) + 0 to satisfy
F = 0, because it would follow from (17) that F < 0 if s** > 0 = s*. Therefore, u-r(w) + 0, or w + u(0) 2 0
as
P-0.
(20)
Assuming that
p > W),
(21)
we can find a positive number, say S > 0, such that
P - V(S)= 0,
(22)
in view of (13). Then, in order to satisfy F = 0, we should have s** > s* as s* +5, because (17) would lead to F > 0 if s** = s* = 5 > 0. That is, s** = u-‘(w) > s* = 5, or w > u(S)
as
s*+s.
(23)
Considering (20) and (23) in addition to (19), we can illustrate the curve representing F(w, s*) = 0 as in Fig. 1. Turning to (9) and denoting the rent differential R(s*) - r(P) by G, we obtain from (2), (5) and (7)
G(w,s*) = B(p - ~(I(s*))‘/~w-(‘-)/~ - Eu-‘/“(w - u(s*))“”
_@))(‘-“)/a& )(1x0+c,-U-)/a ss*(p 1-b’ut;,.* 0
(24)
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OPEN CITY WITH EXTERNALITIES
We can readily see that the curve representing G(w, 9) = 0 is continuous for 0 < s* < 5 and w > u(O), since hv/dr*Ic,e = - (aG/%*)/(aG/Elw) and ClG/aw # 0. Furthermore, when s* = 0, we have G(w, 0) = 0, yielding W(‘-+yW
- u(()))‘/a = BE-'u""(P
- V(0))""X,f~"r~c~o > 0. (25)
This implies that as
w > u(0)
s* + 0.
(26)
On the other hand, when s* = S; we have G(w, S) = 0, giving &yw
- u(j))‘/a
j(p - v(s))(‘-4/y& /0
x0 + c,-U-)/a [
1
-b’atp
= 0. (27)
That is, w+u(S)
as
s*+s.
(28)
Thus, in the (s*, w) plane the curve representing G(w, s*) = 0 is located above the curve representing F(w, s*) = 0 when s* = 0, and the former is below the latter when s* = S. As a result, the two curves must intersect at least once, as illustrated in Fig. 1. This proves the existence of a set of values, w > u(0) and 0 < s* < S, satisfying 8’(w, s*) = 0 and G(w, s*) = 0 simultaneously. Finally, conditions (11) and (12) will be ensured by assuming that the manufacturing bid rent curve is more elastic than the residential bid rent curve at every location s s -~-r’(s) -~-R’(s) s s R(s) > s 4s)
for all s .
(29)
In view of (3) and (7), condition (29) is equivalent to the following: 1
V(s)
a P - V(s)
-- 1
u’(s)
a w - u(s)
+2a
1 t, + s -
s*
>o
for all s. (30)
This is a modified version of von Thiinen’s well-known condition, stating that the ratio of marginal transport cost to land for the manufacturing sector should be greater than that for the residential sector minus a certain number which is associated with externalities.8 In summary, we have 8Herc we implicitly assume that t0 + s - s* > 0 for 0 < s < s**, so that the third term in (30) is always positive. Note that (30) is weaker than the usual von Thbn condition which involves only the first two terms in (30).
146
MIYAO, SHAPIRO, AND KNAPP
proved the following proposition: PROPOSITION 1. Under the assumptions (13), (21) and (30) in the present Cobb-Douglas case with zero opportunity cost of land, there exists a spatial equilibrium, i.e., a set of values w > v(O), 0 < s* < S ands** > s*, satisfuing all the equilibrium conditions (8)- (12).
4. UNIQUENESS AND STABILITY We are now in a position to examine the uniqueness property of the equilibrium in our model. As will be shown below, the assumptions which have been made so far are not enough to ensure the uniqueness, and we need an additional assumption whose validity depends on the degree of external diseconomy. First, it is straightforward to see from (24) aG/i3w < 0.
(31)
However, from (24) we have aG as*=
-- R(P) a
V(s*) + r(s*) P - V(s*) a
+ br(s*) a
(P - V(S*))(‘-~)‘~ x, + x
v’(s*) w - v(s*) 2 o >
9
(32)
according as
v’(s*)
MEL
-- 1
a P - v(s*)
v’(s*)
a w - v(s*)
-- b (P - V(s*))(‘-a)‘u x0 + x a
1 o < *
If the expression M in (33) is non-negative, then aG/as*
I 0,
(34)
and therefore in view of (31)
h/dr*l,,,
= - (aG/as*)/
(aG/aw)
I 0,
(35)
meaning that in the (s*, w) plane the curve representing G( w, s*) = 0 has a non-positive slope everywhere. This ensures the uniqueness of the equilibrium as seen in Fig. 2. PROPOSITION 2.
