Skyscraper Height

Abstract

This paper investigates the determinants of skyscraper height. First a simple model is provided where potential developers desire not only profits but also social status. In equilibrium, height is a function of both the costs and benefits of construction and the heights of surrounding buildings. Using data from New York City, I empirically estimate skyscraper height over the 20th century. Via spatial regressions, I find evidence for height competition, which increases during boom times. In addition, I provide estimates of which buildings are economically “too tall” and by how many floors.

Appendices

Appendix A: Proofs

Lemma 1

For $\lambda \in \left[ 0,\overline{\lambda }\right]$ , there is a unique value of h, h ^∗, such that $h^{\ast }=\arg \max_{h\in R_{+}}u_{i}\left( h\right).$

Proof

For λ = 0, $h^{\ast }=\arg \max_{h\in R_{+}}\pi \left( h^{\ast }\right) ,$ which is unique by definition. For λ > 0, at h = 0, $\pi \left( 0\right) +\lambda F\left( 0\right) =0.$ Recall $F\left( h\right)$ bounded by one. Further, since profit is single-peaked and strictly concave, there exits and $\tilde{h},$ such that, $\pi \left( \tilde{ h}\right) +\bar{\lambda}=0.$ Thus for $h\in \left[ 0,\tilde{h}\right] ,$ there must be a global maximum, since the contribution of $\lambda F\left( h\right)$ to utility is bounded and adding $\lambda F\left( h\right)$ to $\pi \left( h\right)$ preserves the single peaked nature of the utility function when $h\in \left[ 0,\tilde{h}\right] .$ □

Lemma 2

h ^∗ > h ^c for $\lambda \in \left( 0,\overline{\lambda }\right] ;$ h ^∗ = h ^c for λ = 0.

Proof

If λ = 0, then h ^∗ = h ^c will be equal since the “status” developer is maximizing the same function as the “competitive” developer. If $0<\lambda \leq \overline{\lambda },$ then given Lemma (1), there exists a unique h ^∗,such that $u^{\prime }\left( h^{\ast }\right) =\pi ^{\prime }\left( h^{\ast }\right) +\lambda f\left( h^{\ast }\right) =0,$ or $\pi ^{\prime }\left( h^{\ast }\right) =-\lambda f\left( h^{\ast }\right) ,$ where $f\left( h^{\ast }\right) >0.$ Given that $\pi \left( h\right)$ is single-peaked, and $F\left( h\right)$ is increasing, the optimal building height is therefore taller than a building where $\pi ^{\prime }\left( h\right) =0;$ that is, h ^∗ will be chosen along the negatively sloped portion of the profit function beyond the peak of $\pi \left( h\right)$.□

Lemma 3

h ^∗ is strictly increasing with λ.

Proof

For a given h ^∗, the utility function is at a global maximum and therefore $u^{\prime }\left( h^{\ast }\right) =0,$ and $u^{\prime \prime }\left( h^{\ast }\right) =\pi ^{\prime \prime }\left( h^{\ast }\right) +f^{\prime }\left( h^{\ast }\right) <0.$ The first order condition gives $\pi ^{\prime }\left( h^{\ast }\right) +\lambda f\left( h^{\ast }\right) =0.$ Taking derivatives via the Implicit Function Theorem gives $dh^{\ast }/d\lambda =-f\left( h^{\ast }\right) /\left[ \pi ^{\prime \prime }\left( h^{\ast }\right) +f^{\prime }\left( h^{\ast }\right) \right] >0.$□

Lemma 4

dh ^∗/dθ > 0.

Proof

As above, $u^{\prime }\left( h^{\ast },\theta \right) =0,$ and $u^{\prime \prime }\left( h^{\ast },\theta \right) =\pi ^{\prime \prime }\left( h^{\ast }\right) +f^{\prime }\left( h^{\ast }\right) <0.$ Via the Implicit Function Theorem $dh^{\ast }/d\theta =-\left( \partial ^{2}\pi /\partial h\partial \theta \right) /\left[ \pi ^{\prime \prime }\left( h^{\ast }\right) +\lambda f^{\prime }\left( h^{\ast }\right) \right] >0,$ since $\partial ^{2}\pi /\partial h\partial \theta >0$, by assumption.

Lemma 5

dh ^∗/dη > 0.

