TOPOGRAPHIC AMENITIES, BUIL ING HLIGHT, AND T

SUPPLY OF URBAN HOL’SIIVC

This ar1v_-lc develops a housing supply model that tre.its ewphcitly the effects of location-,pecific amcnitw. The model cmp:oys a procluctwn funcrmn I-I which housing services are produced not

only by structures. but nlsc by access lo and VIC\S~ of I~xatwwpecific amenities. Housing supply

IS measured by building height. The model is tested rbit11 data from the city of Chicago. Access

to and view of Lake Mlshipan are found to have a signlfi i;nt eflect on the height of buildings.

1. Introduction

A key assumption in urban land use models such as those of Alonso f 1964), Mills (1967). and Muth (1969) is that the city lies on a featureless plain. Although this assumption leads to simple yet powerful analytical models, It precludes the possibil;ly of explaining the substantial impact of a city’s topography on land use patterns. For example, the city of Chicago lies on an essentially featureless plain; yet it is Impossible to explain the pailtt:rn of housing prices and density within the city without taking into account the pervasive influence of Lake Michigan. This paper develops and tests an

urban land use model that incorporates topographic features. It assumes that some topographic features, ;uch as lahes or parks, may be regarded as

location-specific amenities, and that housing units with a view of or accessibility to these amenities provide additional housing services.

182 R. Poflnrd, Topographic amenities, building height. and supply of urban housing

The use of building height as a direct measure of housing output offers a number of advantages. Since building height is directly observable. ali the pitfalls associated with the measurement of the physical stock of housing are avoided. In addition, the inclusion of building height as a determinant of housing output allows one to satisfactorily treat the value of visual amenities,

Preview. In the next section a housing supply model is developed in which housing producers respond to housing prices that are determined not only by accessibility to the Central Rusiness District (CBD), but by accessibility to and views of location-specific: amenities. A production function is employed that incorporates tile effect of building height on the productivity of inputs in producing housing. Assuming profit maximization by competitive housing producers yields equilibrium conditions for the optimal height of buildings. The equilibrium conditions are used to analyze the effects of amenities on the optimal height of bmldings.

In the following section the model is applied to estimate the effect of the Lake Michigan amenity on housing in the City of Chicago. Parameters of the model are estimated and discussed and the predictions of the model concerning building height are compared with a direct measure of building height.

In the fmal section the quantitative importance of amenitie> on housing expenditures and the supply of housing is discussed.

2. The basic model

The predominant uses of urban land are for residential, industrial or commercial purposes. According to traditi’Jna1 theory, competitive bidding among potential users will asure that land is allocated to the activity commanding the highest rent. The model developed below deals only with residential land use, so it implicitly assumes that the most productive use of all land considered is in housing.

The underlying purpose of the model to be developed below is to generate empirically testable propositions concerning the effects of amenities on residential building heights, Since it would be necessary to eventually adopt specific functional forms in order to derive empirically testable propositions, these specific forms will be introduced directly into the theoretical iierivation of the model.

2.1. The price of housing services

Since the principal focus of this paper is on the supply of housing. a formal model of the demand for housing will not be de\:elo

pr~e function for housing ~,ill be assumed that is somewhat more specific. though entirclj, consistent with those common in the literature.

In conventional models. where all employment takes place in the CBD, the

price of d unit of housing services depends only on the distance to the CBD. I\, i.e., P= P (A). in this present study it will be assumed that ind viduals are willing to pay not only for accessibility to the CBD, bu’t also for accessability to and a view of a location-specific amenity, i.e., P=P~k..s.~,r~. ‘.v’ is the distance of a housing unit from a location-specific amenity. ‘z’ is the floor of

a particular housing unit. designed to capture the idea that the higher the floor the better the view afforded. ‘1.’ is the O-1 variable indicating the

absence or presence of a yicw of an amenity from the housing unit. The specific housing price functton employed in tilis model ib’

This formulation assumes that the price of a unit of housing serbices declines exponentially with distance from the CBD and amenity.’ The remaining terms in the price equation are intended to capture the effects af views. The breadth of view’ increases Lvith the height of a housing unit. A constant elasticity. z. \vill be assumed between the price of housing and the floor of the unit. In addition to breadth of view, housing units that provide a ~‘I~YV of the amcnlty ;tre assumed to command a premium of ;’ percent.

combine lar,d and non-land inputs. The intensity of factor use ill producing housing is measured by Ihe ratio of non-land to land inputs. In actuality. most of the variation III fiuztor intensity is accomplished by variation in the height of buildings, alrhough height per se is not explicitly treated in these models. Since the primary concern of this paper is the height of buildings. a producticln function will he rmpinyed rhat treats height explicrtly.

