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Land use and zoning in the central business district
Regional
Science
and Urban
LAND
Economics
USE AND
14 (1984)
521-532.
North-Holland
ZONING IN THE CENTRAL DISTRICT*
BUSINESS
Arthur M. SULLIVAN Graduate
School
of Administration,
Received
January
University 1984, final
of California,
version
received
Davis, March
CA 95616,
USA
1984
A model of a small, open central business district (CBD) is used to derive the conditions that define the market-equilibrium and rent-maximizing CBD. It is shown that, in general, the market equilibrium CBD radius differs from the rent-maximizing CBD radius. Land-use zoning will, under certain conditions, increase the aggregate net return on CBD land.
1. Introduction
This paper considers the allocation of land in the production area of a small, open city. The market-equilibrium central business district (CBD) is compared to the rent-maximizing CBD. It is shown that the marketequilibrium CBD radius will not, in general, correspond to the radius that maximizes the aggregate net return on CBD land. A land-use zoning policy that alters the CBD radius will, under certain circumstances, increase the aggregate net return on CBD land. Several researchers have analyzed the effects of industrial land-use zoning. The analyses focus on the effects of industrial zoning in the presence of pollution externalities. See, for example, Stull (1974), Ohls et al. (1974), and Mills (1979). In this paper, the effects of industrial zoning on land value are explored in the absence of such externalities. The divergence between the market-equilibrium and rent-maximizing radii is caused by the inframarginal effects of land consumption by marginal firms (those that locate at the fringe of the CBD). In the market equilibrium, the CBD is expanded as long as the marginal firm is able to outbid non-business land users for the marginal plot of land. In the absence of inframarginal effects, the market equilibrium, by allocating marginal land to the highest bidder, will maximize the aggregate net return on land. In the presence of inframarginal effects, however, the allocation of marginal land to the highest bidder will not, in general, lead to the maximization of the aggregate return on land. *The
author
01660462/84/$3.00
gratefully 0
acknowledges 1984, Elsevier
the assistance Science
Publishers
of Edwin
Mills
and Ken
B.V. (North-Holland)
Small.
522
A.M.
Sullivan,
Land
use and zoning
in CBD
The inframarginal effects of marginal land consumption are of two types. First, the expansion of the CBD increases aggregate export production, which decreases production costs as external scale economies are realized. As production costs decrease, land prices increase, since land is the residual claimant. Consequently, the marginal firm affects the price of inframarginal land (land in the interior of the CBD). Second, the expansion of the CBD increases commuting distances, causing wages, which compensate laborers for commuting costs, to increase. As the labor costs of inframarginal firms increase, land prices decrease, since land is the residual claimant. Because of these two inframarginal effects, the fact that the marginal firm can outbid other land users does not mean that its occupation of the land (the market equilibrium outcome) will increase the aggregate net return on land; if the negative impact on inframarginal firms (caused by increasing commuting distances) is large ‘relative to the positive impact (caused by scale economies), the marginal increase in land rent will be dominated by the inframarginal decreases in land rent.
2. The model In this section, we describe a model of a CBD. Firms in the CBD produce an export good with inputs of capital, labor, and land. Workers commute within the CBD, with commuting time coming at the expense of either leisure time or labor time. A negatively sloped wage function makes workers indifferent among all workplaces. An endogenous land bid-rent function of the export industry determines the market-equilibrium CBD radius, as well as the levels of aggregate export production and aggregate input demands. The supply of labor to the CBD is assumed to be perfectly elastic at an exogenous utility level 8. The wage paid for a worker employed at the outer edge of the CBD (the base-wage) is also exogenous. The assumption of an exogenous base-wage of the city allows us to focus attention on the impact of changes in the CBD radius on wages within the CBD. The wage changes occur in the absence of the conventional positive relationship between aggregate labor demand and wages. As the CBD grows, an infinite supply of labor is available at an exogenous base-wage, yet labor costs increase. An alternative assumption is that the base-wage of the city increases with the CBD radius. That is, the aggregate labor-supply curve is upward sloping, so that increases in labor demand bid up the wage paid at the edge of the CBD. In such a model, an expansion of the CBD would increase wages in the CBD as: (a) intra-CBD commuting distances increased, and (b) the basewage of the city increased. In this paper, we focus on the first of these effects; the extensions required to consider the second labor-market effect are straightforward. The important observation is that, given the first effect, wage
A.M.
Sullivan,
Land
use and zoning
in CBD
523
rates increase in a model in which the supply of labor to the CBD is perfectly elastic. At the center of a flat, featureless plain is a transport node through which all export output passes. The relative positions of all CBD locations will be described in a single variable, u, which equals the radial distance to the central export node. The use of a one-dimensional index of location follows from the assumption that circumferential travel is costless. The export industry, the sole occupant of the CBD, is perfectly competitive in input and output markets. The export good is produced with labor and land. The production technology is summarized in a fixed coefficients production function that exhibits external economies of scale: q(u) = l(u). c)(Z) = t(u). c)(Z), q(u) l(u) t(u) 4(Z) z
where
(1)
=output per firm in ring 24, =labor per firm in ring u, =land per firm in ring u, = scale-economy variable, = aggregate export output; and 4’(Z) > 0.
