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Optimal lot size and configuration
JOURNAL
OF URBAN
ECONOMICS
26,
90-109 (1989)
Optimal Lot Size and Configuration PETERF. COLWELLANDTIMSCHEU* Ofice of Real Estate Research, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, and *Department of Finance, Western Michigan University, Kalamazoo, Michigan 49008
Received November 251986; revised February lo,1987 The developer chooses lot size and shape to maximize profit subject to institutional constraints. The price structure for lots is exogenous and Cobb-Douglas in form. Costs of land development are assumed to be both proportional to frontage and to lot area. The constraints are the minimum lot area from zoning and the minimum lot depth from subdivision ordinances. Numerical and empirical results support this theory of optimal development. One implication is that lower frontage cost from technological change may have led to wider-shallower lots, a result also consistent with lower land rent in a conventional urban model. 0 1989 Academic Press. Inc.
I. INTRODUCTION Most urban landscapes are characterized by a few highly visible features. Foremost among these is the clustering of tall buildings around a small number of nuclei. Another such feature is the discrete, rather than gradual, changes in building heights as distance to a nucleus increases. Still another of these visually obvious features is the spatial variation in the relative proportions of residential lots. Older and more centrally located lots tend to have a smaller frontage to depth ratio than newer and more peripherally located lots. This latter feature has been virtually ignored in the literature in contrast to the wealth of research on density gradients and discontinuities in density gradients. Land developers must consider the way lot size and configuration affect value as well as how they affect costs when deciding on the optimal lot design. In addition, zoning ordinances and subdivision regulations may constrain the choice. The purpose of this paper is to structure the problem of choosing lot size and configuration as one of maximking profit subject to institutional constraints. It is assumed that the price structure for lots of *The authors are ORER Professor of Real Estate, University of Illinois, and Assistant Professor of Finance, Western Michigan University, respectively. In part, this research was supported by the Office of Real Estate Research, University of Illinois at Urbana-Champaign. The authors benefited from comments by Roger E. Cannaday, James R. Follain, Larry Neal, Robert W. Resek, and an anonymous referee. All remaining errors are the responsibility of the authors. 90 0094-1190/89 Copyright All rights
$3.00
0 1989 by Academic Press, Inc. of reproduction in any form reserved.
OPTIMAL
91
LOT SIZE
various sizes and configurations is given as are the costs of developing lots and the institutional constraints. Thus, the partial equilibrium question of the developer’s choice is addressed in this paper. An empirical section follows the theoretical and institutional sections so as to provide justification for the relative sizes of frontage and depth elasticities used in the theoretical lot value function. A numerical section is provided to simulate developer choice in the context of specific parameter magnitudes. Finally, a section considers a modification to the theoretical model which allows the model to be developed in the context of competitive market equilibrium. Two assumptions are made for the purpose of tractability. First, all lots are assumed to be rectangular in shape. Second, comer lots are assumed to be valued in the same way as interior lots. That is, the fact that comer lots have frontage on a second side is ignored. II. THE UNCONSTRAINED
OPTIMUM
This section begins by examinin g the lot value function and then turns to the development cost function. Next, value and cost are brought together in a profit function. The conditions for profit maximization are derived. Finally, the implications of changing parameters for optimal lot size and configuration are explored. Lot Value The value of a lot is assumed to be a Cobb-Douglas function of frontage and depth [6]. There are several reasons for this assumption. First, on an empirical basis, the Cobb-Douglas function has been shown to be superior in this application to a variety of well-known functions. This superiority has been demonstrated by Box-Cox and Box-Tidwell functional form analysis [3]. Second, there is substantial historical precedent for using the Cobb-Douglas function in this application. This function has been used to value urban lots for about a century. The context in which such functions are used is called Depth Rules [6]. Third, the historical competitors of the Cobb-Douglas function tend to be somewhat odd (e.g., shifted log functions and segments of parabolas spliced to still other functions) and admit less readily to estimation. Fourth, there are stories, perhaps apocryphal, of very substantial data sets having been used to estimate Cobb-Douglas Depth Rules [4]. Thus, it is possible that what is lost in generality by using the Cobb-Douglas function rather than a general form may be more than compensated by specific results that have a good chance of being correct. There is probably greater reason to use the Cobb-Douglas form in this application than in most of its applications. The value of a lot is given as v = aF4Y
9
(1)
92
COLWELL AND SCHEU
where V = lot value, F = frontage, D = depth, a = the value of the first square foot of lot area, /3 = the frontage elasticity of value, y = the depth elasticity of value, and 0 < y < fi < 1. Under certain conditions, this function is consistent with well-known depth rules such as the HoffmanNeil, Reeves, Harper, Hobbs, and Milwaukee rules which have been recognized by courts and utilized by assessors for nearly a century [4, 61. If the depth elasticity of value is in the neighborhood of .5 to .585 and the frontage elasticity of value is unitary, the value function resembles these depth rules closely. The frontage rule of Ross suggests, however, that the frontage elasticity may be less than unitary [6]. Both frontage and depth elasticity are assumed here to be less than unitary for the sake of the model’s tractability. The model requires that the frontage elasticity be greater than the depth elasticity. There are plausible explanations for the relative magnitudes of the frontage and depth elasticities. For commercial and industrial property, frontage is associated with exposure, access to customers, and corporate identity. There may be similar arguments for residential property related to status, displays of wealth, or demonstration effects. However, the explanation may be quite different for residential property. If there is diminishing marginal utility from spatial separation between households, roads and zoning setbacks already provide substantial separation in the depth dimension and would cause the marginal benefit of frontage to exceed that of depth over the ranges one is likely to encounter. This question of relative elasticities is investigated empirically in a later section of this paper. Development Costs
Land developers’ costs are primarily the cost of raw land and the cost of providing access, vehicular and pedestrian access as well as access to utility services. The length of streets, sidewalks, and utilities (i.e., sewer, water, electric and gas) associated with developing a lot is essentially equal to the frontage of the lot.’ Thus, one component of development cost is proportional to the frontage of the lot. The other component of development cost is proportional to lot area. Thus, the total cost function may be expressed as C = SFD + #F,
where C = total development costs, and FD = lot area. The constant of proportionality on FD is primarily the sum of the cost of raw land per square foot, the prorated cost of land and improvements for a side street, ‘Developers may be reimbursed to some extent, For the installation of certain utilities. For example, these items can include underground electrical and gas service.
OPTIMAL
LOT
93
SIZE
and the prorated cost of the land and improvements for the crosswalk that may go between blocks. The constant of proportionality on frontage is primarily the total cost per front foot for land and improvements for streets as well as for utility development costs for which the developer is responsible. In addition, the costs of surveying, planning, and platting the subdivision may be related to both frontage and area. Projit The developer’s profit per lot is the lot value minus the costs of developing the lot. Thus, profit may be expressed as s = aFBDr - 6FD - #F 3
(3)
where v = profit per lot. The first-order conditions for profit maximization are r1 = cuj3F’+‘D’
- SD - II, = 0
(4)
r2 = ayFpD7-’
- 6F = 0.
(5)
From (4) and (5) it is simple to show that 6D +# -SF=--
/3D 6F ’
(6)
which says that the marginal rate of substitution in costs must equal the marginal rate of substitution in value. That is, an isovalue curve in frontage and depth space must be tangent to an isocost curve in that space. See Fig. 1 for the expansion path of isocost and isovalue tangency points. Eq. (5) can be rewritten
s I’=
V/FD’
which means that the depth elasticity must equal the ratio of the cost of raw land per square foot to the value of developed lots per square foot. The value of developed lots per square foot is another Cobb-Douglas function of frontage and depth: V/FD = o~F~--~D~-~. The slope of the contours of this value per square foot function is
-(l - ,@D/((l - Y)F),
(8)
94
COLWELL
AND
FIGURE
SCHEU
1
which is less steep than the slope of the isovalue curves because of the relative frontage and depth elasticities specified earlier. Condition (7) requires that the profit maximixi ng lot size and configuration be found along one special value per square foot contour. This is the one where value per square foot equals the ratio of the cost of land per square foot to the depth elasticity (i.e., value per square foot must equal a constant). See Fig. 1 for an illustration of this condition. So far, two distinct requirements have been offered for the profit maximizing lot size and configuration. These requirements are that frontage and depth be on the expansion path and that they be on a specifk isovalue-persquare-foot contour. Thus, the profit m axinking size and configuration are found at the intersection of these two functions. See Fig. 1. Although it is not needed for the determination of the profit maximizing lot size and configuration, additional insights may be obtained by focusing on (4). First, (4) can be rewritten
6 p = V/FD
-!L
+ V/F ’
which means that the frontage elasticity must equal the same ratio of cost to value as the depth elasticity plus the ratio of the cost of development per front foot to the value of a developed lot per front foot. Thus, the frontage elasticity exceeds the depth elasticity by the ratio of the cost of develop-
OPTIMAL
95
LOT SIZE
ment per front foot to the value of a developed lot per front foot. Second, (9) can be rewritten
v,FD=$+&.