184 R. Pollard, Topographic amenities, building bright, and .mpply vf urban housing

input, N, and the height of buildings, H. The marginal products of land and non-land inputs a~:,: assumed positive; the marginal product of height is discussed below. It will simplify matters with only a slight loss in generality to assume that the height component of the production function is separable from the land and non-land components. The production function may then be written Q - G(L, N)F(H). Given the possibility of replicatmg buildings it seems reasonable to further simplify matters by assuming constant returns to scale for land and non-land inputs for buildings of a given height. These modifications permit the production function to be expressed as a function of the ratio of non-land to land inputs, II, or Q = Lg(n)F(H). The assumption of constant returns to scale implies that g’(n) > 0 and g”(n) < 0. The specification of the height component of this production function requires a further explanation.

An apparent feature of multi-story buildings is that the construction costs per square foot of living space increases with height. This is plainly illustrated in the case of a two-story building. The addition of a second story requires a stairway that reduces living space on both floors as well as entailing a capital expenditure. For high-rise buildings the incremental cost of additional floors is not so clear. Foundation costs and +ndow-wall costs do not increase in proportion to the number of floors. The heterogeneity of buildings and the rapid technological change in construction techniques makes it difficuit to provide firm estimates. The available evidence suggests that after the few initial floors, additional floors can be added at constant cost.4 However the cost of living space increaes with the number of floors because additional space must be devoted to structural supports. ele\,ator shafts, and mechanical equipment.“.

To incorporate these properties into the production function the height component will be written as H-l’, where /I is an elasticity, assumed constant, that measures the rate of change of the proportion of usable floor area with respect to changes in the height of buildings. Using these assumptions the complete production function may now be written as

Using ey. (2) a supply function for housing st:r\riccs can now bc JCriv8ed by considering the production d&ion of an cntrcprcneur in ;I ,.:ompctiti\~c housing market. At any location the tntrepreneur must decide the height of

“In a comptiter simulation study of a hypothetical high rise building, Thompscn (1966, p. 50) found that construction costs per floor remained Amos ~onstwt.

‘Hoch (1969. p. 90).

building to construct. I[< f oar ;irea anti the level d non-land inputs per floor.

The total profit. r:*. from a building is. of course. equal to total revenue less

total Loht. The total rei-enue from a buildin_g can be found by summing the

revenue for each floor. (It ~111 simplify matters to treat bullding height as a

continuous variable so that the summation process can be done by

integration.) The revenue function is then given by

Substituting in the specific function:ll fi)rms fov I’ and Q gifcn ’ y eqs. I I )

and (2) into (3) yields

where P,,-(1 (I -tr))Pz.

The total cost for a building at any locatio*, is the sum ;,f the land and

non-land costs. Land is wumed to sell for I’ per unit. The price of non-land

Inputs P, is :lswmed constant throughout the city and equal to ur,ity. The

profit from ;1 burldIng may then he Lvrittcn as

As shrill be ncjted from r‘q. (3) total prclfits ;irc directI>, ploportional to the

land area used 111 the buildlns; thus the l ptlrnal ;t:u of 3 bullding ih

indeterminant. Ho\vcvcr. the primary concern here 11r with the hei&t of

buildings and this does not require a detcrminatiolr of building area. By

recasting the profit function in terms of profit4 per unit of land. 71. it is

poGhlc to \oI\c for the crptimal Ic\cl rjf land an nt~ll-li~nd 111putls per floor

186 R. Pollard, Topographic amenities, building height, and supply of urban housirtg

n _gr(n)p,e-gkk-d,s+rl~H’+d-B_H=O, ” (64

11H=(1+01-_)POe-8kk-bss+y”g(,)H”-B-n=0. (6b)

The second-order condition for a maximum requires that

d =(w--_) [

g”(n In 1 ~- (w-p) g’(4 1 >o.

