Bombs, Boundaries and Buildings
A Regression-Discontinuity Approach to Measure the
Costs of Housing Supply Restrictions
Hans R.A. KOSTER, VU University Amsterdam* Jos VAN OMMEREN, VU University Amsterdam Piet RIETVELD, VU University Amsterdam
This version: March 16, 2011.
Abstract
Many cities apply planning policies that restrict housing supply by protecting building stock and by imposing height restrictions. These restrictions distort housing consumption and therefore reduce land rents. We aim to estimate the (gross) costs of these policy restrictions. To avoid endogeneity issues with respect to supply restrictions, we employ a (semiparametric) regression-discontinuity approach using a World War II bombing boundary within the city of Rotterdam. In the bombed area, where fewer restrictions apply, house prices are about 10 percent higher. We demonstrate that current building height restrictions and subdivision rules imply large (gross) costs. As we find no evidence that building height increases external costs, it appears that the current level of restrictions almost certainly reduce welfare. For example, allowing for new buildings that are only 1.60 meters taller and that contain more optimally-sized apartments increases social welfare by about 50 percent.
Key words: housing supply restrictions; height restrictions; spatial discontinuity design; semiparametric methods; bombing.
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*Corresponding author. Department of Spatial Economics, VU University, De Boelelaan 1105 1081 HV Amsterdam, e-mail: [email protected] paper benefited from a NICIS-KEI research grant. NVM (Dutch Association of Real Estate Agents), WDM, the Rijksmonumentenregister, the Kadaster and the Department of Infrastructure and Environment are gratefully acknowledged for providing data. We thank Eric Koomen for kindly providing data on building heights and Ronnie Lassche for his assistance with geographical data. We also thank Jaap de Vries, Paul Koster and Wouter Vermeulen for useful comments.
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1. Introduction
Most American and European cities apply housing supply restrictions to protect a building stock that offers amenities to inhabitants and tourists. Many of these amenities are external benefits, which may justify why cities apply these restrictions. However, studies such as Cheshire and Sheppard (2002), Glaeser et al. (2005b) and Cheshire and Hilber (2008) show that there are also substantial costs associated with supply restrictions. These studies generally focus on between-city differences in regulatory constraints. We focus on within-city regulatory differences. It is fundamental to distinguish between within-city and between-city differences in restrictions. Inter-between-city differences lead to higher house prices in the city with a more stringent policy, but also to lower land rents because regulation restrictiveness increase developers’ costs (Ihlanfeldt, 2007; Glaeser and Ward, 2009). In contrast, we will see that theory implies that intra-city differences, which prevent households from consuming the optimal type of house, lead to lower house prices and lower land rents in areas with more restrictions, given that houses in different areas are perfect substitutes. One of the main empirical difficulties of measuring the welfare implications of housing supply policies is that supply restrictions, such as minimum lot sizes, subdivision rules and height restrictions, are difficult to measure because constraints are often implicit. So, city-level data with detailed information on regulations is often not available, let alone data on intra-city differences in regulations (Quigley and Raphael, 2005; Glaeser and Ward, 2009).
In this paper, we estimate the gross costs of within-city regulatory constraints regarding apartment size (subdivision rules) and building height using a spatial regression-discontinuity design. We focus on the city of Rotterdam, the Netherlands. A large part of the city centre was bombed in World War II, so we use the bombing boundary as an exogenous difference in housing restrictions. After World War II, the city centre of Rotterdam was completely redeveloped and nowadays it is the only Dutch city with an American-style central business district with high-rise buildings (Koomen et al., 2008). A large part of the area that is not bombed is a conservation area, hosting many listed buildings. Regulations are considerably more stringent in the latter area. We will show that in the bombed area apartments are about approximately 35 percent larger at the bombing boundary, suggesting that apartment sizes are too large in the not bombed area. We will also show that buildings in the bombed areas, which are almost always multi-family buildings, are
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about 40 percent higher at the boundary. To our knowledge, this is one of the first studies that is able to investigate the impact of intra-city regulation differences, as one generally does not observe a city with different regulations (Glaeser et al., 2005a; 2005b; 2006; Aura and Davidoff, 2007).1
We use parametric and semiparametric estimation procedures and show, in line with theory, that floor space prices are about 10 percent higher in the bombed area, implying substantial costs of planning restrictions. Our results suggest that previous studies likely provide an underestimate of the costs of building height restrictions, if these restrictions go together with restrictions on house size, which is frequently the case (e.g. Gyourko et al., 2008).
The semiparametric approach implies that we estimate a partially linear hedonic price function where the control variables are nonlinearly related to the dependent variable. This approach is not yet much applied in empirical applications, although allowing for nonlinearities in the control variables near the boundary is fundamental (Angrist and Pischke, 2009). We find no evidence of (extreme) household sorting at the bombing boundary, which facilitates the identification of the effect of restrictions. We also examine the external effects of building height, but do not find any negative effects, suggesting that building height restrictions generate few external benefits. As we may not capture all benefits induced by current regulation, we will indicate how large the benefits must be to justify the current restrictions.
This paper continues as follows. In Second 2, we outline theory and discuss the estimation methodology. Section 3 considers the data. In Section 4 we present the results and Section 5 concludes.
2. Model and methodology
2.1 TheoryConsider a small area in a city with many locations that are perfect substitutes.2 Households either locate at
a certain reference location and receive utility from floor space (or apartment size) and a composite commodity , so , or reside at another location where they receive reservation utility .
1 One exception is the study of Pollalowski and Wachter (1990), which investigates the impact of differences in land
use controls on house prices within localities in Montgomery County.
2 It is important to note that the use of a closed-city model is less accurate when describing the effects of intra-city
regulatory differences. Because we only focus on a small area, any reduction in supply due to housing supply restrictions is small compared to total supply of housing and is unlikely to induce any increase in apartment prices.