(10)
Thus, the value per square foot must also be a particular function of depth. This requirement is also shown in Fig. 1. The second-order conditions for a maximum are
NP- w <0 ’
(11)
< 0,
(12)
F2 =22
=
and TllT22
-
TlZ2
=
&&I(1
-
P)
-
Y(’
PM’ 0.
03)
Conditions (11) and (12) require that the frontage and depth elasticities be less than unitary (i.e., /3, y < 1). Condition (13) utilizes the notion that 6 = yV/FD from (5) and can be further simplified to yield the requirement that the frontage elasticity must be greater than the depth elasticity (i.e., /3 > y), which is apparent from the first-order conditions. Solving (5) for the profit maximizing frontage as a function of depth and solving (4) and (5) for the profit maximizing depth yields F = [ ayDY-‘/S]l’(l-p),
(14)
and
D = W(P/u
- 1).
05)
Thus, the unconstrained optimal frontage and depth can be found in (14) and (15). This solves both the problem of optimal lot area and the problem of the optimal lot configuration in the absence of institutional constraints. Comparative
Statics
The land developer is, of course, interested in the way in which changes in parameters from the value and cost functions (i.e., across submarkets and through time) can change optimal lot size and configuration.
96
COLWELL
AND
SCHEU
A change in the value of the first square foot (i.e., when F and D both equal unity) has a radically different impact on the optimal depth and frontage: c?D -= aa
o. 7
aF
z
yDY-‘FP
= (1 - /?)S ’ O”
Thus, an increase in (Y has no effect on the optimal depth but causes optimal frontage to increase. While it would be expected that an increase in 1y would cause F to increase, it is unexpected that (Ywould have no impact on D. As suggested earlier, one way to view the profit maximizing choice is in the context of isovalue and isocost curves shown in Fig. 1. The profit maximizing combination of F and D is on the expansion path. Note that the expansion path is a horizontal line. Part of the explanation for this is the asymmetric treatment of F and D in the cost function. The expansion path would not be a horizontal line in the context of a cost function which treats F and D symmetrically. Another part of the explanation relates to the CobbDouglas form of the value function. When the value of the first square foot, (Y, changes, the isovalue map is unchanged, although each isovalue curve would relate to a different level of value after the change. This is a result of having chosen the Cobb-Douglas form for the value function. Thus, the expansion path and the profit maximizing D would be unchanged as a result of changing the value of the first square foot. A change in the frontage elasticity, /?, has opposite effects on optimal depth and frontage:
In F1-fi
(1 - p)’
(1 -Y)
JD
- (1 - /?)D ap
1
’ ”
Thus an increase in fl causes optimal depth to decrease as the isovalue curves become steeper, but it also causes optimal frontage to increase. Obviously, the lot frontage to depth ratio increases as the frontage elasticity increases. However, the impact of frontage elasticity on lot size is not obvious. This impact is 8FD dD -=F,p+Dq=F JP
I~F
aD (Y -P> -gql-p)
+
D In F1-@ (1-p)2
1
‘OS
OPTIMAL
LOT
97
SIZE
Thus, an increase in the frontage elasticity results in an increase in optimal lot size. A change in the depth elasticity, y, moves both optimal frontage and depth in the same direction: C?D -Tq=
b2(P/Y
aF au=
F Y(l - PI
p+
> 0; - v $1
l+ylnD-
- y)D-‘1
> 0.
An increase in y causes optimal depth to increase as the isovalue curves become flatter, but it also causes frontage to increase. As a result of these two derivatives having the same signs, it becomes interesting to know the effect of depth elasticity y on the lot configuration (i.e., F/D). The derivative of this ratio with respect to y is dF/D ay
= Fv
+ l/Day
ilF
i@ aF = FG + l/Day > 0.
This derivative is positive, so F/D increases as y increases. A change in the cost of frontage, I), causes optimal depth and frontage to move in opposite directions: aD v=
1
~(p/~ - 1) ’ O;
aF -=-
(1 - Y)F g
a4
(1 -P)D
< 0
a+
.
As 1c, increases and the isocost curves become steeper, optimal depth increases, while optimal frontage decreases. Thus, as street building technology has changed from bricks and blocks to continuously poured cement, thereby lowering the real #, new lots would be expected to have become wider and less deep. As a result of the opposite signs on these two derivatives, one would like to know the direction of the impact on the size of lots. The derivative of FD with respect to +!I is
84
aFD aF -=D,J,+F,JI=-
aD
[p-u1Fi?
a+
.
Thus, lot size increases as the cost of frontage decreases. Recall that it has been assumed that /3 > y and /3 -C 1. Finally, an increase in the cost of land area, 6, causes the optimal depth to decrease as the isocost curves become less steep. The impact on the
98
optimal
COLWELL
AND
SCHEU
frontage is less certain:
dD ---
as -
p(pJ
The sign of aF/l@
-1)
aF -=as
is determined
as
aF as
if l/a
--SO
(l$)
[ f+7-
1
(1 - Y) aD as .
2 - (l - ‘) aD -as.D
In Section V, it will be shown that magnitudes for the parameters that make empirical sense cause the sign of aF/a& to be negative. Assuming that capital costs the same everywhere, urban areas with higher surrounding land values should have smaller residential lots (i.e., smaller D and F), ceteris paribus. Similarly, platted lot sixes should, to some extent, fluctuate counter to fluctuations in surrounding agricultural land values within a single urban area. This has important implications for current urban growth. We should not only see an expansion of urban areas as a result of depressed agricultural land prices (i.e., a downward shift in the agricultural bid-rent curve), but we should also see a decrease in density (i.e., an increase in the sizes of new lots). The question that remains regarding the impact of the cost of land area is its impact on the configuration of lots. The derivative of F/D is aF/D
as
= F a1/D
a6
1 aF +imr
’ D2(l
- P)
The sign is aF/D
as
>
<
o
if
b/Y+-
1
_
(1
-
Y)
aD >
1
D(l - p) 88 < D6(1 - 13) ’
Again, the magnitudes of the parameters determine the sign. Magnitudes which produce a negative sign will be explored in Section V. Taking the negative sign as correct, it appears that higher prices of raw land would be associated with narrower-deeper lots. One might be tempted to try to explain the spatial distribution of frontage to depth ratios on the basis of land rent declining with increasing distance to the urban center. An alternative hypothesis is that the rising ratio as distance increases can be attributed to decreases in frontage prices through time as the lots were platted and developed.
OPTIMAL
III.
99
LOT SIZE
INSTITUTIONAL
CONSTRAINTS
Land developers face a number of institutional constraints. These constraints arise primarily as a result of zoning ordinances and subdivision regulations. Perhaps most obviously, there are minimum lot sizes. Less obviously, there are minimum lot depths which work in a similar way to the requirements placed on the builder for minimum backyards and front setbacks. For example, a new streets and subdivision ordinance may allow for a minimum of 260 feet between the centers of streets. After subtracting 60 feet required for right of way from the center of the streets, the land developer is left with a minimum of 200 feet for lots. With lots back to back, each lot must be at least 100 feet deep. Land developers face regulations related to maximum frontage and depth or, in other words, the frequency of side streets and crosswalks through the block. However, these regulations are not likely to be binding constraints. For example, side streets may be no more than 1200 feet apart if there is a lo-foot crosswalk right of way in the middle of the block. This leaves a maximum frontage and a maximum depth of 565 feet on each side of the crosswalk after allowing for right of way on the streets. Builders face additional constraints. These relate to the placement and size of the structure on the lot. Included in these constraints are floor to area ratios, open space ratios, and sideyard requirements in addition to the backyard and front setback requirements mentioned earlier. Private constraints (i.e., deed restrictions and covenants) may include minimum house size and more rigorous setbacks. The constraints that are the focus of this paper are the minimum lot area and the minimum depth faced by land developers. The constraints are FD 2 Ati,
the minimum
lot area?
and D2Dh,
the minimum
The resulting feasible set is illustrated
lot depth.
in Fig. 2.