By rearranging this second-order condition additional insights into the model are possible. The assumption of constant returns to scale implies that

g”(n)n cx - p --=F g’(n) ’ ONL

where oNL is the elasticity of substitution between land and non-land inputs. The second-order conditions may then be rewritten

d = (ct-pv -& 1 I 1 b-0.

Thus, the second-order condition for a maximum reduces to the condition that the elasticity of substitution be less than unity.’

The second term is the marginal cost of adding non-land inputs: it is the price of a unit of non- land inputs (taken to be unity) times the number of floors, H.

Similarly, eq. (6b) provide, the first-order conditions for the optimal number of floors. The first term on the left-hand side is the marginal revenue product of adding a story, holdmg constant non-land inputs per floor. This consists oi two parts. The First is the revenue received on the extra story from adding an additional story with g(a)/1 c U~II> of housmg services HIIICII can be rented ;lt a price,

The second is the loss of revenue from the first ff stories given hq the chanpc 111 ber\~ccs per Floor, -/Ig(r~)ll 8-1. times the number of floors. If. rinrcs the average prlcc per Iltt:~,

P,* exp { - S,k - 6,s + yu)H”.

The second term is the marginal cost of an additional flop. II.

-‘The elasticity of substitution defined ab~lve differs sltghtly from ctrnvcntionz! measures. It is the elasticity between land and non-land inputs alnnp ;I housing in general.

pi\en &ror of htqlsing rather than for

R. Pollard, Topographic amenities, bui1dir.g herghr, and supply of urban housing 187

For the purpose of subsequent manipulations it will be useful to re-arrange the first-order conditions. First by dividing the left- and right-hand sides of (6a) by the left- and right-hand sides of (6b) and then re-arranging terms, (7a) be obtained,

g”(W -------=1+X-[,. g(n)

Second by solving (6b) for H, (7b) may be obtained,

(74

Ub)

An idea of the numerical value of the elasticity of the proportion of housing space that is usable with respect to height, B. can be obtained from the formula for factor shares. The factor share accruing to land can be expressed as

Hn __~l-- --

Poe- itk-6,sgi12)jy1 +x-B

1 1

-_ sl ____- ----- = fl- -J*

P,e-d’*-d*M1) n *“-”

Assuming for example, a factor share for land of 10 percent and, as results estimated presented below suggest, a value of 0.07 for ~1, the implied value for p is 0.17.

2.4 fmplicatior~s of the model

The implications of the model for building height may be demonstrated by examining the properties nf the equilibrium conditions expressed in eqs. (7a) and (7b). In particular height gradients, height differentials due to a view, and he&h! differentials due to a view of an amenity will be considered.

,?.J.l. Ifeiglrt ~mdienr.5 for hfw.~ing

The height gradient for h0usir.g structures may be found by taking the differential of eq. (7b),

Height gradients are seen to be proportional to price gradients. Since price gradients are assumed to be exponentially declining with distance from the CBD, this result is analogous to the standard result that the value of output per unit of land declines exponentially with distance from the CBD. Note that

the factor of tiroportionality between height and price gradients depends on “II -fl, a term which reflects the marginal revenue and marginal cost of an additionai fioor.

In this formulation, the presence of a view of the amenity has no influence on height gradients. This result is a consequence of the particular functional form assumed, and will be tested below.

’ 4.2. Height qj/trrrltiuls due to ritw* pwUiL~ a.

In this model the influence of a view on the price

captured in two separate terms: Y, the elasticity of

of housing services is the price of housing

services with respect to height captures the effect of the breadth of v,iew. and ;’ is the premium offered for a view of a specific amenity regardless of height. The presence of either premium increases the optimal height of buildings.

The effect of the height premium on optimal building height can be demonstrated by differentiating eq. (7a) with respect to x,’

dH -H ---------jiogH+.~_I_,j+

dx x-i1

Eq. (7b) implies that buildings with a view of the amenity will be

(111

times as tall as those without such a view.