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Households maximise utility subject to a budget constraint , where denotes the price of residential floor space and the price of is normalised to one.3 In the reference location, the optimum
amount of floor space consumed is and the indirect utility function is then . In equilibrium, households have the same utility, so . In the reference location, restrictions prevent households from consuming the optimal apartment size, so the consumption of floor space is restricted:
. Given , the equilibrium price given a restriction, , is defined by the following relationship . Given supply restrictions, so , the price is always lower than inducing lower land revenues for landlords.4 This is fundamental, as revenues from land may be
interpreted as a measure of social welfare in this model (see Brueckner, 1990). Apartment size restrictions are often combined with other restrictions such as building height constraints. It may be shown that land rents are then even lower. We will test these implications within our empirical framework.
In Appendix A, we illustrate the most important implications of theory. Given a Cobb-Douglas utility function and profit-maximising developers that construct apartments facing a constant returns to scale Cobb-Douglas production function with land and building height as inputs, we obtain an explicit expression for land rent. The model predicts, for example, that when apartment size is 50 percent above its optimal value, floor space prices and land rents are respectively 8.5 and 15 percent lower. When building height is half its optimal value, land rents are 19 percent lower.
2.2 Empirical model
The most straightforward methodology to investigate the impact of house supply restrictions would be to regress the floor space price on the stringency of supply constraints, controlling for housing attributes, such as number of rooms and construction year, and neighbourhood attributes, such as height of nearby buildings (as in Ihlandfeldt, 2007). According to theory the price should be higher in areas with less supply restrictions, ceteris paribus. However, it is very likely that not all control variables, in particular relevant
3 For simplicity, we ignore external amenities (although we will control for amenities in the empirical application) and
commuting costs (in a small area commuting costs are more or less equal for all households). In reality, we may also expect that households have heterogeneous preferences. Households with a stronger preference for space will sort themselves in larger apartments. However, in our empirical application we will show that there is little sorting with respect to household characteristics (income, family size) along the boundary.
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neighbourhood attributes, are observed and, hence, the estimates may suffer from omitted variable bias (Black, 1999). Glaeser et al. (2005b) and Ihlanfeldt (2007) argue that policy constraints may be endogenous (e.g. whether local residents form lobby groups (that influence policy) may depend on the mix of type of dwellings in the neighbourhood). Furthermore, hedonic price models do generally not identify supply effects only, because the curvature of the price function is a combination of supply and demand (Ekeland et al., 2004).
To address these problems, one may only include observations that are close to a boundary that represents an exogenous difference in a regulatory policy (see Black, 1999; Bayer et al., 2007). We regress the logarithm of residential floor space price (per square meter) at location on structural attributes (e.g. the log of apartment size) and neighbourhood variables, a dummy indicating whether an area was bombed in 1940 and transaction year dummies :
(1) ,
where , and are parameters to be estimated and is a matrix with housing and neighbourhood variables . Black (1999) points out that this methodology is equivalent to calculating differences in mean apartment prices on opposite sides of the boundary, conditional on control variables (see also Imbens and Lemieux, 2008).
As is well known, a hedonic price function is a combination of demand and supply functions. So, in the hedonic price function we control for apartment size to take into account that the costs of supplying larger apartments falls with floor space (decreasing costs to scale), that the marginal utility of apartment size is usually diminishing and, importantly, that households may differ in their preferences regarding apartment size. Planning restrictions may prevent households from consuming their optimal apartment size. As these restrictions are discontinuous at the boundary, while the hedonic price function with respect to apartment size is continuous, we are able to identify the effect of restrictions at the boundary. Our analysis allows for household sorting based on apartment size, but household sorting usually reduces the effect of apartment size restrictions on prices. More specifically, in the extreme case of perfect household sorting, the price effect of apartment size restrictions will be zero (so there is always a household for which the restricted
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size is the optimal one). In the other extreme case of no household sorting, the price effect will obtain its maximum.5 See Appendix B for a more detailed explanation based on diagrams.
We use a weighted regression estimator to estimate (1), as suggested by Hahn et al. (2001). This implies that observations near the boundary get more weight than observations farther away:
(2) ,
where denotes the weight of location . We will start our empirical analysis with a uniform weight scheme:
(3) ,
where is an indicator function that equals one when the condition is true (and zero otherwise), and is some threshold distance from the boundary. To correct for a potential bias that is larger when observations are further away from the boundary (for example, because of omitted unobserved factors or location-specific preferences), it is sometimes preferred that observations near the boundary get more weight. We therefore also use a bisquare weight scheme, which is often used in spatial applications:
(4) ,
so, properties closer to the boundary receive more weight than distant ones.
Angrist and Pischke (2009) argue that it is very important to allow for nonlinear effects of control variables, especially around the boundary. Otherwise, the jump in the dependent variable at the boundary may not be caused by the treatment, but simply corresponds to a misspecification of the hedonic price function. We therefore estimate the following price function:
(5) ,
where is some function of housing attributes and neighbourhood characteristics. We estimate this function by local linear regression techniques for all observations within .6 Locally weighted regression
is the most common nonparametric approach to analyse spatial data, as it allows for a flexible functional
5 We find little evidence of households sorting at the boundary, so we max expect a measurable price effect of
restrictions.
6 Angrist and Pischke (2009) suggest that may be estimated using polynomials. Given our large number of
covariates, this is not feasible, as this will lead to the so-called curse of multidimensionality. Local methods have a lower asymptotic bias than the Nadaraya-Watson estimator and a lower asymptotic variance than the Gasser-Müller estimator (Bajari and Kahn 2005). Lee and Lemieux (2010) suggest that regression-discontinuity estimation should not rely on one particular method or specification. In our analysis we test three types of specifications, so we investigate whether our results are sensitive to the choice of specification.