Constrainted Optimum The problem is to choose positive values of F and D so as to maximize a = aFpDY - 6FD - I/IF subject to FD 2 Ati,, DzDti.
100
COLWELL
AND
SCHEU
D
F FIGURE
The Lagrangian
2
function is
L= aFBDDy- 6FD-t/F+X,(FD-Ati)+h,(D
-D,,).
(16)
Application of the Kuhn-Tucker conditions, which are necessary and sufficient for maximum profit (i.e., due to the concavity of the profit function), yields
g
=/3ctFB-IDY-
SD -t,b-X,D
i3L -JD =yaF@-'6F-A,F-X, FD-A,,>0 D-D,,>0 A, 2 0 A, 2 0.
10 I 0
It is necessary to explore two important cases, the case of neither constraint binding haying been taken up in Section II. These two cases (i) where the minimum lot area is the binding constraint and (ii) where the minimum depth is the binding constraint. Case 1. The optimal frontage and depth would be
F-
[$//aA~&3
- y)]l~B-u-l),
(17)
and
D=A,,JF.
(18)
OPTIMAL
As the minimum changes as
LOT
lot area constraint
101
SIZE
increases, the optimal
frontage
Thus, an increase in A,, would cause frontage to increase. Similarly, increase in A,, would cause depth to increase as
an
As a result of the signs of the above two derivatives being positive, the interesting question is the direction of the impact of an increase of A,, on the frontage to depth ratio: aF aF/D -=aA,,a+(yAeF a&in =
1
al/D
2/4&8)/h3+Y) 1 _ ;
[ a(;-
J
+ @-&8)/(1-fl+Y)
-l’(l-B+y)
_
1 1 j
B, yA-.-Y)/(‘~+‘)l
The sign is if A(p-Y)/(l-b+Y)
$
mm
-1-B
Y
*
As 6 approaches /3 at magnitudes greater than .5 the sign is positive, while at magnitudes below .5 the sign is negative. With /3 close to unity, it is unlikely that this sign would be negative. Thus, the F/D should be expected to increase as the minimum lot area constraint increases. Cuse 2. The optimal frontage and depth are
F = [ “%i&&+]
-+‘),
and D = Oh,,,
(20)
102
COLWELL AND SCHEU
As the minimum according to
depth constraint increases, the optimal frontage changes
aF
6 SD,, + Ic,I ’
ao,,=
The sign of this derivative is determined aF ao,,
i O
ifDti5
as v+ s(1 -y).
Again, the sign depends on the magnitudes of the parameters. However, some things can be said about the key parameters. Although 4 and 6 are likely to vary greatly through space, the ratio of 1c,to S is important for the sign of aF/aD& These two parameters are likely to move together, because S, the price of raw land, is imbedded in #, the price of frontage (i.e., right-of-way costs are part of #). Thus, #/S is not likely to be volatile. Our judgment is that at depth elasticities (y) below .3, one can assume that $/S is insufficiently large to cause dF/aD,, to be positive. However, the probability that this derivative is negative declines as the initial minimum depth requirement decreases and as the depth elasticity increases. At depth elasticities at or above those of traditional depth rules (i.e., .5 to .585), one can be quite sure that the derivative is positive. A presentation of all the comparative statics in the constrained cases would be extremely complex and of questionable value. Thus, an examination of the impact of changing parameters on the constrained optimum is saved for the numerical analysis in Section V. IV. AN EMPIRICAL TEST The structure of this paper rests on the notion that lot value is a Cobb-Douglas function of depth and frontage and that the frontage elasticity is greater than that of depth. If the elasticities were the same, unconstrained optimal lot configuration would not be an interesting question, because it would always pay to substitute depth for frontage given the relative costs. Similarly, the model yields absurd results if the elasticity on depth is greater than the elasticity on frontage. Thus, it is necessary to support the structure of this paper by adding empirical support for the necessary relative elasticities. The regression model is 1nP = lncll + /3[m FD] + (y - /3)[ln D] + {(Y - 70) + e,
(21)
where P = lot price, and Y = year of sale. The central hypothesis is that
OPTIMAL
103
LOT SIZE
the coefficient on the log of depth, (y - j?), is signiticantly negative. See Kowalski and Colwell [3] for a similar approach but one which includes more explanatory variables. The first data set consists of 20 residential land sales between 1970 and 1974 in Champaign-Urbana, Illinois. All sales were outside a l-mile ring around the urban center (i.e., Campustown for the University of Illinois) [6] so as to eliminate most vestiges of the value gradient and many other spatial complexities. The results show that the model explains a respectable proportion of the variance in selling prices with the coefficients having the expected signs and magnitudes.
In P = 3.274 + .867 (1nFD) - (SiZJ)(lnD) (1.887)
(.308) n = 20
+ (.;C$ (Y - 70).
.
R= = .3645
df=16
The annual rate of appreciation is estimated to be 5.48, but it is not significantly different from zero. Note that the coefficient on the log of area (i.e., the frontage elasticity) is .867. While this is less than unitary elasticity, it is not significantly less. However, one would not expect frontage elasticity to be much less than unitary. Therefore, the test for a signiticant difference here was doomed to failure by the effect of the small sample size on the standard errors. The principal hypothesis being tested is confirmed by the results. The coefficient on the log of depth is -.574 (i.e., the depth elasticity is .293 = .867 - .574). The fact that the coefficient of the log of depth is significantly negative (at the 90% level of significance) means that the depth elasticity is signitlcantly less than that for frontage. The model was tested on a second data set out of concern for the robustness of the results given the small sample size and the stark simplicity of the model specification. The second data set consists of 23 sales of vacant land from 1975 to 1982 in Bloomington-Normal, Illinois [6]. Again, the sample is limited to properties which are at least 1 mile from the urban center (i.e., State Farm World Headquarters). The results are very similar to the Champaign-Urbana results with regard to the signs and relative magnitudes of the coefficients.
lnP = 4.263 + .828 (lnFD) - (.444)(lnD) (1.645)
(.192)
n = 21
.
df=17
where MOS = month of sale (e.g., l/75
+ (.OOji)(MOS),
R= = .764 = 0, 2/‘75 = 1, and 2/‘76 = 13,
104
COLWELL
AND
SCHEU
etc.). The frontage elasticity is .828 (i.e., compared to .867 for Champaign-Urbana), while the calculated depth elasticity is .38 (i.e., a bit higher than the .293 for Champaign-Urbana). Although a slightly lower frontage elasticity and a slightly higher depth elasticity yield a narrower gap in Bloomington-Normal, the required relative magnitudes are maintained. Again, the frontage elasticity is not significantly different from unity. Also, the depth elasticity is not significantly different from the frontage elasticity even though its magnitude is on the order of what would be expected. Another difference between the two data sets seems to be that the explanatory power of the model is about twice as high for Bloomington-Normal as for Champaign-Urbana. The evidence is that the required magnitudes for the elasticities were estimated for two distinct samples. While the statistical significance of the results for each sample can be questioned, the reproducibility of the parameter estimates is striking. Estimates in Kowalski and Colwell [3] confirm this reproducibility. V. NUMERICAL
RESULTS
Utilizing the coefficients from the Champaign-Urbana regression as well as estimates of cost parameters provides an opportunity for viewing the results in quantitative terms. Figure 2 is produced using the following estimates for 1974: 8 = 32.78,
,tf = 7= &= I) = A,, =
assuming in 1974 (i.e., exp(3.274 + .054(74 - 70))),
.867, .293, .25, 37, 6500,
and
Dti = 100. The magnitudes of the cost parameters 6 and # were selected so that the results would initially be “in the ballpark” (i.e., below the constraints with no extreme values for F and 0). The implication of the magnitude of S is that costs, including raw land costs but excluding the frontage street and utility costs, were $10,890 per acre. The magnitude of Ic, represents the cost of building half a street and the sidewalk plus utilities per front foot (i.e., $37/front foot).