The combined effects of accessibility to ;rnd ;I vicu, 01 ;III amcn~ty c:tn bc illustrated by examining building heights along a street tltitt cman ‘(c‘s fr~~m it location specific amenity. See fig. 1.

The tallest buildings are those IOCi~t~d iLLI_j!jaCCllt 10 kItId CjlTCl-ill)I cl 1 IC\\ 01 the amenity. Moving away from the amenity building heights d,:clinc at a rate of ci,;(x-{I). At the point designated S, a view of the amenity is ~0

longe, possible and there is a discrete decline in building height of (e ” ‘I m-“’

- 1) percent. Buildings continue to decline at a rate of &,;(x-~) until the

point at which accessibility to the amenity ceases to exert any influence on

the price of housing. Beyond this point. designated s(,. the height gradient is flat.

Distance to amenity K

3. Empirical results

The model presented above qlclds txpliclt predictions about the rel;ition of yice and height gradients for housing. and pr”~ and height differentials dw

to a iie\t of an amenity. In this sicction thtx 3arameters are estimated using

data collected from the city of Chicago. and the predictions of the model are tested.

190 R. Pollard, Topographic nmenities, building height, and supply of urban housing

rent and size were recccded, as well as whether it has a view of the Loop or Lake Michigan. In some cases air-conditionin g, l-eat, utilities, and carpeting were included in the rent, while in others the; -,vere not. It was necessary therefore to adjust rents for the services inch~deJ.iO

3.2. Basic resdts

Eq. (1) specifies how the price of a unit of housing services is affected by accessibility to the CBD and accessibility to and a view of amenities. The parameters of this price equation are the basis for the predicted height gradients in eq. (9): and the predicted height differentials due to a view in eq. (11). Befo re this price equation can be estimated, a number of modifications are necessary. In particular, the empirical counterpart of a unit of housing services must be defined. and consideration must be given to the presence of multiple amenities.

The observations described above consist of apartment rents. which are, of course, the product of price times quantity of housing services. To determine the price of housing services it is necessary, therefore, to have a measure of the quantity of housing services provided by an apartment.

The simplest assumption that could be made is that the quantity of housing services yielded by an apartment is proportional to its area. i.e..

Q APT= F;(SQFT). However, upon reflection this seems to be an oversimplification. Apartments are not just living space. Every apartment, regardless of size, must contain a kitchen and bathroom which produce services in addition to mere living space. Therefore, it seems reasonable to assume for apartments as a whole, the quantity of housing set-Jices increases less than in proportion to the area of the apartment. To incorporate this assumption the quantity of housing services flowing from an apartment may be written as QAPT= k(SQFT)“, where vc 1 is the elasticity of housing services with respect to apartment size.

Another factor to be considered is that buildings are capital goods whose flow of services declines over time. Therefore, it is necessary to adjust the flow of services emanating from an apartment for the age of the structure. One of the many possible assumpti0n.s is that buildings depreciate at a constant percentage per year. I1 The quantity of housing services yielded by an apartment may now be written as

Q APT = ~~(SQF~)‘e-"""~'l~

‘*The cost of carpeting was estimated to be lp per square foot per month. The following costs in terms of square foot per month were determined on the basis of datj prGded by Commonwealth ‘Edison: heat ?c? utllittes 2, air conditioning jc.

“Several other specifications of the age term wer cons~drrrd. hug thr: ~0\Ae5s-of-fit test suggests that this form IS the most appropriate.

Ha\.ing defined the measure of housing serbices. it is necessary now to

spectf; how the presence of multiple amenities will affect its price. The

dominant amenity iq the sample area is. of course. Lake Michigan. Howexr.

casual empiricism suggests that two other important amenities are a view of

the Loop and accessibility to Lincoln Park, an extensive city facility located

along the Lakefront. Since the effects of amenities are captured in the

exponential terms of the price equation, it can be easily modified to account

for additional amenities.“’

Llsing the modified price eq. ( 13). the equation for building height i 10 and

I 1 ) may he kvritten tin log form)

192 R. Pollard, Topographic amenities, building height, and supply qf than housing

.

Table 1

Rent acd height regressmns.