Factor substitution is impossible, but the productivity of all inputs increases as aggregate output rises (a consequence of the greater factor specialization possible in a larger city). Each unit of output requires (1/4(Z)) units of labor and land. Using capital letters to denote inputs and output per unit width of ring u,
Q(u)= L(u). WI = T(u). 4(z) = I(u). 4(Z),
(2)
where I(u) equals the common input quantity in ring u. The expression for profit in ring u, using (2), is n(u) = Ph(u) * I(u). 4(Z) - PL(u) . I(u) -P,(u)
. I(u),
where
(3)
!;2‘ =P,-teeaL, = = tQ PL(u) = P=(U) =
PQ
the exogenous price of Q, transport costs per Q per unit distance, the price of labor at u, the price of land at u.
Since the export industry is perfectly competitive, profit is zero at all locations. The zero-profit condition implicitly defines the price of land at u as the price such that firms at u make zero profit. Rearranging (3),
PT(U)= {Ph(u).4(Z) - M41.
(4)
A.M.
524
Sullivan,
Land
use and zoning
in CBD
Labor for export production is supplied at an exogenous base-wage. Commuting within the CBD requires t, units of time per unit of distance. In equilibrium, laborers must be indifferent among all workplaces. The equilibrium price of labor decreases as the distance to the export node increases, since as u increases, intra-CBD commuting times decrease. The optimization program for residents is as follows:
max u [Y(u),JF(41, w.JFl
(5)
\I
subject to
y = PL(4. JL(4,
(6)
J = JL(U)+ JF(U)+ JT(U),
(7)
JT(U)
Y J, 4 J J, J, t.J
= tJ.
(%z -
u),
where
(8)
= disposable income (consumption of commodities), =labor time (hours per week), = the radius of the CBD, =exogenous maximum leisure time per week, =leisure time per week, = total intra-CBD commuting time, = time cost of commuting per unit distance. The first-order conditions for utility-maximization
!$Jy-i=o, gLJ,--M,(u)=O,
‘are as follows: (9)
(10)
F
where 2 is the symbol for the Lagrangian expression 2(I:JF,A), and il is the Lagrangian multiplier. The manipulation of (9) and (10) gives the following expression for the wage function:
Pt(u) =-.L7F(4
(11)
U,(u)
The wage equals the monetized value of the marginal utility of leisure time. The solution to the optimization program, at a given U, is
A.M.
In equilibrium,
Sullivan,
Land
use and zoning
in CBD
525
utility is constant over u,
dU=O=
UrdY+
U,dJ,
or
$=2+%.
(13)Y (14)
Using (8) and (13), apI.
au.JL(u)+z.P,(u)= aI’, -=
PL(u) . ( - dJ,/&
au
- dJ,/du)
-(aJ,/&),
w. PL(4.(aJ,lW du= JL(~ ’ From (lo), (aJ,/iYu) = -t,,
(15)
(16)
J,(u)
From (9), (dJ,/,CJu+aJ,/du)=
86. -=au
or
-P,(u)-%,
so (17)
so
PA4 *t, Jd4 .
(18)
The wage gradient is negative. A one-unit increase in u reduces intra-CBD commuting time by t,, and increases either labor time or leisure time by the same amount. The value of the reduction of commuting time is t, . PL(u), since the value of time (labor or leisure) equals the wage. To maintain constant utility over a, disposable income falls by PL(u). t,, and the wage falls by the change in income divided by J=(u). The wage function PL(u) is conditional on the CBD radius, u,. For a given workplace u, an increase in u, lengthens the commuting distance, i.e., J=(U) increases with u,. To maintain a constant utility level, wages at all u increase with u,: PL(u,), the base wage, is exogenous, but for u
ah --= -tnyb(Z)-~ au {
. 1
(19)
We will assume that tQ is sufficiently large that the rent gradient is negative,
A.M.
526
Sullivan,
Land
use and zoning
in CBD
i.e.,
The supply of labor is endogenous in two ways. First, labor supply per laborer (hours per week) is endogenous. Every laborer chooses labor and leisure time in accordance with (11); since the wage varies over u, labor supply per laborer will vary over -u. Second, the supply of laborers (city residents) is perfectly elastic at an exogenous utility level 8. The equilibrium number of laborers is determined by the demand side of the labor market. The most important variable in the model is the radius of the CBD, u,. The radius detines the border separating the CBD from the non-business district. Since land is occupied by the highest bidder, u, is defined as the u at which the bid-rent of the business sector equals the exogenous bid-rent of the non-business sector,
PT(u,)= RA.