OPTIMAL
LOT
CONSTRAINED
FIGURE
105
SIZE
BY A,in
ONLY
3
Using (14) and (15), the unconstrained optimum is a frontage of 85.035 feet and a depth of 75.547 feet. This unconstrained optimum produces a slightly smaller lot area (i.e., 6424.1 square feet) than required for R2 zoning in the City of Champaign (i.e., 6500 square feet) [2]. It also produces less depth than is implicitly required by Champaign’s New Streets and Subdivision Ordinance (i.e., 100 feet) [l]. See Fig. 3 for the size and configuration of the optimal lot. The value of this lot is $5481, and the profit is $729. Using the lot area constraint and (17) and (18), the optimum is a frontage of 85.724 feet and a depth of 75.824 feet. This result does not satisfy the minimum depth requirement. Again, see Fig. 3 for the size and configuration of the optimal lot subject to the minimum lot area constraint. The value of this lot is $5526, and the profit is just pennies less than that for the unconstrained optimal lot. Using the minimum depth constraint and (19) yields an optimum frontage of 72.263 feet in combination with a depth of 100 feet. This exceeds the minimum lot area (i.e., 7226 versus 6500) and thus is the constrained optimum. This is also shown in Fig. 3. In this case, it is unnecessary to go to a corner solution with a frontage of 65 feet and a depth of 100 feet in order to satisfy both constraints. The value of this lot is $5168 and the profit is only $687. Changes in parameters of the value function result in changes in constrained optimal lot size and configuration. For example, as (Y declines ceteris paribur, optimal frontage declines until, at (Y= 32.3, the corner solution (F = 65; D = 100) becomes optimal. Similarly, as p declines, optimal frontage declines until, at p = 0.864, the developer is driven to the
106
COLWELL
AND
SCHEU
comer solution. The comer solution occurs for changes in depth elasticity at y = 0.289. Changes in cost function parameters also result in changes in constrained optimal lot size and configuration. If the parameter S increases, optimal frontage decreases until 6 = 0.259 where the comer solution obtains. As 6 decreases beyond 0.19, the area and depth constraints no longer bind. Changes in # are a bit more complex. As 4 increases, optimal frontage decreases until Ic, = 37.9 where the comer solution obtains. Continuing to increase # causes the minimum lot size constraint to bind at 4 = 42. Additional increases in $J cause optimal depth to increase and optimal frontage to decrease along the minimum lot size constraint. VI. HEDONICS
AND
EQUILIBRIUM
It is useful to review what has gone before with the tools of hedonic theory [5]. Figure 4 shows cost and value as functions of lot size, a metaphor for frontage and depth. The economic theory has identified the size which maximizes the difference between value and cost, profit. This profit maximizing size is shown as s* in Fig. 4. The impact of each parameter of the value and cost functions on size has been identified. Finally, the consequences of introducing a minimum lot size constraint, such as s^in Fig. 4, have been explored. What has not been done is to consider equilibrium conditions in a competitive market as would be conventional with hedonic theory. This was not done because it is unnecessary for the results that have been obtained. However, the model developed to this point is incapable of representing a sensible competitive (i.e., zero profit) equilibrium. This limitation results from the fact that both the value and cost functions have zero vertical
s*
FIGURES
SIZE
OPTIMAL
107
LOT SIZE
S*
SIZE
FIGURE 5
intercepts, so equilibrium at zero profit can only be found at zero size. This limitation can be overcome by introducing a fixed cost component, C = 5 + 6FD + #F, where $ = f&d cost per lot. The rationale for such a cost component is to represent the cost of provisions taken for utility hookups and/or to represent the cost of zoning and plat approval. The question then is what fixed cost is consistent with competitive equilibrium, the parameters of the value function, and the other parameters of the cost function. The answer is the profit obtained earlier for the unconstrained model 5 = $729. The cost function then becomes C = 729 f .25FD + 37F. Of course, utilizing this cost function and the value (i.e., hedonic) function used earlier yields zero profit at a frontage of 85.035 feet and a depth of 75.547 feet, the same dimensions as with the earlier unconstrained model. This shift of the cost function is illustrated in Fig. 5 using the size metaphor. VII.
CONCLUSIONS
This paper has shown that it is theoretically and empirically possible to develop a model of optimal lot size and configuration. The constituent parts
108
COLWELL
AND
SCHEU
of this model are a value function, a cost function, and a set of institutional constraints. It was necessary to utilize a lot value function resembling a family of depth rules which developed over the past hundred years. The function is Cobb-Douglas with the frontage elasticity of value larger than the depth elasticity of value. These relative elasticities were shown to exist by way of empirical investigations of land prices in Champaign-Urbana and Bloomington-Normal, Illinois. The costs of land development were divided into two components: one related to lot area and one related to frontage. The lot area component includes, most importantly, the cost of raw land. The frontage component primarily includes the cost of front streets. Comparative static analysis of the unconstrained optimum reveals a number of interesting results. First, changes in parameters of the value function have rather diverse effects on frontage and depth. An increase in the value of a lot’s first square foot causes frontage to increase but has no effect on depth. An increase in frontage elasticity causes frontage to increase, depth to decrease, and lot area to increase. On the other hand, an increase in depth elasticity causes both frontage and depth to increase while the frontage to depth ratio increases. Next, the effects of changes in parameters of the cost function are more intuitively appealing. For example, an increase in the cost of frontage causes depth to increase, frontage to decrease, and lot area to decrease. Thus, it may be worthwhile to pursue the hypothesis that a secular decline in the cost of frontage (e.g., streetbuilding) may be responsible for the consistent spatial pattern of lot configurations found in many older urban regions. Finally, an increase in the cost of land area causes depth, frontage, and the frontage to depth ratio to decrease. Thus, one should expect declines in the density and depth to frontage ratio of new developments as a result of the recent radical declines in the cost of raw land. The institutional constraints which provide a focus for this paper are the minimum lot size and the minimum lot depth. The minimum lot size is part of a zoning ordinance. The minimum lot depth, on the other hand, is implicit in subdivision regulations which specify the minimum distance between streets. With the minimum lot size constraint binding, it was shown that an increase in this constraint would cause frontage, depth, and the frontage to depth ratio to increase. With the minimum depth constraint binding, an increase in this constraint would cause frontage to decrease. The numerical results show that the model is capable of credibly representing land developer behavior. For example, it is not uncommon to find lots with depths exactly equal to the minimum depth but with substantial variation in frontage. An institutional rigidity such as this may be the source of the notion of a “standard depth” used so extensively in the depth
OPTIMAL
LOT SIZE
109
rule literature. Perhaps the most interesting finding from the numerical analysis is that the comer solution is the constrained optimum over a broad range of parameter magnitudes. That is, it should not be unusual to find lots which just meet both the minimum depth and minimum size constraints. In the final section, the theory is recast in the framework of a hedonic model in competitive equilibrium. It is possible to restructure the cost function so as to include fixed costs to produce positive lot size in the face of zero profit. This change in the model does not affect the size or the configuration of lots. REFERENCES 1. Champaign New Streets and Subdivision Ordinances, Council Bill No. 78-01, Article 4 1-48 (1978). 2. City of Champaign, IL, “Amended Zoning Ordinance of 1965,” Revised Draft 40-65 (1980). 3. J. G. Kowalski and P. F. Colwell, Market versus assessed values of industrial land, J. Amer. Real Estate Urban Econom. Assoc., 14, 361-373 (1986). 4. S. L. McMichael, “McMichael’s Appraising Manual,” Prentice-Hall, New York (1951). 5. S. Rosen, Hedonic prices and implicit markets: Product differentiation in pure competition, J. Polk Econom., 82, 34-55 (1974). 6. T. Scheu, “Site Valuation and Optimal Development,” unpublished dissertation, University of Illinois (1985).