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form and interactions between the control variables (McMillen and Redfearn, 2010). So, one estimates for each location a weighted regression based on a multivariate kernel. Then, a kernel may be employed that is based on housing attributes and neighbourhood characteristics (see e.g. Bajari and Kahn, 2005):
(6) ,
where is the kernel weight of in the local regression of and is a kernel function of a chosen bandwidth and the difference between the attributes . Most studies use the same kernel function for all variables. However, as is noted by Racine et al. (2006) it is preferable to use a different kernel function for continuous and dichotomous variables. For continuous variables we use a conventional Gaussian kernel function:
(7) .
For dichotomous variables we employ the following kernel function, following Racine et al. (2006):
(8) ,
The choice of the values of bandwidths (one for continuous and the other for dichotomous variables) is important here (Sain et al., 1994; Bajari and Kahn, 2005). Lower bandwidths lead to a lower mean-squared error, but to higher variance of the estimator. Larger bandwidths may create a larger bias when the underlying function is nonlinear (Fan and Gijbels, 1996). We use a cross-validation procedure to determine the bandwidths (see Racine et al., 2006). A cross-validation score (CV) is then defined as:
(9) ,
where denotes the fitted value of with the observation of omitted from the calibration process. We choose the bandwidths that minimise the cross-validation score. Sain et al. (1994) argue that cross-validation usually leads to undersmoothing of the kernel (so too low bandwidths), so the control variables’ specification is too flexible and the effect of bombing then will be an underestimate. We therefore interpret the semiparametric estimate as a lower bound estimate.
Equation (5) is partial linear, as the treatment and the transaction year dummies are assumed to be linearly related to . We therefore employ the procedure proposed by Robinson (1988) to estimate (5). First, we regress , and on nonparametrically. Then, we regress the residual of on
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the residuals of and the residuals of the time dummies. This leads to -consistent estimates for and . The last step is to regress nonparametrically on to obtain a consistent estimate of .
3. Regional context and data
3.1 Bombing of RotterdamThis analysis focuses on Rotterdam, the second largest city of the Netherlands. The current city’s population is 584,060 (the wider urban area contains about 1.2 million inhabitants). In the beginning of World War II, Rotterdam was bombed (by German air forces). The bombing only took 20 minutes, but due to the bombing and fires the old city centre was completely destroyed, including 25,000 houses and 11,000 other buildings. About 850 people died and 80,000 inhabitants became homeless (about 12 percent of the city’s population at that time). Although the impact was not as large as in some Japanese and German cities due to Allied bombings, Figure 1 shows that almost no building within the bombed area survived. After World War II, the city of Rotterdam decided not to rebuild this area, but to completely redevelop the whole area.
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3.2 Planning restrictions
The Dutch government is strongly involved in the planning and regulation of residential locations, both in terms of volume, house type as well as the location of residential development. One of the legal responsibilities of the local government is to develop land use plans and preserve the existing housing stock. Regulatory restrictions generally encompass height restrictions, zoning restrictions, preservation of open space and prevention of land use fragmentation. Subdivision of a property into several apartments is allowed in Rotterdam, but the building’s exterior often has to remain intact, which makes subdivision usually prohibitively expensive. We emphasise that height restrictions imply implicit size restrictions as height restrictions decrease the profits of redeveloping a complete building. The national government provides subsidies in order to stimulate local governments to maintain the building stock, especially in conservation areas.7 Regulations with respect to construction and adding extensions to existing buildings
are more stringent in the latter areas, using the argument that cultural heritage located in these areas improves the quality of life (see Municipality of Rotterdam, 2007).
Regulatory constraints are less restrictive (and binding) in the bombed area in two measurable ways. First, apartment size differences between the two areas are substantial: apartments are about 35 percent smaller in the bombed area. Second, height restrictions are less stringent in the bombed area, as the average building is about 40 percent taller (about 4.9 meters) in the bombed area. The buildings destroyed in World War II are frequently replaced by buildings that are taller. However, sometimes a bombed building is not replaced by a new structure.8
We emphasise that the conservation area policy likely creates external benefits, but outside the conservation area, destroying older buildings and increasing building height by one or two floors will unlikely lead to substantial, if any, external costs. This is particularly so for buildings in specific streets (e.g. streets that are broad, so views are not constrained, or when the ground floor is used for shopping). So, our costs estimates of a marginal change in policy in Section 4 likely reflect overall welfare effects.
7 A conservation area contains buildings that are protected by the national government because of special
architectural and historic interest. In our data analysed later on, 39 percent of the properties in the not bombed area are located in a conservation area. In the bombed area it is only 4 percent.
8 As a supplement we include some photos that illustrate adjacent buildings that are bombed and not bombed. It is
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3.3 Data
The first dataset used contains information on the current boundary of the bombing area. Figure 2 shows that the boundary’s current measurement is exactly identical to that in Figure 1.
We also use a property dataset from the NVM (Dutch Association of Real Estate Agents). It contains information on a large majority (about 75 percent) of (owner-occupied) apartment transactions between 1985 and 2007.9 For 50,777 transactions we know the transaction price, the exact address, and a wide
range of structural attributes such as floor space size (in square meter, excluding gardens), number of rooms, garden, and construction year. We also know whether the apartment is in a listed building. Construction year controls for a range of difficult to observe amenities of apartments (e.g. building quality, size of windows, architectural style etc.). Furthermore, and this is less common, we enrich the property dataset with information on the building height of buildings using building information obtained from the Department of Infrastructure and Environment (Koomen et al., 2009).