The equilibrium whose existence is ensured in Proposition
1 is unique, .if the expression A4 in (33) is non-negative for al/possible values ofw>v(O)andO
OPEN CITY
WITH
147
EXTERNALITIES
0
> 5’
-s
FIGURE 2
Note that the additional assumption M 2 0 may be violated if the degree of external diseconomy, b, is sufficiently large relative to the elasticity of the utility function with respect to residential land, a.’ Next, we shall investigate the stability property of the equilibrium by assuming a dynamic adjustment process of the wage rate and the inner boundary. It turns out that we need exactly the same additional assumption as in Proposition 2 in order to ensure the dynamic stability of the equilibrium for any positive speedsof adjustment. In particular, we assume that the wage rate is adjusted in response to the excessdemand for labor as I+ = +[ F(w,s*)]
with
+‘( ) > 0
and
$48 = 0, (36)
where the dot denotes differentiation with respect to time. On the other hand, the inner boundary is adjusted in response to the rent differential at that boundary asi’ S* = I//[ G(w,s*)]
with
a 1> 0
and
l)(O) = 0. (37)
Here we shall focus on local stability in a small neighborhood of the equilibrium (G, i*) by linearizing the system (36) and (37): r#J’(O)aF/aw c#a’(o)aF/as* $‘(O)aG/tIw 1+5’(0)tIG/b*
I[ I W-8
s*
-
s^*
’
(38)
Considering (18) (3 l), (32) and (33), we can show that the equilibrium is locally stable for any positive speeds of adjustment r+‘(O)and 4’(O), if the 91t should also be noted that (30) implies M > 0, leading to uniqueness, if there is no externality, i.e., b = c = 0. ‘OFor this kind of boundary adjustment, see Miyao [2].
148
MIYAO, SHAPIRO, AND KNAPP
expression M in (33) is non-negative; because in that case the trace of the matrix in (38) is negative, i.e., c#‘(O)i3F/~w + pY(O)tIG/b* < 0,
(39)
and the determinant is positive, i.e., +‘(O)$‘(O)[ (tW/aw)(
aG/as*)
- (i3G/aw)( W/b*)]
> 0.
(40)
PROPOSITION 3. The equilibrium whoseexistence is proved in Proposition 1 is locally stable for any positive speerisof adjustment +‘(O) and q’(O), if the expressionM in (33) is non-negative at the equilibrium.
Finally, it should be noted that if the degree of external diseconomy, 6, is so large as to give M < 0, for any possible values of w and s*, then there will exist a set of positive speedsof adjustment cp’(0)and #‘(O) (the latter is sufficiently larger than the former) such that the trace becomes positive, i.e., #(O)W/8w
+ \Cl’(O)aG/ih* > 0,
(41)
for all possible equilibria. This means the following: PROPOSITION 4. Zf the expressionM is negativefor allpossible values of w and s*, then there exist a set of positive speed of adjustmentfor which none of the equilibria will be loca& stable.
The results obtained above have particular significance for the applied work on hedonic prices. As is well known, the hedonic price approach to evaluating environmental quality is to discover the functional relationship between housing prices and externalities. It has been shown by Polinsky and Shave11(1976) that hedonic prices should properly be interpreted only within the context of a spatial equilibrium model. The disturbing feature of our results is the implication that the hedonic price may not be unique and may not even be observed, as the uniqueness and stability of the underlying spatial equilibrium may not be ensured without some prior restrictions on the degree of externality. Although this fact does not deny the applicability of the hedonic price approach, one must be careful in using it for making a policy argument in the context of spatial equilibrium with external diseconomies. ACKNOWLEDGMENTS The authors wish to thank the editor and a referee for helpful comments and suggestions.
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149
REFERENCES 1. J. V. Henderson, Externalities in a spatial context, J. Public Econ., 7, 89-110 (1977). 2. T. Miyao, Dynamics and comparative statics in the theory of residential location, J. Icon Tbeofy, 11, 133- 146 (1975). 3. A. M. Polinsky and S. Shavell, Amenities and property values in a model of an urban area, J. Public Econ, 5, 119- 129 (1976). 4. A. M. Polinsky and D. L. Rubinfeld, The long-run effects of a residential property tax and local public services, J. Urban &on., 5, 241-262 (1978). 5. R. M. Solow, On equilibrium models of urban location, in “Essays in Modem Economics” (M. Parkin, Ed.), Longman, London (1973). 6. D. A. Starrett, Fundamental nonconvexities in the theory of externalities, J. Econ Zheoty, 4, 180- 199 (1972). 7. W. J. Stull, Land use and zoning in an urban economy, Amer. Econ Rev., 64, 337-347 (1974).