(21)
Once the radius of the CBD is determined, determined,
aggregate export output can be
2 = 7 Q(u’) du’.
(22)
0
Because the export good is produced function,
with a fixed-coefficient
Lz?= T* = Z/+(Z),
production
(23)
where the A superscript indicates aggregate input quantities. 3. The rent-maximizing
CBD radius
In this section we derive a rule for the CBD radius that maximizes aggregate net return on CBD land. The rule directs the central planner to increase u, up to the point at which the marginal net benefit of an additional CBD ring (increased land rent) equals zero. By comparing the rentmaximizing rule to the market-equilibrium condition (21), we identify two reasons for divergence between the market-equilibrium and rent-maximizing radii. The qualitative difference between the equilibrium CBD radius and the rent-maximizing radius depends on the magnitude of the positive inframarginal effect (external scale economies) relative to the magnitude of the negative inframarginal effect (increasing commuting distances).
A.M.
Sulliuan,
Land
use and zoning
in CBD
527
In the rent-maximizing CBD, the central planner has the objective of maximizing the aggregate net returns of the CBD. The net return is defined as the excess of CBD land rent over the opportunity cost of CRD land, R,. The net return in ring u is R(u)={P,(u)-
RA). T(u).
(24)
The aggregate net return of the CBD is RT = us”R(u’) du’>
(25)
0
RT=T {I’;.
I(u). 4(Z) -Pp,(u’; u,) .L(u’) - RA. T(u’)} du’.
(26)
0
The planner chooses u, to maximize R T. The first-order condition for a maximum sets the derivative of RT with respect to u, to zero. Because of the assumption of fixed factor proportions, I(u), K(u), L(u), and T(u) do not vary with u,. Using Leibnitz’s rule to differentiate (26), g=R(u,)+ x
uxi?R(u’) i rdu’, x
(27)
or o={P,(u,)-R,}T(u,)+~P~(u)4(u)~-.--du’
-p(d). Rearranging
az a.2 au,
a&
ww;au 4) du,. x
(28) gives the expression for the rent-maximizing
(28)
CBD border,
PT(u:) = RA - -
(29) Using E, to denote the absolute value of the second element of the righthand side of (29), and E, to denote the third component, (29) becomes PT(u;)+El-E2=RA.
(30)
A.M.
528
Sullivan,
Land
use and zoning
in CBD
The rule for the rent-maximizing radius differs from the marketequilibrium condition. In the rent-maximizing CBD, PT(@) will not necessarily equal R,. As shown below, E, equals the inframarginal reduction of land rent caused by increasing commuting distances, and E, is the land subsidy required for the internalization of external scale economies. Depending on the relative magnitude of the two inframarginal effects, the rentmaximizing radius may be greater than, less than, or equal to the equilibrium radius. Consider the second component of the right-hand side of (29), El= &.
A Ph(u).I(~)$&.~du’.
x
(31)
Using (22), El
=Q(@ 84 ':s P(o(u)*I(u)du’. -.-. T(@) 82 0
(32)
E, equals the marginal external benefit of land consumption at u,. An additional unit of land consumption means that (1/7(u,)) of the ring is occupied. The ratio Q(u,): T(u,) gives the change in aggregate production (dZ) from a one-unit change in land consumption. In ring u, output increases by the change in labor productivity (a+/aZ). dZ, multiplied by total input quantity I(u). The value of the increase in output (in ring u) equals the change in output times the net price, &(a). The aggregate change in the value of output is the integral, from 0 to u,, of the per-ring increase in value of output. Because land is the residual claimant, E, equals the change in land rent that occurs as a result of a unit increase in land consumption at u,. Consider first a land-tax policy to internalize external scale economies. Returning to (30), suppose E, = 0, and we start with the market-equilibrium border uz. A unit land subsidy equal to E, will cause an increase in u, and an increase in the net return on land. An increase in u, causes two changes in the aggregate returns of the CBD. First, R(u) is less than zero for rings beyond u,O: negative returns occur in each additional ring. Second, as the land area of the CBD increases, Z increases, causing a decrease in production costs at all u
A.M.
Sullivan,
Consider the third component
Land
use and zoning
of the right-hand
in CBD
529
side of (29) (33)
Because the utility level is exogenous, an increase in u, will increase PL(u): as radial commuting distances increase, workers demand higher wages at each IL The expression for (aP,(u)/du,) is derived by the same process used for ww4, dY _-.--U, du,Uy
dJ, du;
(34)
The analog of (17) is (35)
Using (8), (36)
Therefore, E,= &.~L(u’).~da’.
