Optimization of Subdivision Development.

Journal of Real Estate Finance and Economics, 3: 195-206 (1990) © 1990 Kluwer Academic Fublishen

Optimization of Subdivision Development ROGER E. CANNADAY Associate Professor of Finance. University of Illinois ai Urbana-Champaign, Urbana, IL 61801 PETER F. COLWELL OREM Professor of Real Estate, University of Illinois at Urbana-Champaign, Urbana. IL 61801

Abstract Optimal loi size and configuration are determined by the maximization of profit per unit of land area. Ixn value is assumed to be a Cobb-Douglas function of frontage and depth. The cost of developing a lot is assumed to be related to lot area, frontage, and a fixed cost parameter. Tlie forms of the value function, the cost function, and the objective function are rationalized. The impacts of changes in the value and cost function parameters on optimal frontage, depth, frontage to depth ratio, and lot area are investigated.

In analyzing the optimization of subdivision development, one should consider the way that lot configuration affects value (sales price) as well as cost of development. To ensure that Uital profit for the subdivision is maximized in the context of lot configuration choice, the objective function should be the maximization of profit per unit of land area. The partial equilibrium question regarding the implications of the profit-maximizing developer for choice of lot configuration is addressed in this article. The model developed in this article abstracts from the land area and other inputs required for cross streets and the resulting corner lot valuation issues. Lot depth is defined to include one-half of the width of the street and alley right-of-way (ROW) along with the depth of the lot itself. It is assumed that the price structure and cost of developing lots of varying sizes and configurations are given and that all lots will have equal dimensions and be rectangular in shape. Recently, Colwell and Scheu (1989) (hereafter C-S) addressed a similar question l^ developing a model with the objective of maximizing profit per lot. Unfortunately, their formulation of the objective function does not ensure that profit for the entire subdivision is maximized since the number of lots in the subdivision may vary as the size and configuration of lots vary. For an interesting comparison, see Edelson (1975) for a partial equilibrium model based on a different set of assumptions. Edelson considers such Actors as income of potential buyers and the relationship between property taxes and public services of alternative jurisdictions in developing his model for optimal subdivision development. This introductory section is followed ty a section on the development of the theoretical model. The next section provides a discussion of the impact of changes in the magnitudes of parameters in the profit function on the optimal size and configuration of lots. This is

196

ROGER E. CANNADAY AND PETER R COLWELL

followed by a section that compares the results of this article with the C-S results. We see that seemingly innocuous changes in the objective function lead to substantially different results. The final section is a summary of major conclusions.

1. Optimal frontage and depth First, functions for lot value and development cost per square foot are developed. Then, value and cost are brought together in a profit-per-square-foot function, and conditions for profit maximization are derived.

IJ. Lot value Assume that the value of a lot is as given in equation (1) of C-S (1989, p. 91): V = oF^D-^

a)

where V F D a

= lot value, = frontage of lot, = depth of lot plus one-half of ROW width, = the value of the first square foot of lot area where lot area is defmed as FD; i.e., if F = D = I, V = a (this definition is simply an artifact of the choice of functional form for the value equation), 0 = the frontage elasticity of value, 7 = the depth elasticity of value, and 0 < 7 < /3 < 1. The value per square foot of lot area is derived by dividing equation (1) by the lot area, FD, as follows: V/FD = oF^-'Zr-'.

(2)

A graphical illustration of the value function is presented in figure 1. Some of the reasons for choosing a Cobb-Douglas form of the value ftinction for equation (1) are summarized in C-S (1989, pp. 91, 92). These reasons include the results of previous empirical studies (some of which utilize functional form analysis) and historical precedent based on the use of various depth rules (for example, Hoffnian-Neill). A comprehensive discussion of depth rules is included in Scheu (1985). The specific value function chosen for this article is intended to represent valueconfiguration relationships only in those neighborhoods and over the range of lot sizes where subdivision takes place. It would be accurate to say that the value function must