Variable

Dist-Loop

D&-Lake

Loopview

Lakeciew

LincPark Age

logfloor tog SQFT

Rent equation Height equation

Coefficient t-statistic CsetIicicnt t-statistic

- 0.038 (-11.2) -0.123 ( - 7.8)

-0.085 ( - 2.2) - 0.854 ( - 5.2)

0.029 (0.8) 0.581 (4.3)

0.070 (2.7) 0.565 (5.1)

0.049 (2.4) 0.409 (4.0) - 0.0029 (-4.2) -0.011 (-4.1)

0.067 (6.2)

0.616 (18.7)

Constant 1.590 (7.1) 3.063 (19.1)

R= 0.858 0.172

iv 232 0.1544

The coefficients of the Loopview and Lakeview terms, S, and iiz. imply that a view of the Loop increases rents by 3 percent and a view of the Lake increases rents by 7 percent. However, the Loopview coefficient is not significant at the 5 percent level. The coefficient of the Lincoln Park term implies that accessibility to the Park increases rents by 5 percent.

The coefficient of the floor variable is an estimate of 2, the elasticity of the premium paid for height. The results imply that a 1 percent increase in the floor of an apartment leads to an increase in rent of about 7 percent. For example, moving from the 20th to the 21st floor is an increase in height of 5 percent; rents increase, therefore, by 0.07 x 0.05 =0.0035 percent, or using the mean monthly rent in the sample of $257.50. about 9061 a month.

The remaining coefficients in the rent regression are concerned with the quantity of housing services yielded by an apartment. The coefficient of the SQFT term is an estimate of the elasticity of the flow of housing seri ices with respect to the area of an apartment. As assumed, it has an c~tima!cd value substantially smaller than unity, The coefficient of the age term, while stgnificant. is only three-tenths of 1 percent, suggesting that tl~~: dcclinc in services flowing from an apartment takec place very slo~vlp.

The results of the height regression appear on the right side 01 the table 1. The building height gradier?ts from the Loop and Lake, 6: and 6; are estimated to be 12 percent and 85 percent respectively. The coefficients of the Loopview and Lakeview terms imply that buildings v ith a view of the Loop are eo.;n = 1.79 times and those with a view of the kakc are e”.‘- = 1.77 times

3% rail as those nithout a xie\v. Buildings with accessibility to Lincoln Park ;tre c ’ Ji = 1.51 times as tall as those lacking accessibility.‘”

Height gradients and view differentials hate been directly estimated in the height regressions. In addition, the model shows how these gradients and differentials can be inferred from the price equation. Hence a test of the predictive ability of the model IS provided by a comparison of the height gradients and view differentials predicted by the model with those directly estimated in the height regression. Due to the non-linearity of the estimating equation, large-sample statistical theory must be used to provide an appropriate test. The test statistic is derived in appendix B.

The Z-statistic which provides the basis for the test of the gradients and differentials predicted by the modei with those directly estimated includes tile

parameter /‘I. Since there is a considerable range in the estimates of the factor share of land, and consequently the value of /3, the test statistic is computed for values of #I corresponding to a factor share for land of 10, 15, and 25 percent. l4 Test results appear in table 2.

For an assumed factor share for land of 10 percent :he assumption of equality can be rejected only for the Loop gradient cct the 5 percent confidence level. If the share of land is 15 percent, only the Fssumption of the equality of the Lakeview differential can be maintained. If he share of land is as large as 25 percent the assumption of equality can ;,e rejected in all instances. Consequently the test for the equality of thv measured and predicted height gradients and l,iew differentials yields inctinclusive results.

13Thc residuals or the rent and height regressions were examined for evident: of violations of the basic dssumptmns of the lmear regression model. A Goldfield-Quandt test, with observations orderec accorJing to distance to the Loop, did not provide evide Ice for a rejection of the hornosccdasti:ity assumption in cithcr the rent or height regresstons. A test for spatial autocorrclation along lmes suggested by Cliff and Ord (1972) revealed no eridencr: of aulocorrclatlon in the height regression, but rexealed the presence of a slight positi\? ;lutacorrelatmn In the rent regressIon. l‘hlc pos~t~\c autocorrelatlon may be attributable to the fact that m wm~ tm~ranc~s thcrc were multiply rct\t obrcr~at~ons fram d slnglu bullding. A test for corrclatw,l of the rcslduals of the rent and hslght rcgresslon did not provide evldencc for a rqrctwn of the mdependcncc ;~rsumptmn