OF URBAN
ECONOMICS
26,
90-109 (1989)
Optimal Lot Size and Configuration PETERF. COLWELLANDTIMSCHEU* Ofice of Real Estate Research, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801, and *Department of Finance, Western Michigan University, Kalamazoo, Michigan 49008
Received November 251986; revised February lo,1987 The developer chooses lot size and shape to maximize profit subject to institutional constraints. The price structure for lots is exogenous and Cobb-Douglas in form. Costs of land development are assumed to be both proportional to frontage and to lot area. The constraints are the minimum lot area from zoning and the minimum lot depth from subdivision ordinances. Numerical and empirical results support this theory of optimal development. One implication is that lower frontage cost from technological change may have led to wider-shallower lots, a result also consistent with lower land rent in a conventional urban model. 0 1989 Academic Press. Inc.
I. INTRODUCTION Most urban landscapes are characterized by a few highly visible features. Foremost among these is the clustering of tall buildings around a small number of nuclei. Another such feature is the discrete, rather than gradual, changes in building heights as distance to a nucleus increases. Still another of these visually obvious features is the spatial variation in the relative proportions of residential lots. Older and more centrally located lots tend to have a smaller frontage to depth ratio than newer and more peripherally located lots. This latter feature has been virtually ignored in the literature in contrast to the wealth of research on density gradients and discontinuities in density gradients. Land developers must consider the way lot size and configuration affect value as well as how they affect costs when deciding on the optimal lot design. In addition, zoning ordinances and subdivision regulations may constrain the choice. The purpose of this paper is to structure the problem of choosing lot size and configuration as one of maximking profit subject to institutional constraints. It is assumed that the price structure for lots of *The authors are ORER Professor of Real Estate, University of Illinois, and Assistant Professor of Finance, Western Michigan University, respectively. In part, this research was supported by the Office of Real Estate Research, University of Illinois at Urbana-Champaign. The authors benefited from comments by Roger E. Cannaday, James R. Follain, Larry Neal, Robert W. Resek, and an anonymous referee. All remaining errors are the responsibility of the authors. 90 0094-1190/89 Copyright All rights
$3.00
0 1989 by Academic Press, Inc. of reproduction in any form reserved.
OPTIMAL
91
LOT SIZE
various sizes and configurations is given as are the costs of developing lots and the institutional constraints. Thus, the partial equilibrium question of the developer’s choice is addressed in this paper. An empirical section follows the theoretical and institutional sections so as to provide justification for the relative sizes of frontage and depth elasticities used in the theoretical lot value function. A numerical section is provided to simulate developer choice in the context of specific parameter magnitudes. Finally, a section considers a modification to the theoretical model which allows the model to be developed in the context of competitive market equilibrium. Two assumptions are made for the purpose of tractability. First, all lots are assumed to be rectangular in shape. Second, comer lots are assumed to be valued in the same way as interior lots. That is, the fact that comer lots have frontage on a second side is ignored. II. THE UNCONSTRAINED
OPTIMUM
This section begins by examinin g the lot value function and then turns to the development cost function. Next, value and cost are brought together in a profit function. The conditions for profit maximization are derived. Finally, the implications of changing parameters for optimal lot size and configuration are explored. Lot Value The value of a lot is assumed to be a Cobb-Douglas function of frontage and depth [6]. There are several reasons for this assumption. First, on an empirical basis, the Cobb-Douglas function has been shown to be superior in this application to a variety of well-known functions. This superiority has been demonstrated by Box-Cox and Box-Tidwell functional form analysis [3]. Second, there is substantial historical precedent for using the Cobb-Douglas function in this application. This function has been used to value urban lots for about a century. The context in which such functions are used is called Depth Rules [6]. Third, the historical competitors of the Cobb-Douglas function tend to be somewhat odd (e.g., shifted log functions and segments of parabolas spliced to still other functions) and admit less readily to estimation. Fourth, there are stories, perhaps apocryphal, of very substantial data sets having been used to estimate Cobb-Douglas Depth Rules [4]. Thus, it is possible that what is lost in generality by using the Cobb-Douglas function rather than a general form may be more than compensated by specific results that have a good chance of being correct. There is probably greater reason to use the Cobb-Douglas form in this application than in most of its applications. The value of a lot is given as v = aF4Y
9
(1)
92
COLWELL AND SCHEU
where V = lot value, F = frontage, D = depth, a = the value of the first square foot of lot area, /3 = the frontage elasticity of value, y = the depth elasticity of value, and 0 < y < fi < 1. Under certain conditions, this function is consistent with well-known depth rules such as the HoffmanNeil, Reeves, Harper, Hobbs, and Milwaukee rules which have been recognized by courts and utilized by assessors for nearly a century [4, 61. If the depth elasticity of value is in the neighborhood of .5 to .585 and the frontage elasticity of value is unitary, the value function resembles these depth rules closely. The frontage rule of Ross suggests, however, that the frontage elasticity may be less than unitary [6]. Both frontage and depth elasticity are assumed here to be less than unitary for the sake of the model’s tractability. The model requires that the frontage elasticity be greater than the depth elasticity. There are plausible explanations for the relative magnitudes of the frontage and depth elasticities. For commercial and industrial property, frontage is associated with exposure, access to customers, and corporate identity. There may be similar arguments for residential property related to status, displays of wealth, or demonstration effects. However, the explanation may be quite different for residential property. If there is diminishing marginal utility from spatial separation between households, roads and zoning setbacks already provide substantial separation in the depth dimension and would cause the marginal benefit of frontage to exceed that of depth over the ranges one is likely to encounter. This question of relative elasticities is investigated empirically in a later section of this paper. Development Costs
Land developers’ costs are primarily the cost of raw land and the cost of providing access, vehicular and pedestrian access as well as access to utility services. The length of streets, sidewalks, and utilities (i.e., sewer, water, electric and gas) associated with developing a lot is essentially equal to the frontage of the lot.’ Thus, one component of development cost is proportional to the frontage of the lot. The other component of development cost is proportional to lot area. Thus, the total cost function may be expressed as C = SFD + #F,
where C = total development costs, and FD = lot area. The constant of proportionality on FD is primarily the sum of the cost of raw land per square foot, the prorated cost of land and improvements for a side street, ‘Developers may be reimbursed to some extent, For the installation of certain utilities. For example, these items can include underground electrical and gas service.
OPTIMAL
LOT
93
SIZE
and the prorated cost of the land and improvements for the crosswalk that may go between blocks. The constant of proportionality on frontage is primarily the total cost per front foot for land and improvements for streets as well as for utility development costs for which the developer is responsible. In addition, the costs of surveying, planning, and platting the subdivision may be related to both frontage and area. Projit The developer’s profit per lot is the lot value minus the costs of developing the lot. Thus, profit may be expressed as s = aFBDr - 6FD - #F 3
(3)
where v = profit per lot. The first-order conditions for profit maximization are r1 = cuj3F’+‘D’
- SD - II, = 0
(4)
r2 = ayFpD7-’
- 6F = 0.
(5)
From (4) and (5) it is simple to show that 6D +# -SF=--
/3D 6F ’
(6)
which says that the marginal rate of substitution in costs must equal the marginal rate of substitution in value. That is, an isovalue curve in frontage and depth space must be tangent to an isocost curve in that space. See Fig. 1 for the expansion path of isocost and isovalue tangency points. Eq. (5) can be rewritten
s I’=
V/FD’
which means that the depth elasticity must equal the ratio of the cost of raw land per square foot to the value of developed lots per square foot. The value of developed lots per square foot is another Cobb-Douglas function of frontage and depth: V/FD = o~F~--~D~-~. The slope of the contours of this value per square foot function is
-(l - ,@D/((l - Y)F),
(8)
94
COLWELL
AND
FIGURE
SCHEU
1
which is less steep than the slope of the isovalue curves because of the relative frontage and depth elasticities specified earlier. Condition (7) requires that the profit maximixi ng lot size and configuration be found along one special value per square foot contour. This is the one where value per square foot equals the ratio of the cost of land per square foot to the depth elasticity (i.e., value per square foot must equal a constant). See Fig. 1 for an illustration of this condition. So far, two distinct requirements have been offered for the profit maximizing lot size and configuration. These requirements are that frontage and depth be on the expansion path and that they be on a specifk isovalue-persquare-foot contour. Thus, the profit m axinking size and configuration are found at the intersection of these two functions. See Fig. 1. Although it is not needed for the determination of the profit maximizing lot size and configuration, additional insights may be obtained by focusing on (4). First, (4) can be rewritten
6 p = V/FD
-!L
+ V/F ’
which means that the frontage elasticity must equal the same ratio of cost to value as the depth elasticity plus the ratio of the cost of development per front foot to the value of a developed lot per front foot. Thus, the frontage elasticity exceeds the depth elasticity by the ratio of the cost of develop-
OPTIMAL
95
LOT SIZE
ment per front foot to the value of a developed lot per front foot. Second, (9) can be rewritten
v,FD=$+&.