(37)
E, equals the inframarginal reduction in land rent caused by increasing intra-CBD commuting distances. A one-unit increase in land consumption at u, increases the radius of the CBD by (l/T(u,)). Commuting time per worker rises by (t,/T(u,)). Commuting time comes at the expense of either labor or leisure time, both of which are valued at the wage P,(u;u,). In order to maintain a utility level 0, the income of workers in ring u rises by P,(u;u,) per unit decrease in leisure time. The wage in ring u rises by the change in income divided by worktime in ring u, JL(u). Total wage costs rise by the wage change times total labor (in hours) in ring u, L(u). E, equals the change in aggregate labor costs, i.e., the integral of per-ring labor-cost increases, from the city center to u,. Since land is the residual claimant, E, also equals the inframarginal reduction of land rent: as labor costs rise, land rents fall. Consider a land-tax policy to internalize the labor-market effects (rising wages) associated with increases in u,. Returning to (32), suppose that El is zero, and we start from the market-equilibrium radius of n!j. A tax equal to E, would decrease the CBD radius and increase the aggregate net return on
A.M.
530
Sullivan,
Land
use and zoning
in CBD
land. A decrease in u, causes two changes in the aggregate returns of the CBD. First, positive returns are foregone: R(u) > 0 for u
(38)
If the negative inframarginal effect (increasing commuting distances) is large relative to the positive externality, E, is large relative to E,, and land near U, is taxed. The taxation of land reduces the radius of the CBD: the rentmaximizing radius is smaller than the market-equilibrium radius. If, on the other hand, E, is large relative to E,, t is negative, meaning that marginal land is subsidized, and the rent-maximizing radius exceeds the marketequilibrium radius. An alternative corrective public policy is land-use zoning. Land within a central circle of radius u,* is zoned for industrial use, while land outside the circle is zoned for other uses. The zoning policy causes a discontinuity in the rent function at u:. If E, is large relative to El, the zoning policy will decrease the CBD radius and increase aggregate CBD land rent. If the reverse is true, the zoning policy will increase the CBD radius while it increases aggregate CBD land rent.
5. Summary and conclusions A simple model of a central business district (CBD) has been used to derive conditions that defme the market-equilibrium CBD radius and the rent-maximizing CBD radius. The rent-maximizing CBD can be attained in a market setting if taxes and subsidies are used to confront the marginal firm with the inframarginal effects of its land-consumption decision. Alternatively, land-use zoning can be employed to either increase or decrease the CBD radius. There are two basic results from the analysis. First, we demonstrate that the consumption of marginal CBD land generates inframarginal changes in the price of land in the CBD. A marginal firm increases the radius of the CBD, which increases the radial commuting distance of CBD laborers. In order to maintain a constant utility level, laborers demand higher wages. As a result, the production costs of inframarginal CBD firms increase, and land rent falls.
A.M.
Sullivan,
Land
use and zoning
in CBD
531
The second result concerns the effects of land-use zoning on the aggregate net return on land. Depending on the relative magnitudes of the positive inframarginal effect (external scale economies) and the negative\,inframarginal effect (increasing commuting distances), a rent-maximizing zoning policy will either increase or decrease the radius of the CBD. Zoning will establish a discontinuity in the rent-distance function at u,. The discontinuity is motivated by the fact that the marginal firm (locating at a,) affects activities in the interior of the CBD: scale economies are realized as aggregate output increases; commuting distances increase as the CBD radius increases, causing increases in wages; in both cases, land rent in the interior of the CBD is affected. If the objective is to maximize the aggregate net return on land, the marginal firm’s bid-rent for land should not simply equal the opportunity cost of land (RA), but should exceed R, (if the negative inframarginal effects dominate) or should fall short of R, (if external scale economies dominate).
Appendix List
J JF(~ JL(~ JT(~ 44 $4 p,(u) PQ
PT(4 q(u) Q(u) R(u) R.4 R’ t(u) t&l tQ
T(u) TA U
ux 0 ux u: 0 Y z d?(z)
Table A.l. of symbols.
=maximum time per period =leisure time at workplace u =labor time at workplace u =commuting time at workplace u =labor per firm in u =labor demand in ring u = aggregate labor demand =price of labor at u = exogenous export price =land rent at u = output per firm in ring u = export output =net return at u = exogenous non-industrial bid-rent = aggregate net return of CBD =land per firm in u = time required to pass through ring u = transport cost of export output = land demand in ring u = aggregate land demand = distance to export node = CBD radius = market-equilibrium CBD radius =rent-maximizing CBD radius =exogenous utility level of city laborers = total income per laborer = aggregate export output = scale-economy variable
532
A.M.