‘l”t’~t~ i%ccrc ;tlbo pcrformuti on the qxc~f~c~at~on IA the model A tc\t IO dctcrmlne t’lc extent o! 111~’ i~ks’s trlflucncc on Iwrlrl~ng haglIt\ dnd wilts ~upp~)rtcd the a\\umptlon th;it the cffcct of thr Lake rxtrnds o\er the cntlre xamplc ;irtx An IntCractIU~ rrrm \hi~, Jncludcd ICI ~n:elt~y~tc the ptr\\lblilt> ol’ dlffwxtlril r<flt tired height gradlent\ rn the prcwxc rl ri ~IPM. T.K lnteractlt~n LCiilb HClC IlOf st~t&tl&&il) s:gn!f:cant IzLIr d,d :!!Clr I~ClU~i~~I1 ~!~I~‘ii~~~lt!) IIILi’t!Liif tbiL’

explanatory power of the model. Finally, alternative functional forms, including the linear and logarithmic were considered for the Loop and Lak:: gradients. Using the residual sum of squares as a measure of goodness-of-fit, they proved inferior to the exponentia: formulation.

**Muth (19?1, p. 25) estimates the factor share of land rc be 10 percent u.htle Smith 11976. p. 403) estimates the share to be 15 to 20 percent and \I1113 1’172. p 110) c~tei evldcncr placmf

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FVitli the exception of the height gradient from the Loop. the test hinges upon the actual factor share of land in housing, a magnitude which unfortunately is not knofvn \vith a sufficient degree of certainty.

4. The importmce ef topographic amenities

Using the regression results presented above it is possible to make some statements about the quantitative importance of topographic amenities, Speci~~all~, the effects of Lake Michigan on housing prices and expenditures, and on the supply of housing will be considered.

The effect,s of amenities on housing prices can be illustrated by considering two housing units that are identical except for their amenity components.

Consider two apartment units that are 16 years old and contain 6% square feet of living area - the mean values in the sample. Both units are located 8 miles from the LOOP. The first unit is located on the first floor of an apartment located one mile from the Lake and consequently does not provide a Lakeview. It has a predicted monthly rent of $173. A similar apartment with a lakeview, located on the tenth floor of a Lakefront building has a predicted monthly rent of $235.

In this example the residents of the Lakefront apartment are paying $62 per month or 26 percent of their housing expenditure for consumption of amenities. Assuming families spend 20 percent of their incl)me on housing, this expenditure for amenities represents 6 percent of annual income. Since there are numerous buildings along the Lake in excess ol 50 sto;.ics, this example cannot be regarded as ext:-eme.

The effect of amenities on the supply of housing - as med.suied by building height - can be demonstrated by considering building ILeights predicted by the height regression,

At the Loop buildings without a view of the Loop or Lake have a predicted height of I’7 floors, those with a view of the Loop or the Lake have a predicted height of 30 floors, and those with a view of both Loop and Lake have a predicted height of 54 floors. Eight miles from the Loop a Lakefront building without a tiew has a prcdictcd height of 6 floors. A building at the ~;ltne location with a lake\ieu has a predicted height of 1 1

tloors. A building quid~stan: from the Lwp but OIIL’ mill: from the Lake ha+. a pre&c;led height of only 3 flotiia. C:oi~s~~ue~itly the presence of cithcr ir Lakeview or accessibility to the Lake leads to a doubling of the predicted height of b*iildings. These results for Lake ichigan are indicative of the substantial impact of topographic amenities on the height of buildings, the supply of housing. and the i;tru:ture of cities.

196 R. Pollard, Topographic amenities, building height, and supply of urban housing

Appendix A: Variable definitions and sample statistics table

Rent - Dist-Loop - Dist-La ke - SQFT --

Floor - Age -- Lakeview - Loopview -- LincPark -

Adjusted monthly apartment rent in dollars. Distance to Loop in miles. Distance to Lake ill miles. Size of apartment in square feet. Floor of apartment. Age of building in years. Dummy variable for apartments with a view of Lake. Dummy variable for apartments with a view of Loop. Dummy variable for apartments with accessibility to Lincoln Park.