(10)
Thus, the value per square foot must also be a particular function of depth. This requirement is also shown in Fig. 1. The second-order conditions for a maximum are
NP- w <0 ’
(11)
< 0,
(12)
F2 =22
=
and TllT22
-
TlZ2
=
&&I(1
-
P)
-
Y(’
PM’ 0.
03)
Conditions (11) and (12) require that the frontage and depth elasticities be less than unitary (i.e., /3, y < 1). Condition (13) utilizes the notion that 6 = yV/FD from (5) and can be further simplified to yield the requirement that the frontage elasticity must be greater than the depth elasticity (i.e., /3 > y), which is apparent from the first-order conditions. Solving (5) for the profit maximizing frontage as a function of depth and solving (4) and (5) for the profit maximizing depth yields F = [ ayDY-‘/S]l’(l-p),
(14)
and
D = W(P/u
- 1).
05)
Thus, the unconstrained optimal frontage and depth can be found in (14) and (15). This solves both the problem of optimal lot area and the problem of the optimal lot configuration in the absence of institutional constraints. Comparative
Statics
The land developer is, of course, interested in the way in which changes in parameters from the value and cost functions (i.e., across submarkets and through time) can change optimal lot size and configuration.
96
COLWELL
AND
SCHEU
A change in the value of the first square foot (i.e., when F and D both equal unity) has a radically different impact on the optimal depth and frontage: c?D -= aa
o. 7
aF
z
yDY-‘FP
= (1 - /?)S ’ O”
Thus, an increase in (Y has no effect on the optimal depth but causes optimal frontage to increase. While it would be expected that an increase in 1y would cause F to increase, it is unexpected that (Ywould have no impact on D. As suggested earlier, one way to view the profit maximizing choice is in the context of isovalue and isocost curves shown in Fig. 1. The profit maximizing combination of F and D is on the expansion path. Note that the expansion path is a horizontal line. Part of the explanation for this is the asymmetric treatment of F and D in the cost function. The expansion path would not be a horizontal line in the context of a cost function which treats F and D symmetrically. Another part of the explanation relates to the CobbDouglas form of the value function. When the value of the first square foot, (Y, changes, the isovalue map is unchanged, although each isovalue curve would relate to a different level of value after the change. This is a result of having chosen the Cobb-Douglas form for the value function. Thus, the expansion path and the profit maximizing D would be unchanged as a result of changing the value of the first square foot. A change in the frontage elasticity, /?, has opposite effects on optimal depth and frontage:
In F1-fi
(1 - p)’
(1 -Y)
JD
- (1 - /?)D ap
1
’ ”
Thus an increase in fl causes optimal depth to decrease as the isovalue curves become steeper, but it also causes optimal frontage to increase. Obviously, the lot frontage to depth ratio increases as the frontage elasticity increases. However, the impact of frontage elasticity on lot size is not obvious. This impact is 8FD dD -=F,p+Dq=F JP
I~F
aD (Y -P> -gql-p)
+
D In F1-@ (1-p)2
1
‘OS
OPTIMAL
LOT
97
SIZE
Thus, an increase in the frontage elasticity results in an increase in optimal lot size. A change in the depth elasticity, y, moves both optimal frontage and depth in the same direction: C?D -Tq=
b2(P/Y
aF au=
F Y(l - PI
p+
> 0; - v $1
l+ylnD-
- y)D-‘1
> 0.
An increase in y causes optimal depth to increase as the isovalue curves become flatter, but it also causes frontage to increase. As a result of these two derivatives having the same signs, it becomes interesting to know the effect of depth elasticity y on the lot configuration (i.e., F/D). The derivative of this ratio with respect to y is dF/D ay
= Fv
+ l/Day
ilF
i@ aF = FG + l/Day > 0.
This derivative is positive, so F/D increases as y increases. A change in the cost of frontage, I), causes optimal depth and frontage to move in opposite directions: aD v=
1
~(p/~ - 1) ’ O;
aF -=-
(1 - Y)F g
a4
(1 -P)D
< 0
a+
.
As 1c, increases and the isocost curves become steeper, optimal depth increases, while optimal frontage decreases. Thus, as street building technology has changed from bricks and blocks to continuously poured cement, thereby lowering the real #, new lots would be expected to have become wider and less deep. As a result of the opposite signs on these two derivatives, one would like to know the direction of the impact on the size of lots. The derivative of FD with respect to +!I is
84
aFD aF -=D,J,+F,JI=-
aD
[p-u1Fi?
a+
.
Thus, lot size increases as the cost of frontage decreases. Recall that it has been assumed that /3 > y and /3 -C 1. Finally, an increase in the cost of land area, 6, causes the optimal depth to decrease as the isocost curves become less steep. The impact on the
98
optimal
COLWELL
AND
SCHEU
frontage is less certain:
dD ---
as -
p(pJ
The sign of aF/l@
-1)
aF -=as
is determined
as
aF as
if l/a
--SO
(l$)
[ f+7-
1
(1 - Y) aD as .
2 - (l - ‘) aD -as.D
In Section V, it will be shown that magnitudes for the parameters that make empirical sense cause the sign of aF/a& to be negative. Assuming that capital costs the same everywhere, urban areas with higher surrounding land values should have smaller residential lots (i.e., smaller D and F), ceteris paribus. Similarly, platted lot sixes should, to some extent, fluctuate counter to fluctuations in surrounding agricultural land values within a single urban area. This has important implications for current urban growth. We should not only see an expansion of urban areas as a result of depressed agricultural land prices (i.e., a downward shift in the agricultural bid-rent curve), but we should also see a decrease in density (i.e., an increase in the sizes of new lots). The question that remains regarding the impact of the cost of land area is its impact on the configuration of lots. The derivative of F/D is aF/D
as
= F a1/D
a6
1 aF +imr
’ D2(l
- P)
The sign is aF/D
as
>
<
o
if
b/Y+-
1
_
(1
-
Y)
aD >
1
D(l - p) 88 < D6(1 - 13) ’
Again, the magnitudes of the parameters determine the sign. Magnitudes which produce a negative sign will be explored in Section V. Taking the negative sign as correct, it appears that higher prices of raw land would be associated with narrower-deeper lots. One might be tempted to try to explain the spatial distribution of frontage to depth ratios on the basis of land rent declining with increasing distance to the urban center. An alternative hypothesis is that the rising ratio as distance increases can be attributed to decreases in frontage prices through time as the lots were platted and developed.
OPTIMAL
III.
99
LOT SIZE
INSTITUTIONAL
CONSTRAINTS
Land developers face a number of institutional constraints. These constraints arise primarily as a result of zoning ordinances and subdivision regulations. Perhaps most obviously, there are minimum lot sizes. Less obviously, there are minimum lot depths which work in a similar way to the requirements placed on the builder for minimum backyards and front setbacks. For example, a new streets and subdivision ordinance may allow for a minimum of 260 feet between the centers of streets. After subtracting 60 feet required for right of way from the center of the streets, the land developer is left with a minimum of 200 feet for lots. With lots back to back, each lot must be at least 100 feet deep. Land developers face regulations related to maximum frontage and depth or, in other words, the frequency of side streets and crosswalks through the block. However, these regulations are not likely to be binding constraints. For example, side streets may be no more than 1200 feet apart if there is a lo-foot crosswalk right of way in the middle of the block. This leaves a maximum frontage and a maximum depth of 565 feet on each side of the crosswalk after allowing for right of way on the streets. Builders face additional constraints. These relate to the placement and size of the structure on the lot. Included in these constraints are floor to area ratios, open space ratios, and sideyard requirements in addition to the backyard and front setback requirements mentioned earlier. Private constraints (i.e., deed restrictions and covenants) may include minimum house size and more rigorous setbacks. The constraints that are the focus of this paper are the minimum lot area and the minimum depth faced by land developers. The constraints are FD 2 Ati,
the minimum
lot area?
and D2Dh,
the minimum
The resulting feasible set is illustrated
lot depth.
in Fig. 2.