Sullivan,
Land
use and zoning
in CBD
References Mills, Edwin S., 1979, Economic analysis of urban land-use controls, in: P. Meiszkowski and M. Straszheim, eds., Current issues in urban economics (Johns Hopkins, Baltimore, MD). Ohls, James C., Richard C. Weisberg and Michelle J. White, 1974, The effect of zoning on land value, Journal of Urban Economics 1,4288444. Stull, William, 1974, Land-use zoning in an urban economy, American Economic Review 64, 337-347.
Science
and Urban
LAND
Economics
USE AND
14 (1984)
521-532.
North-Holland
ZONING IN THE CENTRAL DISTRICT*
BUSINESS
Arthur M. SULLIVAN Graduate
School
of Administration,
Received
January
University 1984, final
of California,
version
received
Davis, March
CA 95616,
USA
1984
A model of a small, open central business district (CBD) is used to derive the conditions that define the market-equilibrium and rent-maximizing CBD. It is shown that, in general, the market equilibrium CBD radius differs from the rent-maximizing CBD radius. Land-use zoning will, under certain conditions, increase the aggregate net return on CBD land.
1. Introduction
This paper considers the allocation of land in the production area of a small, open city. The market-equilibrium central business district (CBD) is compared to the rent-maximizing CBD. It is shown that the marketequilibrium CBD radius will not, in general, correspond to the radius that maximizes the aggregate net return on CBD land. A land-use zoning policy that alters the CBD radius will, under certain circumstances, increase the aggregate net return on CBD land. Several researchers have analyzed the effects of industrial land-use zoning. The analyses focus on the effects of industrial zoning in the presence of pollution externalities. See, for example, Stull (1974), Ohls et al. (1974), and Mills (1979). In this paper, the effects of industrial zoning on land value are explored in the absence of such externalities. The divergence between the market-equilibrium and rent-maximizing radii is caused by the inframarginal effects of land consumption by marginal firms (those that locate at the fringe of the CBD). In the market equilibrium, the CBD is expanded as long as the marginal firm is able to outbid non-business land users for the marginal plot of land. In the absence of inframarginal effects, the market equilibrium, by allocating marginal land to the highest bidder, will maximize the aggregate net return on land. In the presence of inframarginal effects, however, the allocation of marginal land to the highest bidder will not, in general, lead to the maximization of the aggregate return on land. *The
author
01660462/84/$3.00
gratefully 0
acknowledges 1984, Elsevier
the assistance Science
Publishers
of Edwin
Mills
and Ken
B.V. (North-Holland)
Small.
522
A.M.
Sullivan,
Land
use and zoning
in CBD
The inframarginal effects of marginal land consumption are of two types. First, the expansion of the CBD increases aggregate export production, which decreases production costs as external scale economies are realized. As production costs decrease, land prices increase, since land is the residual claimant. Consequently, the marginal firm affects the price of inframarginal land (land in the interior of the CBD). Second, the expansion of the CBD increases commuting distances, causing wages, which compensate laborers for commuting costs, to increase. As the labor costs of inframarginal firms increase, land prices decrease, since land is the residual claimant. Because of these two inframarginal effects, the fact that the marginal firm can outbid other land users does not mean that its occupation of the land (the market equilibrium outcome) will increase the aggregate net return on land; if the negative impact on inframarginal firms (caused by increasing commuting distances) is large ‘relative to the positive impact (caused by scale economies), the marginal increase in land rent will be dominated by the inframarginal decreases in land rent.
2. The model In this section, we describe a model of a CBD. Firms in the CBD produce an export good with inputs of capital, labor, and land. Workers commute within the CBD, with commuting time coming at the expense of either leisure time or labor time. A negatively sloped wage function makes workers indifferent among all workplaces. An endogenous land bid-rent function of the export industry determines the market-equilibrium CBD radius, as well as the levels of aggregate export production and aggregate input demands. The supply of labor to the CBD is assumed to be perfectly elastic at an exogenous utility level 8. The wage paid for a worker employed at the outer edge of the CBD (the base-wage) is also exogenous. The assumption of an exogenous base-wage of the city allows us to focus attention on the impact of changes in the CBD radius on wages within the CBD. The wage changes occur in the absence of the conventional positive relationship between aggregate labor demand and wages. As the CBD grows, an infinite supply of labor is available at an exogenous base-wage, yet labor costs increase. An alternative assumption is that the base-wage of the city increases with the CBD radius. That is, the aggregate labor-supply curve is upward sloping, so that increases in labor demand bid up the wage paid at the edge of the CBD. In such a model, an expansion of the CBD would increase wages in the CBD as: (a) intra-CBD commuting distances increased, and (b) the basewage of the city increased. In this paper, we focus on the first of these effects; the extensions required to consider the second labor-market effect are straightforward. The important observation is that, given the first effect, wage
A.M.