OPTIMIZATION OF SUBDIVISION DEVELOPMENT

197

V/FD

Figure I. Value per square foot.

look very different if land assembly is the norm. These issues have been explored as plattage versus plottage by Colwell and Sirmans (1978, especially pp. 514-516). The value function in this application must reflect plattage, a value increment from subdivision. We assume that both the frontage and depth elasticities arc less than tinity in the context of subdivision. Furthermore, we assume that frontage elasticity is greater than depth elasticity. These assumptions are supported by the empirical results of C-S and Kowalski and Colwell (1986) (hereafter K-C). The economic rationale for the relative m^nitudes offrontageand depth elasticities should have both demand and supply dimensions. Regarding the demand side, assume there is diminishing marginal utility from the spatial separation of households. Institutional constraints prcTvide substantial separation in the depth dimension. This would cause the marginal benefit of frontage to exceed that of depth over the ranges that one is likely to encounter. On the supply side, the relative elasticities must reflect the relative costs of subdividing in the two dimensions. Suppose very deep lots were subdivided by puncturing their depth with streets. What would the value be of each new subdivided lot? Given equation (1) for the value of each initial deep lot, each new lot is worth \~^V where X equals the number of smaller lots. The total value of the smaller lots created by subdividing a larger one is \^~''V. The amount of increase in the value (the plattage value) is \^~'^V - V. Therefore, X'"'*' - 1 is equal to the proportionate increase. Now suppose very wide lots were subdivided in terms of frontage. Again, given eqtiation (I) for the value of each initial wide lot, each new lot is worth \~^V. In a manner similar to that for subdividing deep lots, it can be shown that X'"^ - 1 is equal to the proportionate increase in value from subdividing each wide lot. The cost of adding frontage (subdividing in d^th) is certainly greater than the cost of dividing up existing frontage. In equilibrium, X'"^ - 1 < X'"'^ — 1 and 1 < X^"'''. Therefore, j3 must be greater than 7 since X is greater than unity.

1.2. Development cost We assume that the development cost of a lot is as given in equation (2) in C-S (1989, p. 92) with the addition of a fixed cost per lot item as follows:

198

ROGER E. CANNADAY AND PETER F. COLWELL

C = 6FD

(3)

where C 6 V' i

= = = =

total development cost per lot, cost per square foot of lot area, cost per front foot, and fixed cost per lot.

Note that while the cost of building on a lot may be a function of depth, the cost of developing the lot is not. For example, one could imagine that deeper lots might require longer driveways. The cost of building longer driveways may influence the price of lots. This affects the lot value function, not the cost function. The cost per square foot of lot area is derived by dividing equation (3) l^ FD:

OFD = 5 + ^/D +

(4)

A graphic illustration of the cost function is presented in figure 2. The first component of development cost, 6, is proportional to lot area and primarily represents the cost of raw land per square foot. The second component, ^, is proportional lo the frontage of the lot and includes costs related to front streets, back all^s, sidewalks, and utilities (water, sewer, etc.). The fixed cost component, ^, includes any costs that are proportional to the number of lots such as marketing, surveying, planning, and/or platting costs not related to frontage or area and possibly part of the cost of obtaining any approvals needed from public agencies. The inclusion of a fixed cost component is necessary for tractability when the objective function is the maximization of profit per unit of land area.

C/FD

Figure 2. Cost per square foot.

199

OPTIMIZATION OF SUBDIVISION DEVELOPMENT

1.3. Developer's profit The developer's profit per square foot is then the value per square foot minus the cost per square foot, as follows: TIFD

- 6 -

=

(5)

A graphic illustration of the profit function is presented in figure 3. The first order conditions for profit maximization are derived as follows: djir/FD) dF

= 0. and

dir/FD)

(6)

= 0.