Figure 2: Rotterdam in 2007
9 In the analysis we refer to houses as apartments, as 95 percent of the observations are apartments located in
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Black (1999) argues that it would be ideal to use a spatial discontinuity design wherein the prices of apartments on opposite sides of a street (that may serve as a boundary) are compared, to overcome omitted variable biases. As this will lead to an extremely small property dataset, we select apartments that are within 500 meter of each side of the bombing boundary.10 This is about 20 percent of transactions in
the Rotterdam municipality, although the selected area is only 3 percent of the total municipality area. After these selections, the dataset consists of 7,593 housing transactions.
In the analysis, we control for a range of locational attributes. In particular, we include dummy indicators whether a property is within 50 metres of a major road, park or open water, within 150 meter of commercial buildings or a metro station.11 Furthermore, we include the distance to the city centre and the
average construction year of building stock within 150 meters of the property. To estimate the external effect of building height of other buildings, we include the height of the tallest building within 150 meter of the property. Furthermore, using a dataset from the Rijksmonumentenregister (listed building register), we control for the number of listed structures (including residential buildings) within 500 meters of each property and include a dummy indicating whether the property is located within a conservation area,
Descriptives are presented in Appendix C. The average price per square meter is € 1,654. It is € 1,572 and € 1,745 in respectively the bombed and not bombed area. So, unconditional on any other attributes, the average price per square meter is about 11 percent lower in the bombed area. This is not too surprising as in the not bombed area, apartments have many price-increasing attributes.12 Furthermore, the average
distance to the boundary is only 233 meters.
10 So, we set the threshold distance to 500 meters. We furthermore exclude transactions with prices that are above
€ 1.5 million or below € 25,000 and have a price per m2 which is below € 250 or above € 5,000. We also delete
observations that refer to properties smaller than 25m2, or larger than 500m2.
11 Land use data are for the year 2000, which is reasonable because the land use pattern has hardly changed between
1985 and 2007. Only the pattern of commercial land use may have changed somewhat, but there are no major changes that may bias our results.
12 For example, on average, there are only 10 listed buildings within 500 meter of the property in the bombed area but
18 in the not bombed area. There are essentially no single-family properties in the bombed area, whereas 11 percent in the not bombed area are single-family properties. The share in garden is also much higher in the not bombed area.
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4. Results
4.1 Graphical analysis
We start our empirical analysis by graphing the data close to the bombing boundary. Similar to Bayer et al. (2007), we regress the variable of interest on 25 meter distance band dummies. We plot coefficients of the distance bands and add a linear trend line on both sides of the boundary. Figures 3-10 present the results. Negative distances on the left hand side indicate locations that are not bombed.
Figure 3: Price per square meter Figure 4: Price per square meter with control variables
Figure 5: Apartment size Figure 6: Garden
Figure 7: Building height Figure 8: Construction year of buildings
7.10 7.20 7.30 7.40 7.50 7.60 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 lo g( p ri ce p er m ²) Distance to boundary Not Bombed Bombed
7.10 7.20 7.30 7.40 7.50 7.60 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 lo g( p ri ce p er m ²) Distance to boundary Not Bombed Bombed
50 75 100 125 150 175 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Si ze in m ² Distance to boundary Not Bombed Bombed
0.0 0.1 0.2 0.3 0.4 0.5 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 G ar d en Distance to boundary Not Bombed Bombed
9.00 12.00 15.00 18.00 21.00 24.00 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Hei gh t i n m Distance to boundary Not Bombed Bombed
1920 1940 1960 1980 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Co n st ru ct io n Y ear Distance to boundary Not Bombed Bombed
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Figure 9: Household income Figure 10: Household size
In line with previous descriptives, Figure 3 shows that the logarithm of floor space price in the bombed area is somewhat lower. In Figure 4 we present the price when controlling for attributes (see the next section for details). Now we find that the price is substantially higher in the bombed area: a substantial discrete change of about 0.16 log points is observed at the boundary. Note also that by controlling for attributes, price changes become rather continuous over space, except at the boundary, indicating that a discontinuity regression is a useful approach in this context.
Figure 5 shows that apartments in the bombed area are substantially smaller: at the boundary, apartments are approximately 35 percent smaller.13 This is an important observation and suggests that
households in the not bombed area occupy too large apartments.14 This result is in accordance with
Ihlanfeldt (2007), who found that apartments are substantially larger in more restricted areas. In Figure 6, we present the share of properties with a garden. The share is substantially lower in the bombed area: the difference at the boundary is 0.18. This suggests also that households in the not bombed area consume too much space.
Our property data imply that the average building height of apartments is 14.8 meters. This is a somewhat misleading measure for building height: taller buildings are overrepresented as they host more apartments. The building-weighted average is substantially smaller and only 10.7 meter (weights are equal
13 Discontinuous control variables do not invalidate the discontinuity design, because we estimate the effect of
bombing conditional on covariates (Imbens and Lemieux, 2008).
14 We may only speculate why these apartments are too large. However, this is consistent with the observation that
after the war, high-income households began to leave cities, while the poor began to inhabit the city centres. Poor households prefer smaller apartments, but the housing stock in the not bombed area still consists of many (large) pre-war structures. Another explanation is a rather steep drop in family size after World War II.
10,000 17,500 25,000 32,500 40,000 47,500 55,000 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Y ear ly In com e in € Distance to boundary Not Bombed Bombed
1.0 1.5 2.0 2.5 3.0 3.5 4.0 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 Hou seh old Si ze Distance to boundary Not Bombed Bombed
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to the inverse of the number of observed apartments in a building). To be more accurate with respect to building height, we use another building dataset, which covers all buildings within 500 meters of the bombing boundary (10,288 buildings).15 We investigate the building height and construction year along the
boundary (we weight by the building’s surface area). Figure 7 shows that there is a substantial discrete jump of building height at the boundary of about 40 percent (about 4.8 meters). Newly constructed buildings adjacent to not bombed properties are usually higher.16 So, developers face fewer height
restrictions in the bombed area. Figure 8 highlights the devastating effect of the bombs: the difference in the average construction year at the boundary is about 32 years. It may also be shown that the other control variables, including the number of rooms and share of apartments with central heating, are continuous at the boundary. The finding that the number of rooms is continuous, whereas apartment size is discrete at the boundary, strongly suggests that households in the not bombed area are constraint to consume too large rooms (rather than too many rooms).