Sullivan,
Land
use and zoning
in CBD
523
rates increase in a model in which the supply of labor to the CBD is perfectly elastic. At the center of a flat, featureless plain is a transport node through which all export output passes. The relative positions of all CBD locations will be described in a single variable, u, which equals the radial distance to the central export node. The use of a one-dimensional index of location follows from the assumption that circumferential travel is costless. The export industry, the sole occupant of the CBD, is perfectly competitive in input and output markets. The export good is produced with labor and land. The production technology is summarized in a fixed coefficients production function that exhibits external economies of scale: q(u) = l(u). c)(Z) = t(u). c)(Z), q(u) l(u) t(u) 4(Z) z
where
(1)
=output per firm in ring 24, =labor per firm in ring u, =land per firm in ring u, = scale-economy variable, = aggregate export output; and 4’(Z) > 0.
Factor substitution is impossible, but the productivity of all inputs increases as aggregate output rises (a consequence of the greater factor specialization possible in a larger city). Each unit of output requires (1/4(Z)) units of labor and land. Using capital letters to denote inputs and output per unit width of ring u,
Q(u)= L(u). WI = T(u). 4(z) = I(u). 4(Z),
(2)
where I(u) equals the common input quantity in ring u. The expression for profit in ring u, using (2), is n(u) = Ph(u) * I(u). 4(Z) - PL(u) . I(u) -P,(u)
. I(u),
where
(3)
!;2‘ =P,-teeaL, = = tQ PL(u) = P=(U) =
PQ
the exogenous price of Q, transport costs per Q per unit distance, the price of labor at u, the price of land at u.
Since the export industry is perfectly competitive, profit is zero at all locations. The zero-profit condition implicitly defines the price of land at u as the price such that firms at u make zero profit. Rearranging (3),
PT(U)= {Ph(u).4(Z) - M41.
(4)
A.M.
524
Sullivan,
Land
use and zoning
in CBD
Labor for export production is supplied at an exogenous base-wage. Commuting within the CBD requires t, units of time per unit of distance. In equilibrium, laborers must be indifferent among all workplaces. The equilibrium price of labor decreases as the distance to the export node increases, since as u increases, intra-CBD commuting times decrease. The optimization program for residents is as follows:
max u [Y(u),JF(41, w.JFl
(5)
\I
subject to
y = PL(4. JL(4,
(6)
J = JL(U)+ JF(U)+ JT(U),
(7)
JT(U)
Y J, 4 J J, J, t.J
= tJ.
(%z -
u),
where
(8)
= disposable income (consumption of commodities), =labor time (hours per week), = the radius of the CBD, =exogenous maximum leisure time per week, =leisure time per week, = total intra-CBD commuting time, = time cost of commuting per unit distance. The first-order conditions for utility-maximization
!$Jy-i=o, gLJ,--M,(u)=O,
‘are as follows: (9)
(10)
F
where 2 is the symbol for the Lagrangian expression 2(I:JF,A), and il is the Lagrangian multiplier. The manipulation of (9) and (10) gives the following expression for the wage function:
Pt(u) =-.L7F(4
(11)
U,(u)
The wage equals the monetized value of the marginal utility of leisure time. The solution to the optimization program, at a given U, is
A.M.
In equilibrium,
Sullivan,
Land
use and zoning
in CBD
525
utility is constant over u,
dU=O=
UrdY+
U,dJ,
or
$=2+%.
(13)Y (14)
Using (8) and (13), apI.
au.JL(u)+z.P,(u)= aI’, -=
PL(u) . ( - dJ,/&
au
- dJ,/du)
-(aJ,/&),
w. PL(4.(aJ,lW du= JL(~ ’ From (lo), (aJ,/iYu) = -t,,
(15)
(16)
J,(u)
From (9), (dJ,/,CJu+aJ,/du)=
86. -=au
or
-P,(u)-%,
so (17)
so
PA4 *t, Jd4 .
(18)
The wage gradient is negative. A one-unit increase in u reduces intra-CBD commuting time by t,, and increases either labor time or leisure time by the same amount. The value of the reduction of commuting time is t, . PL(u), since the value of time (labor or leisure) equals the wage. To maintain constant utility over a, disposable income falls by PL(u). t,, and the wage falls by the change in income divided by J=(u). The wage function PL(u) is conditional on the CBD radius, u,. For a given workplace u, an increase in u, lengthens the commuting distance, i.e., J=(U) increases with u,. To maintain a constant utility level, wages at all u increase with u,: PL(u,), the base wage, is exogenous, but for u
ah --= -tnyb(Z)-~ au {
. 1
(19)
We will assume that tQ is sufficiently large that the rent gradient is negative,
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i.e.,
The supply of labor is endogenous in two ways. First, labor supply per laborer (hours per week) is endogenous. Every laborer chooses labor and leisure time in accordance with (11); since the wage varies over u, labor supply per laborer will vary over -u. Second, the supply of laborers (city residents) is perfectly elastic at an exogenous utility level 8. The equilibrium number of laborers is determined by the demand side of the labor market. The most important variable in the model is the radius of the CBD, u,. The radius detines the border separating the CBD from the non-business district. Since land is occupied by the highest bidder, u, is defined as the u at which the bid-rent of the business sector equals the exogenous bid-rent of the non-business sector,
PT(u,)= RA.