(7)

Solving equation (6) for the profit-maximizing frontage as a function of depth, and solving equation (7) for the profit-maximizing depth as function of frontage yields , and D =

Solving

(8) (9)

-I

equation

D = 1

(6)

^

Figure 3, Profit per square foot.

for 1 \

depth

as

a

function

of

frontage

yields (10)

200

ROGER E. CANNADAY AND PETER F. COLWELL

Setting equation (10) equal to equation (9) and solving for F yields

(1 -m

^ ^

Substituting the right side of equation (11) for F in equation (10) yields:

r t>-v 1 "T 7T-r r . La(l - )5)'-^03 - 7/J

(12)

It is necessary to examine the second order conditions to ensure that we have found a maximum rather than a minimum. The Hrst two second order conditions for a maximum are: = a{^ - 1)03 - DF^-'D-r-'

_ . ^

^ ' 3

2^

< o, and

JL

(13)

0.

(14)

Rearranging terms in equation (13), it can be shown that F~^£)"'[a()3 - l)(/3 - 2)F^D''-2^]

< 0

(15)

only if:

£ > (g - m - iw Since aF^D"^ is equal to K according to equation (1), inequality (16) can be simplified as follows: V.

(17)

Dividing both sides of inequality (17) l^ V and rearranging other terms yields g) .

(18)

Assume that the frontage elasticity, 0, is in the range of 0.6 to 0.9 based on the empirical results of K-C (1986, p. 368) and C-S (1989, pp. 103, 104). Then the second derivative shown in equation (13) will be negative when fixed development cost per lot is greater than somewhere between 5 percent and 28 percent of lot value.

OPTIMIZATION OF StJBDIVISION DEVELOPMENT

201

Continuing with an examination of the second order conditions by rearranging terms on the right side of equation (14), it can be shown that (7

7

0

(19)

only if: ^f+

f ^ (1 - 7)(2 - 7) 2 '

V

.20)

Using a depth elasticity, 7, in the range of 0.25 to 0.4 based on earlier empirical results (C-S, 1989, pp. 103, 104; and K-C, 1986, p. 368), the second derivative shown in equation (14) will be negative when the sum of the fixed development cost and frontage related cost per lot is greater than somewhere between 48 percent and 66 percent, respectively, of lot value. Casual empiricism (i.e., familiarity with actual subdivision cost) suggests that this condition is met. Finally, to ensure a saddle point has not been found, the cross partial derivative is needed, as follows:

f^

oFdD

(21)

a(y

For the points found by equations (11) and (12) to represent a maximum, the second derivatives represented by equations (13) and (14) must both be negative and the following (the determinant of the Hessian) must be positive, that is:

dF" J

I

dD^ j

I dFdD

Let T||, T22, and F , ; represent the three derivatives in equation (22). The first two second order conditions fbr a maximum require that ir^i and ^22 both be negative. Thus, the product of T|, and ir22 must be positive. We know that TJJ must be positive. Therefore, a maximum requires that jriiT22 be greater than T^J. This is likely to be the case if/3 and 7 are in the range supported by previous empirical evidence. However, as )3 approaches one, TTJJ is likely to be greater than ^11^22. Assuming all three second order conditions necessary for a maximum are met, the optimal frontage and depth can be found by equations (U) and (12). In the absence of institutional constraints, this solves the problems of optimal lot area and optimal lot configuration, based on rectangular lots of equal dimensions and the other simplifying assumptions noted previously.

2. Comparative statics Observers of urban development would like to know the impact of changes in the magnitudes of parameters in the profit function across submarkets or through time. The goal

202

ROGER E. CANNADAY AND PETER F. COLWELL

of this section is to determine, to the extent possible, the direction of the impact of parameters changing on the optimal size and configuration of lots. A change in the value of the first square foot, a, has different impacts on optimal frontage and depth. This may be seen by taking partial derivatives of equations (11) and (12) as follows: ^

= 0; and

^= da

(23)

--ct''D< 7

0.