Proof

$u^{\prime }\left( h^{\ast },\eta \right) =\pi ^{\prime }\left( h^{\ast }\right) +\lambda f\left( h^{\ast },\eta \right) =0,$ and $u^{\prime \prime }\left( h^{\ast },\eta \right) =\pi ^{\prime \prime }\left( h^{\ast }\right) +f^{\prime }\left( h^{\ast },\eta \right) <0.$ Via the Implicit Function Theorem, $dh^{\ast }/d\eta =-\lambda f^{\prime }\left( h^{\ast },\eta \right) /\pi ^{\prime \prime }\left( h^{\ast }\right) +f^{\prime }\left( h^{\ast },\eta \right) >0,$ since $f^{\prime }\left( h^{\ast },\eta \right) =\partial ^{2}F/\partial h\partial \eta >0,$ by assumption.□

Lemma 6

l ^∗ is monotonically increasing in λ; l ^∗ = l ^c, when λ = 0.

Proof

Given the optimal height for a plot, h ^∗, land value is given by $l^{\ast }=\pi \left( h^{\ast }\left( \lambda \right) \right) +\lambda _{i}F\left( h^{\ast }\left( \lambda _{i}\right) \right) -r.$ By the Implicit Function Theorem, $dl^{\ast }/d\lambda =F\left( h^{\ast }\right) >0$ . If λ = 0, land value is simply given by $l^{\ast }\left( \lambda _{i}\right) =\pi \left( h^{c}\right) -r,$ since h ^c maximizes profit.□

Lemma 7

Given the monotonicity of l ^∗, the minimum and maximum land values for a plot of land is l ^c and $\bar{l}^{\ast }=\pi \left( h^{\ast }\right) +\overline{\lambda } F\left( h^{\ast }\right) -r,$ respectively.

Proof

If λ = 0, $l^{\ast }=l^{c}=l\left( h^{c}\right) =\pi \left( h^{c}\right) -r,$ where $l^{\prime }\left( h^{c}\right) =\pi ^{\prime }\left( h^{c}\right) =0.\;$As discussed in lemma (4), for λ > 0, since l ^∗ is strictly increasing in λ, and λ has a maximum of $\bar{\lambda},$ no developer would be willing to pay more than $\bar{l}^{\ast }$□

Lemma 8

Given that $\lambda \sim U\left[ 0,\overline{ \lambda }\right] ,$ the pdf for land valuations is given by $k\left( l^{\ast }\right) =\left\{ \begin{array}{c} \frac{1}{\overline{\lambda }F\left( h^{\ast }\right) },l\in \left[ l^{c}, \bar{l}^{\ast }\right] \\ 0,\rm{ otherwise} \end{array} \right\} ;$ with a cdf of K $\left( l^{\ast }\right) =\frac{l^{\ast }-l^{c}}{ \overline{\lambda }F\left( h^{\ast }\right) },\,l\in \left[ l^{c},\bar{l} ^{\ast }\right]$ .

Proof

Recall that $l^{\ast }\left( \lambda \right) =\left[ \pi \left( h^{\ast }\right) -r\right] +\lambda F\left( h^{\ast }\right) .$ Since $\lambda \sim U\left[ 0,\overline{\lambda }\right] ,$ $g\left( \lambda \right) =\frac{1}{\overline{\lambda }}.$ The pdf of $l^{\ast }\left( \lambda \right)$ follows from the formula for a change of distribution for a function of a random variable that has a uniform distribution: $k\left( l^{\ast }\right) =\frac{d\lambda }{dl^{\ast }}g\left( \lambda \right) =\frac{ 1}{F\left( h^{\ast }\right) }\frac{1}{\overline{\lambda }}$. Note that $l^{c}=\pi \left( h^{\ast }\right) -r.$ The cdf follows from K $\left( l^{\ast }\right) =\frac{1}{\overline{\lambda }F\left( h^{\ast }\right) } \int_{l^{c}}^{l^{\ast }}dx=\frac{l^{\ast }-l^{c}}{\overline{\lambda }F\left( h^{\ast }\right) }$.□

Proposition 1

Given each agent’s land valuation function, and that the status parameter has a $U\left[ 0,\overline{\lambda }\right]$ distribution, there exists a unique, symmetric equilibrium of the land auction game such that each agent’s bid is given by $\beta \left( l^{\ast }\right) =\frac{\left( N-1\right) l^{\ast }+l^{c}}{N}$ .