(21)
Once the radius of the CBD is determined, determined,
aggregate export output can be
2 = 7 Q(u’) du’.
(22)
0
Because the export good is produced function,
with a fixed-coefficient
Lz?= T* = Z/+(Z),
production
(23)
where the A superscript indicates aggregate input quantities. 3. The rent-maximizing
CBD radius
In this section we derive a rule for the CBD radius that maximizes aggregate net return on CBD land. The rule directs the central planner to increase u, up to the point at which the marginal net benefit of an additional CBD ring (increased land rent) equals zero. By comparing the rentmaximizing rule to the market-equilibrium condition (21), we identify two reasons for divergence between the market-equilibrium and rent-maximizing radii. The qualitative difference between the equilibrium CBD radius and the rent-maximizing radius depends on the magnitude of the positive inframarginal effect (external scale economies) relative to the magnitude of the negative inframarginal effect (increasing commuting distances).
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In the rent-maximizing CBD, the central planner has the objective of maximizing the aggregate net returns of the CBD. The net return is defined as the excess of CBD land rent over the opportunity cost of CRD land, R,. The net return in ring u is R(u)={P,(u)-
RA). T(u).
(24)
The aggregate net return of the CBD is RT = us”R(u’) du’>
(25)
0
RT=T {I’;.
I(u). 4(Z) -Pp,(u’; u,) .L(u’) - RA. T(u’)} du’.
(26)
0
The planner chooses u, to maximize R T. The first-order condition for a maximum sets the derivative of RT with respect to u, to zero. Because of the assumption of fixed factor proportions, I(u), K(u), L(u), and T(u) do not vary with u,. Using Leibnitz’s rule to differentiate (26), g=R(u,)+ x
uxi?R(u’) i rdu’, x
(27)
or o={P,(u,)-R,}T(u,)+~P~(u)4(u)~-.--du’
-p(d). Rearranging
az a.2 au,
a&
ww;au 4) du,. x
(28) gives the expression for the rent-maximizing
(28)
CBD border,
PT(u:) = RA - -
(29) Using E, to denote the absolute value of the second element of the righthand side of (29), and E, to denote the third component, (29) becomes PT(u;)+El-E2=RA.
(30)
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The rule for the rent-maximizing radius differs from the marketequilibrium condition. In the rent-maximizing CBD, PT(@) will not necessarily equal R,. As shown below, E, equals the inframarginal reduction of land rent caused by increasing commuting distances, and E, is the land subsidy required for the internalization of external scale economies. Depending on the relative magnitude of the two inframarginal effects, the rentmaximizing radius may be greater than, less than, or equal to the equilibrium radius. Consider the second component of the right-hand side of (29), El= &.
A Ph(u).I(~)$&.~du’.
x
(31)
Using (22), El
=Q(@ 84 ':s P(o(u)*I(u)du’. -.-. T(@) 82 0
(32)
E, equals the marginal external benefit of land consumption at u,. An additional unit of land consumption means that (1/7(u,)) of the ring is occupied. The ratio Q(u,): T(u,) gives the change in aggregate production (dZ) from a one-unit change in land consumption. In ring u, output increases by the change in labor productivity (a+/aZ). dZ, multiplied by total input quantity I(u). The value of the increase in output (in ring u) equals the change in output times the net price, &(a). The aggregate change in the value of output is the integral, from 0 to u,, of the per-ring increase in value of output. Because land is the residual claimant, E, equals the change in land rent that occurs as a result of a unit increase in land consumption at u,. Consider first a land-tax policy to internalize external scale economies. Returning to (30), suppose E, = 0, and we start with the market-equilibrium border uz. A unit land subsidy equal to E, will cause an increase in u, and an increase in the net return on land. An increase in u, causes two changes in the aggregate returns of the CBD. First, R(u) is less than zero for rings beyond u,O: negative returns occur in each additional ring. Second, as the land area of the CBD increases, Z increases, causing a decrease in production costs at all u
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side of (29) (33)
Because the utility level is exogenous, an increase in u, will increase PL(u): as radial commuting distances increase, workers demand higher wages at each IL The expression for (aP,(u)/du,) is derived by the same process used for ww4, dY _-.--U, du,Uy
dJ, du;
(34)
The analog of (17) is (35)
Using (8), (36)
Therefore, E,= &.~L(u’).~da’.