(24)

Thus, an increase in a has no effect on the optimal frontage but causes the optimal depth and optimal lot size to decrease and the optimal frontage to depth ratio to increase. The results found when considering changes in frontage elasticity are similar to those found when maximizing profit per lot (C-S, 1989, p. 96). Tkkdng partial derivatives witb respect to a change in tbe frontage elasticity, j3, yields:

?l = H^ ~ y^ > 0; and 3/3 (1 - 0y^k 'n-

+ in

~

(25)

- —•—

.

(26)

Given the likely magnitudes of the parameters in equation (26), the result is almost certain to be negative. Thus, a change in 0 is likely to have opposite effects on optimal frontage and depth. An increase in j3 would cause optimal frontage to increase but cause optimal depth to decrease, thereby increasing the lot frontage to depth ratio. The impact of a change in fix>ntage elasticity on optimal lot size is not as obvious. This impact can be determined by the following:

3)3

30

9|8

Substituting from equations (25) and (26) yields

dFD

FD[.

^|^ , „ \ - a

7

It is not clear what tbe impact on optimal lot size of a change in )3 would be if 7 and 5 are in the range supported by previous empirical evidence. However, as /3 approaches unity, an increase in /? will cause the optimal lot size to increase. As /3 approaches 7, an increase in /3 will cause the optimal lot size to decrease.

OPTIMIZATION OF SUBDIVISION DEVELOPMENT

203

A change in depth elasticity, 7, has a similar impact on optimal frontage and depth. This may be seen by taking partial derivatives as follows:

37 dD dy

< 0;

(29)

- inD];

(30)

(1 D 7

dFD ^ FD

dy

7 L/3 - 7

dy

J

dy

^7

(1 7D L

; and

0- y

(31) -

^,. ^ .. . •

(32)

An increase in depth elasticity results in a decrease in the optimal frontage. For likely magnitudes of the parameters in equations (30) and (31), the results are almost certain to be negative. Therefore, an increase in depth elasticity also is likely to result in a decrease in the optimal depth and the optimal lot size. It is not clear what the impact on optimal frontage to depth ratio would be for the empiricaUy supported range of 0 and 7. However, as 0 approaches unity or as 0 approaches 7, the derivative in equation (32) becomes negative. A change in tbe cost of frontage, ^, has different impacts on optimal frontage and depth. These impacts can be determined by the following: 3^ _ _ d\l^

f(^ (1 -

< 0; and

T)

dD 0D ^ ^ -- = ~ > 0.

(33) ,-., (34)

As ^ decreases, optimal frontage increases and optimal depth decreases, thereby increasing the optimal frontage to depth ratio. This is consistent with the notion that as street and/or sewer building technology improves, decreasing 0, new lots would be expected to have more frontage and less depth. Since the signs on the above two derivatives are opposite, it is not immediately clear what the impact of a change in cost of frontage is on tbe optimal lot size. The partial derivative of lot area, FD, with respect to l/^ is as follows: dip

yyl.

(1

y)D

-

The direction of change in optimal lot size will depend upon whether the optimal frontage before the change is less than or greater than j ^ ~ ^ twhich is equal to {y!0)F*}. Since we have assumed that 7 is less than 0, the optimal frontage will be greater. Therefore, as 4/ increases, optimal lot size increases.

204

ROGER E. CANNADAY AND PETER F. COLWELL

A change in tbe fixed cost of development, ^, has similar impacts an optimal frontage and depth. This can be determined hy taking the following derivatives:

d(F/D) ^ _ ni - 0) , 03-7) ^ D{i - m' As fixed costs increase, optimal frontage, depth, and lot size increase. It is the number of lots that decreases. The direction of change in optimal frontage to depth ratio will depend upon whether the optimal frontage before the change is less tban or greater than ^ ^ ^ [which is equal to (7/(1 - (3))F*]. Given the likely magnitudes of 7 and /3, the optimal frontage will be less. Therefore, as i increases, the optimal frontage to depth ratio increases. Finally, change in the cost of land per square foot, 5, has no impact on optimal size or configuration. This is so because 6 does not appear in equations (11) or (12), the equations for optimal frontage and depth. While the mathematical rationale is obvious, there appears to be no obvious economic rationale.