It may be argued that household sorting may reduce, or even nullify, the impact of local regulatory constraints, as households that prefer larger apartments will sort themselves in areas with larger properties (Bayer et al., 2007). We therefore investigate the differences in household income and household size, the two main indicators of housing demand, around the boundary in Figures 7 and 8.17 It
appears there is no substantial sorting as household income and household size are approximately continuous at the boundary. This suggests that a specific type of household prefers to live close to the boundary: the descriptives indicate that households mainly consist of childless couples with a slightly below national average income (about 13 percent).
15 Descriptives of this dataset (including both residential and commercial buildings) are presented in Table C2 in
Appendix C. It is for example shown that the buildings’ surface area is much larger in the bombed area, the average difference is 195.19 square meters. The building height, weighted by surface area, for the bombed and not bombed area is respectively 16.38 and 11.53 meters. The weighted average construction year is respectively 1964 and 1939.
16 See the Supplement for some clear illustrations.
17 We use a survey of WDM, a marketing service provider, which is held between 2004 and 2009 and contains 973
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4.2 Discontinuity regression
Table 1 presents the results for two linear specifications (with uniform and bisquare weights) and a semiparametric specification. For the semiparametric specification we present the coefficient of bombing and its standard error and for the control variables, which are nonlinearly related to the dependent variable, the means and standard deviations of the coefficients. The average coefficients of the semiparametric estimation are similar to the coefficients of the parametric regression for most control variables, which increases confidence in the parametric regressions.18
Table 1: Results of discontinuity regression (dependent variable is log of price per m²)
Parametric Regression Semiparametric
Regression
Uniform weights Bisquare weights
Location Bombed in 1940 0.148 (0.009) *** 0.135 (0.010) *** 0.045 (0.014) *** Apartment attributes Size (log) -0.097 (0.017) *** -0.109 (0.018) *** -0.349 0.185 Terraced 0.099 (0.020) *** 0.099 (0.022) *** 0.017 0.105 (Semi-)Detached 0.363 (0.033) *** 0.410 (0.052) *** 0.064 0.137 Number of Rooms -0.004 (0.004) 0.000 (0.004) 0.013 0.032 Parking 0.134 (0.017) *** 0.142 (0.022) *** 0.079 0.145 Garden 0.075 (0.008) *** 0.078 (0.009) *** 0.071 0.060 Central Heating 0.065 (0.007) *** 0.057 (0.008) *** 0.063 0.052 Construction Year 1945-1959 -0.032 (0.007) *** -0.035 (0.008) *** 0.001 0.077 Construction Year 1960-1970 0.038 (0.017) ** 0.064 (0.021) *** 0.030 0.166 Construction Year 1971-1980 0.024 (0.009) ** -0.019 (0.011) * 0.051 0.131 Construction Year 1981-1990 -0.013 (0.009) -0.047 (0.010) *** 0.024 0.138 Construction Year 1991-2000 0.185 (0.011) *** 0.189 (0.014) *** 0.281 0.187 Construction Year ≥ 2001 0.302 (0.017) *** 0.322 (0.022) *** 0.321 0.144 Building Height (log) 0.032 (0.007) *** 0.045 (0.008) *** 0.051 0.156 Listed Building 0.127 (0.018) *** 0.059 (0.038) -0.002 0.032
Neighbourhood attributes
Max Building Height <150m (log) 0.030 (0.006) *** 0.021 (0.007) *** 0.023 0.120 Conservation Area 0.077 (0.009) *** 0.127 (0.011) *** 0.067 0.124 (Listed Structures <500m)/10 0.004 (0.000) *** 0.004 (0.000) *** 0.006 0.009 Distance to Centre 0.131 (0.008) *** 0.092 (0.009) *** 0.056 0.172 Major Road <50m 0.026 (0.006) *** 0.067 (0.007) *** 0.014 0.126 Park <50m 0.357 (0.026) *** 0.339 (0.037) *** 0.130 0.216 Water <50m 0.020 (0.009) ** 0.035 (0.010) *** 0.063 0.114 Metro Station <150m -0.026 (0.010) *** -0.017 (0.010) * -0.082 0.131 Commercial Land Use <150m -0.008 (0.008) 0.014 (0.009) 0.002 0.089 Construction Year Buildings <150m (3th order polyn.) Yes Yes Yes
Transaction Year Dummies (22) Yes Yes Yes
Number of Observations 7,593 7,593 7,593
R² 0.753 0.763 0.924
NOTE: Robust standard errors are between parentheses. Coefficients are significant at *0.10, **0.05 and ***0.01 levels. For the semiparametric specification, we present the coefficients’ mean and standard deviation, except for Location Bombed in 1940.
18 Figure C1 in Appendix C1 demonstrates that when the bandwidths for continuous and dichotomous variables are
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All specifications show that the price of residential floor space in the bombed part is statistically significant different from that in the not-bombed area, with estimates that vary between 4.5 to 14.8 percent. As emphasised before, the semiparametric estimate of 4.5 percent based on the cross-validation procedure is interpretable as a lower-bound estimate. When we, for example, double the bandwidths, the effect of bombing increases to 9.2 percent.19 Recall that properties in the bombed area are 35 percent
smaller than in the bombed part where regulatory constraints are less restrictive. So, our findings are in line with theory, which states that for apartments closer to their optimal size, prices are higher (see Appendix A). We will use a 10 percent value for the welfare analysis that will follow.