Constrainted Optimum The problem is to choose positive values of F and D so as to maximize a = aFpDY - 6FD - I/IF subject to FD 2 Ati,, DzDti.
100
COLWELL
AND
SCHEU
D
F FIGURE
The Lagrangian
2
function is
L= aFBDDy- 6FD-t/F+X,(FD-Ati)+h,(D
-D,,).
(16)
Application of the Kuhn-Tucker conditions, which are necessary and sufficient for maximum profit (i.e., due to the concavity of the profit function), yields
g
=/3ctFB-IDY-
SD -t,b-X,D
i3L -JD =yaF@-'6F-A,F-X, FD-A,,>0 D-D,,>0 A, 2 0 A, 2 0.
10 I 0
It is necessary to explore two important cases, the case of neither constraint binding haying been taken up in Section II. These two cases (i) where the minimum lot area is the binding constraint and (ii) where the minimum depth is the binding constraint. Case 1. The optimal frontage and depth would be
F-
[$//aA~&3
- y)]l~B-u-l),
(17)
and
D=A,,JF.
(18)
OPTIMAL
As the minimum changes as
LOT
lot area constraint
101
SIZE
increases, the optimal
frontage
Thus, an increase in A,, would cause frontage to increase. Similarly, increase in A,, would cause depth to increase as
an
As a result of the signs of the above two derivatives being positive, the interesting question is the direction of the impact of an increase of A,, on the frontage to depth ratio: aF aF/D -=aA,,a+(yAeF a&in =
1
al/D
2/4&8)/h3+Y) 1 _ ;
[ a(;-
J
+ @-&8)/(1-fl+Y)
-l’(l-B+y)
_
1 1 j
B, yA-.-Y)/(‘~+‘)l
The sign is if A(p-Y)/(l-b+Y)
$
mm
-1-B
Y
*
As 6 approaches /3 at magnitudes greater than .5 the sign is positive, while at magnitudes below .5 the sign is negative. With /3 close to unity, it is unlikely that this sign would be negative. Thus, the F/D should be expected to increase as the minimum lot area constraint increases. Cuse 2. The optimal frontage and depth are
F = [ “%i&&+]
-+‘),
and D = Oh,,,
(20)
102
COLWELL AND SCHEU
As the minimum according to
depth constraint increases, the optimal frontage changes
aF
6 SD,, + Ic,I ’
ao,,=
The sign of this derivative is determined aF ao,,
i O
ifDti5
as v+ s(1 -y).
Again, the sign depends on the magnitudes of the parameters. However, some things can be said about the key parameters. Although 4 and 6 are likely to vary greatly through space, the ratio of 1c,to S is important for the sign of aF/aD& These two parameters are likely to move together, because S, the price of raw land, is imbedded in #, the price of frontage (i.e., right-of-way costs are part of #). Thus, #/S is not likely to be volatile. Our judgment is that at depth elasticities (y) below .3, one can assume that $/S is insufficiently large to cause dF/aD,, to be positive. However, the probability that this derivative is negative declines as the initial minimum depth requirement decreases and as the depth elasticity increases. At depth elasticities at or above those of traditional depth rules (i.e., .5 to .585), one can be quite sure that the derivative is positive. A presentation of all the comparative statics in the constrained cases would be extremely complex and of questionable value. Thus, an examination of the impact of changing parameters on the constrained optimum is saved for the numerical analysis in Section V. IV. AN EMPIRICAL TEST The structure of this paper rests on the notion that lot value is a Cobb-Douglas function of depth and frontage and that the frontage elasticity is greater than that of depth. If the elasticities were the same, unconstrained optimal lot configuration would not be an interesting question, because it would always pay to substitute depth for frontage given the relative costs. Similarly, the model yields absurd results if the elasticity on depth is greater than the elasticity on frontage. Thus, it is necessary to support the structure of this paper by adding empirical support for the necessary relative elasticities. The regression model is 1nP = lncll + /3[m FD] + (y - /3)[ln D] + {(Y - 70) + e,
(21)
where P = lot price, and Y = year of sale. The central hypothesis is that
OPTIMAL
103
LOT SIZE
the coefficient on the log of depth, (y - j?), is signiticantly negative. See Kowalski and Colwell [3] for a similar approach but one which includes more explanatory variables. The first data set consists of 20 residential land sales between 1970 and 1974 in Champaign-Urbana, Illinois. All sales were outside a l-mile ring around the urban center (i.e., Campustown for the University of Illinois) [6] so as to eliminate most vestiges of the value gradient and many other spatial complexities. The results show that the model explains a respectable proportion of the variance in selling prices with the coefficients having the expected signs and magnitudes.
In P = 3.274 + .867 (1nFD) - (SiZJ)(lnD) (1.887)
(.308) n = 20
+ (.;C$ (Y - 70).
.
R= = .3645
df=16
The annual rate of appreciation is estimated to be 5.48, but it is not significantly different from zero. Note that the coefficient on the log of area (i.e., the frontage elasticity) is .867. While this is less than unitary elasticity, it is not significantly less. However, one would not expect frontage elasticity to be much less than unitary. Therefore, the test for a signiticant difference here was doomed to failure by the effect of the small sample size on the standard errors. The principal hypothesis being tested is confirmed by the results. The coefficient on the log of depth is -.574 (i.e., the depth elasticity is .293 = .867 - .574). The fact that the coefficient of the log of depth is significantly negative (at the 90% level of significance) means that the depth elasticity is signitlcantly less than that for frontage. The model was tested on a second data set out of concern for the robustness of the results given the small sample size and the stark simplicity of the model specification. The second data set consists of 23 sales of vacant land from 1975 to 1982 in Bloomington-Normal, Illinois [6]. Again, the sample is limited to properties which are at least 1 mile from the urban center (i.e., State Farm World Headquarters). The results are very similar to the Champaign-Urbana results with regard to the signs and relative magnitudes of the coefficients.
lnP = 4.263 + .828 (lnFD) - (.444)(lnD) (1.645)
(.192)
n = 21
.
df=17
where MOS = month of sale (e.g., l/75
+ (.OOji)(MOS),
R= = .764 = 0, 2/‘75 = 1, and 2/‘76 = 13,
104
COLWELL
AND
SCHEU
etc.). The frontage elasticity is .828 (i.e., compared to .867 for Champaign-Urbana), while the calculated depth elasticity is .38 (i.e., a bit higher than the .293 for Champaign-Urbana). Although a slightly lower frontage elasticity and a slightly higher depth elasticity yield a narrower gap in Bloomington-Normal, the required relative magnitudes are maintained. Again, the frontage elasticity is not significantly different from unity. Also, the depth elasticity is not significantly different from the frontage elasticity even though its magnitude is on the order of what would be expected. Another difference between the two data sets seems to be that the explanatory power of the model is about twice as high for Bloomington-Normal as for Champaign-Urbana. The evidence is that the required magnitudes for the elasticities were estimated for two distinct samples. While the statistical significance of the results for each sample can be questioned, the reproducibility of the parameter estimates is striking. Estimates in Kowalski and Colwell [3] confirm this reproducibility. V. NUMERICAL
RESULTS
Utilizing the coefficients from the Champaign-Urbana regression as well as estimates of cost parameters provides an opportunity for viewing the results in quantitative terms. Figure 2 is produced using the following estimates for 1974: 8 = 32.78,
,tf = 7= &= I) = A,, =
assuming in 1974 (i.e., exp(3.274 + .054(74 - 70))),
.867, .293, .25, 37, 6500,
and
Dti = 100. The magnitudes of the cost parameters 6 and # were selected so that the results would initially be “in the ballpark” (i.e., below the constraints with no extreme values for F and 0). The implication of the magnitude of S is that costs, including raw land costs but excluding the frontage street and utility costs, were $10,890 per acre. The magnitude of Ic, represents the cost of building half a street and the sidewalk plus utilities per front foot (i.e., $37/front foot).