3. Comparison with the C-S results In order to dramatize the substantial differences between the results of C-S and this article (C-C), the major results are summarized together. This includes the objective function and the comparative statics.

3.1. Objective Junction The focus of C-S is to maximize profit per lot based on the following objective function: T = aF^D"' - 5FD - V-F. In contrast, the focus of this article is to maximize profit per square foot based on: TTIFD

= oF^-'Zr-' - 8 -

The resulting repressions for optimal frontage and depth are quite different as are the comparative statics.

OPTIMIZATION OF SUBDIVISION DEVELOPMENT

205

3.2. Comparative statics The direction of the impact of changes in parameters on the optimal size and configuration of lots is different between C-S and C-C for more than two-thirds of the relationships considered. Theresultsof the analyses of comparative statics are contrasted directly in table 1. Table 1- Comparison of C-S and C-C results.

3D

dF C-S

+

36

cc

C-S

C-C

C-S

C-C

C-S

C-C

0

0

-

+

-f-

-1-

-

-t-

-

-

+

+

-

+

?

+ 0

0

+ + -

+

HA

-

^7 -

-

-

0

NA

3FD

diFJD)

NA

NA

? + 0

The comparative statics results are radically different for four of the parameters. In the C-S model, a change in value of the first square foot, a, has no impact on optimal depth, but an increase in a causes optimal frontage to increase. In the C-C model, a change in a has no impact on optimal frontage, but an increase in a causes optimal depth to decrease. The results found in C-S and C-C are generally opposite for changes in depth elasticity, 7. In the C-C model, change in the cost of land per square foot (5) has no impact on optimal size or configuration. In the C-S model, however, an increase in 6 causes all the lot dimensions and ratios examined to decrease. In the C-C model, an increase in fixed cost (^) causes all the lot dimensions and ratios considered to increase while the C-S model does not include a fixed cost component.

4. Conclusions This article develops a model of optimal lot size and configuration based upon the maximization of profit per unit of land area. To make development of the theoretical model tractable, a fixed cost per lot item is included in the cost function. The results have become more complex and most often different from those found when maximizing profit per lot. Nevertheless some things are clear. Changes in the magnitude of parameters such as frontage and depth elasticity have reasonably predictable impacts on most aspects of optimal lot size and configuration. An increase in the magnitude of the parameter for value of the first square foot has no effect on the optimal frontage, but causes the optimal depth and lot size to decrease and the optimal frontage to depth ratio to increase. Changes in magnitudes of cost parameters have a clear-cut impact on most aspects of optimal lot size and configuration. Surprisingly, change in the cost of land per square foot

206

ROGER E. CANNADAY AND PETER F, COLWELL

has no impact. Change in the cost of frontage has opposite impacts on optimal frontage and depth and change in the fixed costs per lot has the same direction of impact on optimal frontage and depth.

Acknowledgments This research was supported in part by the Office of Real Estate Research, University of Illinois, Urbana-Champaign. The authors would like to express their appreciation to their graduate research assistant, Tai-le Yang, for his helpful comments.

References Colwell, Peicr F. and Schcu, Tim. "Opumal Lot Size and Configuration." Journal of Urban Economics (March 1989), 90-109. Colwell, Peter F., and SimiAns, C.F "Area, Time, Centrality and the ^ u e of Urban Laod." Land Economics (November 1978), 514-519. Edelson, Noel M. "The Developer's Problem, or How to Divide a Piece of Land Most Profitably." Journal of Urban Economics (October 1975), 349-365. Kowalski. Joseph G., and Colwell, Peter F "Market Versus Assessed \Wues of Industrial Land." AREUEA Journal (Summer 1986), 361-373. Scheu, Tim. "Site V^uation and Optimal Development." Unpublished dissertation, University of Dlioois, 1985.