In addition to analysing the price effect of apartment size restrictions, it is also insightful to investigate how these supply restrictions capitalise into land rents. In order to do so, we use information about marginal construction costs of floor space. The marginal construction costs per square meter are about € 1,320 per square meter up to 6 floors (after that it increases more than proportional to € 2,000 for the 22st
floor).20 The average height of a floor in our sample is 2.74 meter. The land rent is calculated as the
difference between the benefits (price of floor space times the number of floors) and construction costs. Table 2 presents several welfare calculations for a square meter of land, assuming a perfect competitive construction sector.21 Welfare losses are also expressed as a percentage of the land rent (in the not bombed
area).
The reference column of Table 2 presents the land rent per square meter for an ‘average’ residential building that is subject to subdivision and height restrictions. This building has a height of 10.15 meters (the average height of residential buildings in the not bombed area). In column (1) of Table 2, we focus on a building that provides apartments of a more optimal size (while holding building height constant), assuming a 0.10 effect of supply restrictions on apartment prices. The welfare costs of restricting apartment size are then 28 percent of the land rent. The costs are still 12 percent when we take our most
19 A factor two is not very large. For example, Bishop and Timmins (2007) multiply their ‘optimal’ bandwidth even
with five and interpret that as the best bandwidth.
20 We obtain information on construction costs from NEN, an institute which provides norms for constructions costs
of buildings. In particular, see NEN 2631, paragraph 3.2. We have verified this information using building costs websites, such as www.bouwkosten.nl.
21 The assumption of perfect competition seems to hold. In Rotterdam, there were 1,429 construction firms in 2006
with on average 11.75 employees. Only 1.6 percent of the firms have more than 100 employees. It is therefore unlikely that this market is controlled by a few large firms (see similarly Glaeser et al., 2005b).
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conservative estimate of size restrictions of 0.045 (see (2)). In (3), we suppose that building height increases to 12.11 meter (the average height of residential buildings in the bombed area), so by only 1.60 meters. The welfare loss of protecting buildings, so of restricting both apartment size and building height, is then 52 percent of the land rent, about twice as high as the welfare effects for (1). The welfare benefits of fewer restrictions in the bombed area (compared to the not bombed area) are in the order of € 1,000 per year per household residing in the bombed area,22 about 2 percent of the annual income, which is
substantial and in accordance with the literature (e.g. Cheshire and Sheppard, 2002; Glaeser et al. 2005b; Glaeser and Ward, 2009).
Table 2: Welfare calculations per square meter of land
Removal of subdivision
restrictions Removal of height restrictions (and subdivision rules)
Reference Average (1) estimate (2) Low estimate (3) +1.60m +1.60m (4) +10.15m (5) Effect of size restrictions 0.100 0.045 0.100 0.000 0.100 Price of floor space € 2,068.26 € 2,275.09 € 2,161.33 € 2,275.09 € 2,068.26 € 2,275.09
Floors 3.70 3.70 3.70 4.42 4.42 7.41
Revenues of land owner € 7,658.46 € 8,424.31 € 8,003.09 € 10,047.04 €9,133.68 €16,848.62 Construction costs € 4,887.76 € 4,887.76 € 4,887.76 € 5,829.26 € 5,829.26 € 9,812.12 Land rent € 2,770.70 € 3,536.55 € 3,115.33 € 4,217.78 €3,304.41 €7,036.50 Welfare loss € 765.85 € 344.63 € 1,447.08 €533.71 €4,265.79
Welfare loss/land rent 28% 12% 52% 19% 154%
Column (4) highlights that, at least for Rotterdam, the costs of housing supply restrictions are severely understated when one ignores apartment size restrictions. This is interesting because previous studies ignored the costs of size restrictions. Given size restrictions, height restrictions will then seem to lower land rents with only 19 percent. When we allow for even higher buildings, let’s say an additional 10.15 meters (twice the average building height in the not bombed area, which is a widely applied height constraint), the welfare loss per unit of land is 154 percent of the land rent (see (5)).23
The effect of average building height in vicinity is small but positive: doubling the height of the tallest building within 150 meter increases the price per square meter with 1.4 to 2.0 percent. So, despite the negative externalities of reduced views frequently suggested in literature, the net external effect of tall
22 The real interest rate was about 3 percent per year in 2007. Landowners then received 3 percent per year of €
1.477 per m² (difference in land rent), times 270,674 m² (surface area of residential buildings in bombed area). Divided by 12,300 (number of households in the bombed area), this makes € 975 per household per year
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buildings appears to be positive.24 Hence, our results indicate that in the area examined for Rotterdam,
housing restrictions reduce social welfare as it is extremely unlikely that negative externalities exceed the costs (reported in Table 2). So, it will be beneficial for society to allow for a replacement of an existing structure that has a suboptimal size by a building that is higher and provides a more optimal size. Especially for neighbourhoods without listed buildings, external effects of taller buildings are either positive or negative and small at the margin, or even positive, and are therefore unlikely to offset the estimated costs. These neighbourhoods are common: in our study area, 30 percent of the apartments in the not bombed area are neither listed nor have any listed structures within 150 meters.