Proof

Note that this proof is adapted and condensed from Krishna (2002), to which the reader is referred for more information. Let’s say there are N bidders, each with private valuation of $l_{i}^{\ast }\left( \lambda _{i}\right) ,$ where $l_{i}^{\ast }$ is strictly increasing in λ. Suppose there exists a symmetric, increasing equilibrium strategy, $\beta \left( l_{i}^{\ast }\right) .$ First, it would never be optimal to bid $b>\beta \left( \bar{l}^{\ast }\right)$, since the agent would win the auction and could have done better by slightly reducing his bid, as he could win and pay less. Second, a bidder with λ = 0, would never submit a bid greater than l ^c, since he would have negative utility if he were to win, thus $\beta \left( 0\right) =l^{c}.$ Bidder i wins the auction when he submits the highest bid; that is when $max_{j\neq i}\beta \left( l_{j}^{\ast }\right) <b.$ Define λ ^N − 1 as the value of λ for the second highest bidder out of N bidders. Since $\beta \left( l^{\ast }\right)$ is increasing, bidder i wins if he has the highest value of $l_{i}^{\ast }$ (i.e., if $\lambda _{i}>\lambda ^{N-1})$ or if $\beta \left( l^{\ast N-1}\right) <b,$ or equivalently if $l^{\ast N-1}<\beta ^{-1}\left( b\right) .$ Agent i ^′ s expected payoff is therefore $K\left( \beta ^{-1}\left( b\right) \right) ^{N-1}\left( l^{\ast }-b\right) ,$ where $K\left( l^{\ast }\right) ^{N-1}$ is the distribution of the second highest order statistic for land values. Taking the first order condition, replacing $b=\beta \left( l^{\ast }\right)$ (at the symmetric equilibrium), and solving for the differential equation given by the FOC, yields the equilibrium function $\beta \left( l^{\ast }\right) =\left[ 1/K\left( l^{\ast }\right) ^{N-1}\right] \left( N-1\right) \int_{l^{c}}^{l^{\ast }}ydK\left( y\right) ^{N-2}dy.$ Given that land values are distributed $U \left[ l^{c},\bar{l}^{\ast }\right] ,$ this bid function is $\beta \left( l^{\ast }\right) =\frac{\left( N-1\right) l^{\ast }+l^{c}}{N}.$

This result is a necessary condition for the optimal strategy, I now turn to showing the sufficient condition: that if the N − 1 bidders follow $\beta \left( l^{\ast }\right) ,$ then it is optimal for agent i to do so as well. Suppose that all agents but bidder i follow the strategy $\beta \left( l^{\ast }\right) .$ Given that the winner has the highest bid, it is never optimal for agent i to bid more than $\beta \left( \ \bar{l}^{\ast }\right) .$ Denote $z=\beta ^{-1}\left( b\right)$ as the value for which b is the equilibrium bid for agent i, that is $\beta \left( z\right) =b$. The expected payoff to agent i from bidding $\beta \left( z\right)$ is $\Pi \left( \beta \left( z\right) ,l^{\ast }\right) =K\left( z\right) ^{N-1}\left( l^{\ast }-\beta \left( z\right) \right) =K\left( z\right) ^{N-1}\left( l^{\ast }-\beta \left( z\right) \right) +\int_{l^{\ast }}^{z}K\left( y\right) ^{N-1}dy,$ where this equality is obtained via integration by parts. This leads to the conclusion that $\Pi \left( \beta \left( l^{\ast }\right) ,l^{\ast }\right) -\Pi \left( \beta \left( z\right) ,l^{\ast }\right) \geq 0,$ regardless of whether z ≥ l ^∗ or z ≤ l ^∗. Thus for agent i, not using $\beta \left( l^{\ast }\right)$ will make the agent no better off, which implies that $\beta \left( l^{\ast }\right)$ is a symmetric equilibrium strategy.□

Appendix B: Data Sources and Preparation

Skyscraper Height, Number of Floors and Year of Completions: Emporis.com.¹⁹

Plot size: NYC Map Portal (http://gis.nyc.gov/doitt/mp/Portal.do); Ballard (1978); http://www.mrofficespace.com/; NYC Dept. of Buildings Building Information System, (http://a810-bisweb.nyc.gov/bisweb/bsqpm01.jsp).

Plot Regularity: Various editions of the Manhattan Land Book (1927, 1955, 2002; see references) and the NYC Map Portal.

Use and Corporate HQ: For each building, one or more articles were obtained from the New York Times at the time of the building’s construction or just after its completion. From this, I ascertained its primary use and the developer. If the developer was a major corporation and the corporation had an equity stake in the building, it was listed as a Corporate Headquarters.