(37)
E, equals the inframarginal reduction in land rent caused by increasing intra-CBD commuting distances. A one-unit increase in land consumption at u, increases the radius of the CBD by (l/T(u,)). Commuting time per worker rises by (t,/T(u,)). Commuting time comes at the expense of either labor or leisure time, both of which are valued at the wage P,(u;u,). In order to maintain a utility level 0, the income of workers in ring u rises by P,(u;u,) per unit decrease in leisure time. The wage in ring u rises by the change in income divided by worktime in ring u, JL(u). Total wage costs rise by the wage change times total labor (in hours) in ring u, L(u). E, equals the change in aggregate labor costs, i.e., the integral of per-ring labor-cost increases, from the city center to u,. Since land is the residual claimant, E, also equals the inframarginal reduction of land rent: as labor costs rise, land rents fall. Consider a land-tax policy to internalize the labor-market effects (rising wages) associated with increases in u,. Returning to (32), suppose that El is zero, and we start from the market-equilibrium radius of n!j. A tax equal to E, would decrease the CBD radius and increase the aggregate net return on
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land. A decrease in u, causes two changes in the aggregate returns of the CBD. First, positive returns are foregone: R(u) > 0 for u
(38)
If the negative inframarginal effect (increasing commuting distances) is large relative to the positive externality, E, is large relative to E,, and land near U, is taxed. The taxation of land reduces the radius of the CBD: the rentmaximizing radius is smaller than the market-equilibrium radius. If, on the other hand, E, is large relative to E,, t is negative, meaning that marginal land is subsidized, and the rent-maximizing radius exceeds the marketequilibrium radius. An alternative corrective public policy is land-use zoning. Land within a central circle of radius u,* is zoned for industrial use, while land outside the circle is zoned for other uses. The zoning policy causes a discontinuity in the rent function at u:. If E, is large relative to El, the zoning policy will decrease the CBD radius and increase aggregate CBD land rent. If the reverse is true, the zoning policy will increase the CBD radius while it increases aggregate CBD land rent.
5. Summary and conclusions A simple model of a central business district (CBD) has been used to derive conditions that defme the market-equilibrium CBD radius and the rent-maximizing CBD radius. The rent-maximizing CBD can be attained in a market setting if taxes and subsidies are used to confront the marginal firm with the inframarginal effects of its land-consumption decision. Alternatively, land-use zoning can be employed to either increase or decrease the CBD radius. There are two basic results from the analysis. First, we demonstrate that the consumption of marginal CBD land generates inframarginal changes in the price of land in the CBD. A marginal firm increases the radius of the CBD, which increases the radial commuting distance of CBD laborers. In order to maintain a constant utility level, laborers demand higher wages. As a result, the production costs of inframarginal CBD firms increase, and land rent falls.
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The second result concerns the effects of land-use zoning on the aggregate net return on land. Depending on the relative magnitudes of the positive inframarginal effect (external scale economies) and the negative\,inframarginal effect (increasing commuting distances), a rent-maximizing zoning policy will either increase or decrease the radius of the CBD. Zoning will establish a discontinuity in the rent-distance function at u,. The discontinuity is motivated by the fact that the marginal firm (locating at a,) affects activities in the interior of the CBD: scale economies are realized as aggregate output increases; commuting distances increase as the CBD radius increases, causing increases in wages; in both cases, land rent in the interior of the CBD is affected. If the objective is to maximize the aggregate net return on land, the marginal firm’s bid-rent for land should not simply equal the opportunity cost of land (RA), but should exceed R, (if the negative inframarginal effects dominate) or should fall short of R, (if external scale economies dominate).
Appendix List
J JF(~ JL(~ JT(~ 44 $4 p,(u) PQ
PT(4 q(u) Q(u) R(u) R.4 R’ t(u) t&l tQ
T(u) TA U
ux 0 ux u: 0 Y z d?(z)
Table A.l. of symbols.
=maximum time per period =leisure time at workplace u =labor time at workplace u =commuting time at workplace u =labor per firm in u =labor demand in ring u = aggregate labor demand =price of labor at u = exogenous export price =land rent at u = output per firm in ring u = export output =net return at u = exogenous non-industrial bid-rent = aggregate net return of CBD =land per firm in u = time required to pass through ring u = transport cost of export output = land demand in ring u = aggregate land demand = distance to export node = CBD radius = market-equilibrium CBD radius =rent-maximizing CBD radius =exogenous utility level of city laborers = total income per laborer = aggregate export output = scale-economy variable
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References Mills, Edwin S., 1979, Economic analysis of urban land-use controls, in: P. Meiszkowski and M. Straszheim, eds., Current issues in urban economics (Johns Hopkins, Baltimore, MD). Ohls, James C., Richard C. Weisberg and Michelle J. White, 1974, The effect of zoning on land value, Journal of Urban Economics 1,4288444. Stull, William, 1974, Land-use zoning in an urban economy, American Economic Review 64, 337-347.