The other control variables have, in general, plausible signs. Households are willing to pay less per square meter when the size of an apartment is larger. Another observation is that the apartment’s building height is positively related to rents: doubling building height leads to a 2.2 to 3.6 percent increase in the price per square meter. Because we do not observe the floor at which the apartment is, this is likely an underestimate of the effect of a nice view.25 Households are also willing to pay to live close to listed
buildings. The effect of a standard deviation increase in the number of listed structures within 500 meters (about 16 structures) leads to an increase in prices of about 1 to 8 percent. Residing in a conservation area leads to a price increase of 7 to 11 percent. We also find a positive coefficient of distance to centre, which indicates that households, conditional on choosing a destination close to the centre, do not prefer to live in the city centre because of crowding effects, such as noise pollution, traffic congestion and increased criminality. Similar findings have been reported for several American cities, where for parts of the city, the price gradient is positive (e.g. Richardson, 1977; Roback, 1982)
24 A small positive external effect of apartment height makes sense as Finally, households appreciate
aesthetically-appealing landmarks (Rotterdam is known for it modern architecture) and negative crowding externalities are internalised using other regulations (e.g. high buildings require private parking to reduce demand for street parking). Another reason for a positive effect might be that higher buildings in the surrounding imply a positive option effect that future building height regulation will be relaxed (Thibodeau, 1990), consistent with the common view that the effective building height regulation depends on the building height in the surrounding (as policies set building height rules relative to existing buildings).
25 In the analysis, we (implicitly) use the logarithm of the expected apartment height, which is half the building height,
rather than the logarithm of the apartment height, which is unobserved. As is well known, this creates a bias towards zero, as we have a measurement-in-variable error problem. In this special case however, the variance of measurement error is known and is equal to the variance of the logarithm apartment height. Given the assumption that the logarithm of apartment height (and therefore also its measurement error) is not correlated to any other explanatory variable in the model, the estimated coefficient is exactly 50 percent of its true value (Verbeek, 2000).
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4.3 Robustness checks
In this section, we provide robustness checks for parametric specifications. Table 3 summarises the main results for the parametric regressions with uniform weights. The results with bisquare weights are very similar (and can be received upon request).
Table 3:Sensitivity analysis for parametric discontinuity regressions
Parametric regression, Uniform weights
(1) (2) (3) (4) (5) (6) (7) (8)
Location Bombed in 1940 0.080)
(0.014) *** (0.009) 0.156)*** (0.009) 0.136)*** (0.011) 0.144)*** (0.014) 0.152)*** (0.009) 0.135)*** (0.009) 0.121)*** (0.009) 0.180)***
Distance to Boundary (2) No No No Yes No No No No
Polynomial of Size (6) No No Yes No No No No No
Control Variables Included Yes Yes Yes Yes Yes Yes Yes Yes Number of Observations 2,113 7,593 7,593 7,593 7,593 7,191 7,593 7,332
R² 0.812 0.751 0.762 0.762 0.708 0.769 0.726 0.757
NOTE: See Table 1.
One may argue that we do not control in a proper way for unobserved effects that may differ in distance to the boundary. For example, Figure 8 suggests that the price per square meter, conditional on the treatment effect, is somewhat higher further away from the boundary. We allow for different effects in both areas. We therefore include two variables, each measuring the distance to the boundary. This hardly influences our results (see (4)).
In our sample, most properties are apartments. Large buildings hosting many apartments may have a disproportionate impact on the identification of the effect of bombing. Hence, we run a weighted regression where the weights are equal to the inverse of the number of transactions per building. Again, the effect is similar (see (5)). When we exclude terraced and (semi-)detached houses, this leads to nearly identical results (see (6)). Excluding relevant control variables, such as ‘listed building’, ‘listed structures <500m’ and ‘conservation area’ does not lead to different conclusions (see (7)), although these variables are heavily correlated with supply restrictions.
Finally, we examine the potential negative bias due to a error in the measurement of the boundary. It may be the case that the boundary is not exactly observed for all apartments. This will be particularly true for apartments that are on the boundary and partially destroyed by the bombs. We therefore exclude all observations within 12.5 meters of the bombing boundary. The effect of restrictions is somewhat higher now (about 18 percent), suggesting that, if anything, our estimates are conservative.
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5. Conclusions
In contemporary cities, spatial planning policies are applied that aim to protect the current building stock, which may offer external benefits to inhabitants and visitors. However, protecting the building stock may also lead to substantial costs as it prevents optimal housing supply. Households that reside in protected buildings often are unlikely to occupy properties that have the households’ preferred size. According to theory, when supply restrictions prevent households from consuming the optimal size, the equilibrium floor space price is reduced. Combined with height restrictions this will lead to suboptimal land rents and a reduction in social welfare. We test this implication for the city of Rotterdam using a (semiparametric) regression-discontinuity approach. A part of the city of Rotterdam was bombed in World War II. Supply restrictions are much more stringent in the not bombed area than in the bombed area. Households residing in the bombed area occupy apartments that are 35 percent smaller and have fewer gardens. At the same time, in the bombed area buildings are 40 percent higher at the boundary.
The results show that prices are about 10 percent higher in the bombed area, where fewer building height restrictions and implicit subdivision rules apply. This positive effect is robust to many different specifications. Furthermore, we find no evidence that taller buildings induce negative externalities. Allowing for new buildings that are only 1.60 meters higher and that contain more optimally-sized apartments increases social welfare by about 50 percent of land rents. This suggests that local governments should consider relaxation of height restrictions and subdivision rules, especially in areas where negative externalities of taller buildings are expected to be minor.
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Appendix A. Impact of building constraints on land rents: a numerical example
Households residing at the reference location maximise a Cobb-Douglas utility function
,
where denotes a preference parameter and , subject to a budget constraint (the price of is normalised to one). The optimal consumption of the composite commodity is then , and the optimal consumption of floor space is . In equilibrium, . The price for floor space is then. When , the price for floor space is .
A developer constructs apartments and obtains price for space provided. The inputs of the production function are land and building height . The profit function is , where is an elasticity parameter, , is the land rent and is the marginal cost of constructing floor space. Given standardised to one, the optimal building height is . When we insert into the profit function and assume zero profits, we obtain an explicit expression for the optimal land rent:
. When building height is restricted, , the land rent is equal to . We assume that the preference parameter is 0.7 (non-housing expenditure is on average 70 percent) and is 0.5. In Figure 1, we see by how much land rents are reduced when
and .