Distance from Core: For each building I obtained the latitude and longitude from http://www.zonums.com/gmaps/digipoint.html. I calculated the distance for each building i = 1,...,458, from its respective core using the formula $d_{i}=\sqrt{\left[69.1691\left( \rm latitude_{i}-\rm latitude_{\rm core}\right) \right]^{2}\!+\!\left[52.5179\left( \rm longitude_{i}-\rm longitude_{\rm core}\right) \right] ^{2}},$ where latitude and longitude were initially measured in degrees. The degrees to miles conversion is from http://jan.ucc.nau.edu/~cvm/latlongdist.html. In NYC, there are two cores: the intersection of Wall Street and Broadway (downtown) and Grand Central Station (42nd Street and Park Ave.). All buildings south of 14th street belong to the downtown core; all buildings on 14th street or above belong to the midtown core.

Depth to Bedrock: For each building, elevation from sea level (in feet) comes from http://www.zonums.com/gmaps/digipoint.html. Depth to bedrock from sea level (in feet) comes from maps provided by Dr. Klaus Jacob, Columbia University. The maps are based on hundreds of borings throughout Manhattan. The depth to bedrock was calculated by subtracting the depth of bedrock from sea level from the elevation from sea level.

Zoning 1916 and 1961: The New York Times was consulted to determine the first buildings completed under the respective regimes.

1916 Height Multiples: Original zoning maps in effect at the time of completion for each building.

The maps were provided by the New York City Department of City Planning.

1961 Maximum Allowable FAR: Original zoning maps in effect at the time of completion for each building. The maps were provided by the New York City Department of City Planning.

Special Districts: Zoning maps from NYC Dept. of City Planning, and articles from the New York Times.

Air Rights: Data about which buildings purchased air rights comes the New York Times, Real Estate Weekly and http://beta.therealdeal.com/front.

Plaza Bonus: Kayden (2000); www.nyc.gov/html/dcp/html/priv/priv.shtml

Real Construction Cost Index (1893–2004): Index of construction material costs: 1947–2004: Bureau of Labor Statistics Series Id: WPUSOP2200 “Materials and Components for Construction” (1982=100). 1893–1947: Table E46 “Building Materials.” Historical Statistics (1926=100) (1976). To join the two series, the earlier series was multiplied by 0.12521, which is the ratio of the new series index to the old index in 1947. The real index was create by dividing the construction cost index by the GDP Deflator for each year.

GDP Deflator (1893–2004): Johnston and Williamson (2007). (2000=100).

Finance, Insurance and Real Estate Employment (F.I.R.E)/Total Employment (1893–2004): 1900–1970: F.I.R.E. data from Table D137, Historical Statistics. Total (non farm) Employment: Table D127, Historical Statistics. 1971–2004: F.I.R.E. data from BLS.gov Series Id: CEU5500000001 “Financial Activities.” Total nonfarm employment 1971–2004 from BLS.gov Series Id: CEU0000000001. The earlier and later employment tables were joined by regressing overlapping years that were available from both sources of the new employment numbers on the old employment numbers and then correcting the new number using the OLS equation; this process was also done with the F.I.R.E. data as well. 1893–1899: For both the F.I.R.E. and total employment, values were extrapolated backwards using the growth rates from the decade 1900 to 1909, which was 4.1% for F.I.R.E. and 3.1% for employment.

Real Interest Rate (nominal rate minus inflation) (1893–2004): Nominal interest rate: 1893–1970: Table X445 “Prime Commercial Paper 4–6 months.” Historical Statistics. 1971–1997 http://www.federalreserve.gov, 1998–2004: 6 month CD rate. 6 month CD rate was adjusted to a CP rate by regressing 34 years of overlapping data of the CP rate on the CD rate and then using the predicted values for the CP rate for 1997–2004. Inflation comes from the percentage change in the GDP deflator.

Population NYC, Nassau, Suffolk, and Westchester Counties (1893–2004): 1890–2004: Decennial Census on U.S. Population volumes. Annual data is generated by estimating the annual population via the formula $pop_{i,t}=pop_{i,t-1}e^{\beta _{i}},$ where i is the census year, i.e., i ∈ {1890,1900,...,2000}, t is the year, and β _i is solved from the formula, $pop_{i}=pop_{i-1}e^{10\ast \beta _{i}}$. For the years 2001 - 2004, the same growth rate from the 1990’s is used.

Equalized Assessed Land Value Manhattan (1893–2004): Assessed Land Values: 1893–1975: Various volumes of NYC Tax Commission Reports. 1975–2003 Real Estate Board of NY. Equalization Rates: 1893–1955: Various volumes of NYC Tax Commission Reports. 1955–2004: NY State Office of Real Property Services. Equalization Rate: 1893–1955: Various reports NYC Tax Commission Reports. 1955–2004: NY State Office of Real Property Services.