Height - Height of building in floors.

Variable Mean Standard deviation Min. Max.

Rent sample

Rent Vist-Loop D&-Lake SQFT Floor

Age Lakerieu Loopt.+ew LincPork

Height sample

Height D&-Loop D&-Lake

Age Lakeciew LooptTiew LincPark

257.50 5.2 0.41

656.8 10.5 16.4 0.147 0.07 0.26

12.5 5.2 0.45

20. I 0.26 0.15 0.20

84.3 2.9 0.26

189.3 11.6 14.6

13.0 3.0 0.28

15.3

124 481 1.0 10.9 0 1.6

300 1300 I 65 I 55 0 I 0 1 0 1

1 70 1.0 10.9 0 1.6 1 55 0 I 0 I 0 I

Appendix B: Test statistic

The model predicts that height gradients Sf’ will be equal to S/(x - {j) and that height differentials due to a view $’ will be equal to yi/(cl - ii). The estimated coefficients of the height gradients, (5” i , and height differentials due to a view, jjy, provide unbiased estimates of the actual height gradients and differentials due to a view. Similarly, &, $i, and 4 provide unbiased estimates of their respective parameters. /I is determined from the factor share of land in the value of housing. Due to the non-linearity of the functions 8J(& -fl) and fi/(?--/?), large sample theory will be used to create a test.”

15The test statistic will be derived for the case of height gradients; the statistic for height differential due to a biew is identical.

R. Pollard, Topographic amenities. building height. and supply of urban housing 197

di/(&-/?) can be said to have a limiting normal distribution with mean SJ(a -B) and variance

where Q denotes the standard errors and p is the correlation coeKcient between 6i and f.16

The hypothesis of the equality of the height gradients predicted b:r the model with those directly measured may now be written

SJ(a-P)-SH=O. (16)

Since 8i/(& -8) and 8” are asymptotically norm4ly distributed the test statistic will be

z= ~i/(jr_8)-d” var[S,I(i-b)--@‘J’ (17)

Assuming a value for fl the coefl?cients of the rent and height regressions in table 1 provide all the information necessary to compute the numerator. Calculation of the denominator is slightly more complicated.

The variance of the difference in the gradients may be written as

var [&/(6-j?)] + var (3F) - 2 cov [St/(& - fl), Sf] .

The variance of $,/(a -/3) has been defined above, and the variance of 8: is available directly from the variance-covariance matrix in appendix C. Due to the unequal sample sizes, the covariance of ~i/(~ -/I) and @’ is not easily estimated. ,?z r\n alternative the covariance is assumf.d to take on its maximum value, even though this may lead to an underestimate of the true variance. When &/(j, -/I) and 8: are perfectly correlated their covariance will be a maximum and equal to

([var (di/(a-pI)J !,var ($f’)]P.

The test statistic may n;,w. $ written as 1

(18)

‘“Theil (1971, pp, 372-384. especially problem 3-6).

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R. Pollard, Topographic amenities. building height, and supply of urban housing 199

Aionso, W.A, 1964, Location and land use (Harvard University Press, Cambridge, MA). CliR A.D. aad J.K. Qrd, 1992, Testing for spatial autocorrelation among regression residuals,

Geographical Analysis 4.269-284. Hoch, I, 1969, The three-dimensional city, in: Harvey S. Perloff, ed., In the quality of the urban

environment, 95-I 38. Mills, ES., 1969, An aggregative model of resource allocation in a metropolitan area, American

Economic Review 59.199-210. Mills, ES., 1992, Studies in the structure of the urban economy (Johns Hopkins Press,

Baltimore, MD). Muth, R.F.. 1969, Cities and housing (University of Chicago Pre;s, Chicago, IL). Muth, RF., 1991, The derived demand for urban residential land, Urban Studies 8, 234-253. Smith, E., 1976, The supply of housing, Quarterly Journal of Pconomics 90, 389-405. The& H T., 1971. Principles of econometrics (Wiley, New York). Thompsen, C., 1966, How high to rise, Appraisal Journal 12, 386-391.