Figure A1: Relation land rent, optimal size and building height
50 75 100 125 150 44 51 58 65 72 79 86 93 100 60 70 80 90 100 110 120 130 140 150 HRas % of H* r as % of r* sRas % of s* 93-100 86-93 79-86 72-79 65-72 58-65 51-58 44-51
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Appendix B. Hedonic equilibrium and identification
In Figure B1 we compare prices per square meter in an area with a restricted house size to an area without this restriction, so where is optimally chosen. Note the similarity with the approach adopted in Rosen (1974). When we have a continuum of preferences of an infinite number of households, we observe that the price decreases from to . So, there are different households, one with utility function and another one with utility function . This price difference is fully attributable to the heterogeneity in marginal willingness to pay for house size. However, when there is only one type of individual (e.g. ), we see that the price decrease is much larger (from to ). By controlling for house size, we measure the price difference attributable to an inefficient consumption of space (from to ), rather than due to household sorting. It is easily observed that without controlling for house size, the effect of supply restrictions would seem to be equal to and therefore would be overstated.
Figure B1: Hedonic equilibrium and identification
s sR π1 u2 π2 A pB B pC pA u1 C 0 5000 -55 25 Pri ce House Size Demand Supply Price Function
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Appendix C. Descriptives
Table C1: Descriptive statistics of property data
Full Sample Sample, Bombed Sample, Not Bombed
Average Std.Dev. Average Std.Dev. Average Std.Dev.
Price per m2 1,655 594 1,572 532 1,745 643.000
Bombed 0.525 0.499 1.000 0.000 0.000 0.000
Distance to Boundary (in km) 0.233 0.154 0.184 0.134 0.288 0.155 Apartment attributes Size (in m2) 99.202 46.915 82.064 23.518 118.122 57.779 Apartment 0.947 0.224 0.999 0.027 0.889 0.314 Terraced 0.042 0.201 0.001 0.027 0.088 0.283 (Semi-)Detached 0.011 0.104 0.000 0.000 0.023 0.150 Number of Rooms 3.488 1.404 3.145 0.855 3.865 1.751 Parking 0.032 0.177 0.019 0.138 0.047 0.211 Garden 0.148 0.355 0.050 0.217 0.256 0.436 Central Heating 0.853 0.354 0.853 0.354 0.854 0.353 Construction Year <1945 0.280 0.449 0.088 0.283 0.492 0.500 Construction Year 1945-1959 0.256 0.436 0.434 0.496 0.059 0.236 Construction Year 1960-1970 0.024 0.153 0.027 0.163 0.021 0.142 Construction Year 1971-1980 0.105 0.306 0.151 0.358 0.054 0.227 Construction Year 1981-1990 0.202 0.401 0.243 0.429 0.156 0.363 Construction Year 1991-2000 0.109 0.312 0.043 0.203 0.183 0.387 Construction Year ≥ 2001 0.024 0.154 0.015 0.120 0.035 0.184 Building Height (in m) 14.837 13.581 15.392 11.786 14.223 15.298
Listed Building 0.015 0.123 0.004 0.061 0.028 0.166
Neighbourhood attributes
Max Building Height <150m (in m) 23.752 19.808 26.935 21.515 20.238 17.061 Conservation Area 0.206 0.404 0.041 0.199 0.388 0.487 Listed Structures <500m 13.621 16.634 10.094 10.338 17.515 20.865 Distance to Centre (in km) 1.273 0.630 0.876 0.422 1.712 0.520
Road <50m 0.290 0.454 0.297 0.457 0.282 0.450
Park <50m 0.010 0.100 0.000 0.000 0.021 0.145
Water <50m 0.164 0.371 0.260 0.438 0.060 0.237
Metro Station <150m 0.063 0.243 0.081 0.273 0.044 0.205 Commercial Land Use <150m 0.667 0.471 0.876 0.330 0.437 0.496 Construction Year Buildings <150m 1941 23 1955 15 1925 20
Number of Observations 7,593 3,984 3,609
Table C2: Descriptive statistics of building data
Full Sample Sample, Bombed Sample, Not Bombed
Average Std.Dev. Average Std.Dev. Average Std.Dev. Unweighted Sample
Building Surface (in m2) 190.802 523.352 353.128 725.572 157.942 465.202
Building Height (in m) 9.805 4.851 11.390 7.927 9.485 3.869
Construction Year 1929 37 1958 23 1923 37
Weighted Sample
Building Height (in m) 13.045 11.191 16.384 13.044 11.534 9.880
Construction Year 1947 40 1964 22 1939 43
Number of Buildings 10,288 1,732 8,556
NOTE: The weight variable is the building’s surface. These data consist of all buildings in the study area, commercial as well as residential buildings.
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Figure C1: Bandwidth choice
0.8 0.9 1.0 1.1 1.2 1.3 0.0200 0.0205 0.0210 0.0215 0.0220 0.0225 0.0230 0.1 0.2 0.3 0.4 Bandwidth hm (continuous variables) CV -s cor e
Bandwidth hm(dichotomous variables)
0.0230-0.0234 0.0225-0.0230 0.0220-0.0225 0.0215-0.0220 0.0210-0.0215 0.0205-0.0210 0.0200-0.0205
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Supplement
(to be published separately online)
Figure S1: Construction years 1894 and 1956 Figure S2: Construction years 1894 and 1957
Figure S3: Construction years 1888 and 1966 Figure S4: Construction years 1894 and 1957
Figure S5: Construction years 1894 and 1997 Figure S6: Construction years 1912 Not Bombed Bombed
Not Bombed Bombed
Not Bombed Bombed
Not Bombed Bombed
Not Bombed Bombed