OPTIMAL
LOT
CONSTRAINED
FIGURE
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SIZE
BY A,in
ONLY
3
Using (14) and (15), the unconstrained optimum is a frontage of 85.035 feet and a depth of 75.547 feet. This unconstrained optimum produces a slightly smaller lot area (i.e., 6424.1 square feet) than required for R2 zoning in the City of Champaign (i.e., 6500 square feet) [2]. It also produces less depth than is implicitly required by Champaign’s New Streets and Subdivision Ordinance (i.e., 100 feet) [l]. See Fig. 3 for the size and configuration of the optimal lot. The value of this lot is $5481, and the profit is $729. Using the lot area constraint and (17) and (18), the optimum is a frontage of 85.724 feet and a depth of 75.824 feet. This result does not satisfy the minimum depth requirement. Again, see Fig. 3 for the size and configuration of the optimal lot subject to the minimum lot area constraint. The value of this lot is $5526, and the profit is just pennies less than that for the unconstrained optimal lot. Using the minimum depth constraint and (19) yields an optimum frontage of 72.263 feet in combination with a depth of 100 feet. This exceeds the minimum lot area (i.e., 7226 versus 6500) and thus is the constrained optimum. This is also shown in Fig. 3. In this case, it is unnecessary to go to a corner solution with a frontage of 65 feet and a depth of 100 feet in order to satisfy both constraints. The value of this lot is $5168 and the profit is only $687. Changes in parameters of the value function result in changes in constrained optimal lot size and configuration. For example, as (Y declines ceteris paribur, optimal frontage declines until, at (Y= 32.3, the corner solution (F = 65; D = 100) becomes optimal. Similarly, as p declines, optimal frontage declines until, at p = 0.864, the developer is driven to the
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comer solution. The comer solution occurs for changes in depth elasticity at y = 0.289. Changes in cost function parameters also result in changes in constrained optimal lot size and configuration. If the parameter S increases, optimal frontage decreases until 6 = 0.259 where the comer solution obtains. As 6 decreases beyond 0.19, the area and depth constraints no longer bind. Changes in # are a bit more complex. As 4 increases, optimal frontage decreases until Ic, = 37.9 where the comer solution obtains. Continuing to increase # causes the minimum lot size constraint to bind at 4 = 42. Additional increases in $J cause optimal depth to increase and optimal frontage to decrease along the minimum lot size constraint. VI. HEDONICS
AND
EQUILIBRIUM
It is useful to review what has gone before with the tools of hedonic theory [5]. Figure 4 shows cost and value as functions of lot size, a metaphor for frontage and depth. The economic theory has identified the size which maximizes the difference between value and cost, profit. This profit maximizing size is shown as s* in Fig. 4. The impact of each parameter of the value and cost functions on size has been identified. Finally, the consequences of introducing a minimum lot size constraint, such as s^in Fig. 4, have been explored. What has not been done is to consider equilibrium conditions in a competitive market as would be conventional with hedonic theory. This was not done because it is unnecessary for the results that have been obtained. However, the model developed to this point is incapable of representing a sensible competitive (i.e., zero profit) equilibrium. This limitation results from the fact that both the value and cost functions have zero vertical
s*
FIGURES
SIZE
OPTIMAL
107
LOT SIZE
S*
SIZE
FIGURE 5
intercepts, so equilibrium at zero profit can only be found at zero size. This limitation can be overcome by introducing a fixed cost component, C = 5 + 6FD + #F, where $ = f&d cost per lot. The rationale for such a cost component is to represent the cost of provisions taken for utility hookups and/or to represent the cost of zoning and plat approval. The question then is what fixed cost is consistent with competitive equilibrium, the parameters of the value function, and the other parameters of the cost function. The answer is the profit obtained earlier for the unconstrained model 5 = $729. The cost function then becomes C = 729 f .25FD + 37F. Of course, utilizing this cost function and the value (i.e., hedonic) function used earlier yields zero profit at a frontage of 85.035 feet and a depth of 75.547 feet, the same dimensions as with the earlier unconstrained model. This shift of the cost function is illustrated in Fig. 5 using the size metaphor. VII.
CONCLUSIONS
This paper has shown that it is theoretically and empirically possible to develop a model of optimal lot size and configuration. The constituent parts
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of this model are a value function, a cost function, and a set of institutional constraints. It was necessary to utilize a lot value function resembling a family of depth rules which developed over the past hundred years. The function is Cobb-Douglas with the frontage elasticity of value larger than the depth elasticity of value. These relative elasticities were shown to exist by way of empirical investigations of land prices in Champaign-Urbana and Bloomington-Normal, Illinois. The costs of land development were divided into two components: one related to lot area and one related to frontage. The lot area component includes, most importantly, the cost of raw land. The frontage component primarily includes the cost of front streets. Comparative static analysis of the unconstrained optimum reveals a number of interesting results. First, changes in parameters of the value function have rather diverse effects on frontage and depth. An increase in the value of a lot’s first square foot causes frontage to increase but has no effect on depth. An increase in frontage elasticity causes frontage to increase, depth to decrease, and lot area to increase. On the other hand, an increase in depth elasticity causes both frontage and depth to increase while the frontage to depth ratio increases. Next, the effects of changes in parameters of the cost function are more intuitively appealing. For example, an increase in the cost of frontage causes depth to increase, frontage to decrease, and lot area to decrease. Thus, it may be worthwhile to pursue the hypothesis that a secular decline in the cost of frontage (e.g., streetbuilding) may be responsible for the consistent spatial pattern of lot configurations found in many older urban regions. Finally, an increase in the cost of land area causes depth, frontage, and the frontage to depth ratio to decrease. Thus, one should expect declines in the density and depth to frontage ratio of new developments as a result of the recent radical declines in the cost of raw land. The institutional constraints which provide a focus for this paper are the minimum lot size and the minimum lot depth. The minimum lot size is part of a zoning ordinance. The minimum lot depth, on the other hand, is implicit in subdivision regulations which specify the minimum distance between streets. With the minimum lot size constraint binding, it was shown that an increase in this constraint would cause frontage, depth, and the frontage to depth ratio to increase. With the minimum depth constraint binding, an increase in this constraint would cause frontage to decrease. The numerical results show that the model is capable of credibly representing land developer behavior. For example, it is not uncommon to find lots with depths exactly equal to the minimum depth but with substantial variation in frontage. An institutional rigidity such as this may be the source of the notion of a “standard depth” used so extensively in the depth
OPTIMAL
LOT SIZE
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rule literature. Perhaps the most interesting finding from the numerical analysis is that the comer solution is the constrained optimum over a broad range of parameter magnitudes. That is, it should not be unusual to find lots which just meet both the minimum depth and minimum size constraints. In the final section, the theory is recast in the framework of a hedonic model in competitive equilibrium. It is possible to restructure the cost function so as to include fixed costs to produce positive lot size in the face of zero profit. This change in the model does not affect the size or the configuration of lots. REFERENCES 1. Champaign New Streets and Subdivision Ordinances, Council Bill No. 78-01, Article 4 1-48 (1978). 2. City of Champaign, IL, “Amended Zoning Ordinance of 1965,” Revised Draft 40-65 (1980). 3. J. G. Kowalski and P. F. Colwell, Market versus assessed values of industrial land, J. Amer. Real Estate Urban Econom. Assoc., 14, 361-373 (1986). 4. S. L. McMichael, “McMichael’s Appraising Manual,” Prentice-Hall, New York (1951). 5. S. Rosen, Hedonic prices and implicit markets: Product differentiation in pure competition, J. Polk Econom., 82, 34-55 (1974). 6. T. Scheu, “Site Valuation and Optimal Development,” unpublished dissertation, University of Illinois (1985).