REGULATION AND THE SHADOW PRICE OF HOUSING: A TEST OF THE NEO-CLASSICAL URBAN ECONOMIC MODEL

Transcription

1 REGULATION AND THE SHADOW PRICE OF HOUSING: A TEST OF THE NEO-CLASSICAL URBAN ECONOMIC MODEL AARON SWOBODA Abstract. Using a simple model of housing production, this paper develops and implements a statistical test of the importance of regulation in explaining the price of housing. The model is implemented using data on over 18,000 new homes constructed in inland Southern California over a ten year period to estimate the gap between extensive and intensive marginal values of land. Results indicate that regulation increases the price of housing by constraining production in 10 of 14 regions studied. In 7 of the 14 regions, the effect on the price of housing is over $40,000. This indicates that regulation constraining the production of housing can have very large wealth effects. 1. Introduction Housing affordability is a major issue facing today s policy makers. Fannie Mae reports that as a result of high housing prices, over 14 million U.S. households spend over 50% of their income on housing services, an increase of 67% since The first step to alleviating the pressure of seemingly ever increasing housing prices is to understand their cause. One hypothesis developed and supported by much of the urban economics literature is that housing is expensive where land is expensive. This literature is characterized by Muth [1969], Mills [1967], and Alonso [1964]. 2 Brueckner [1983] extends the literature to include yard space as an urban amenity. The conclusions Advisor: Dave Sunding. This work has greatly benefitted from the comments of my peer reviewers, Ricardo Cavazos and James Manley, as well as comments from seminar participants. Ben Arnold of Charles River Associates was immensely helpful with data management, and of course, all mistakes are mine. 1 FannieMae [2003] 2 Brueckner [1987] provides a great synthesis of the standard model and literature in the Handbook of Regional and Urban Economics. 1

2 2 AARON SWOBODA of these models state that land must be valued equally at both the intensive and extensive margins. If not, profit maximizing producers could increase their profits by reallocating land towards the higher valued use. This means that in equilibrium the value of land for an additional unit of yard space should equal the marginal value of using the land to provide more housing. This result is appealing for its intuition and simplicity; however, constraints on the market, such as those imposed through regulation, will cause it to fail. This paper presents a simple model showing that any regulation constraining the quantity of housing available in the market drives a wedge between the value of land at the extensive, more housing, and intensive margin, more yard space. These constraints may come in many forms, be it zoning rules, restrictive growth policies, or even some environmental regulations, but the results are the same; in regulated markets the extensive margin value of land will be greater than the intensive margin value. By testing whether the value of land is greater at the extensive margin, we can test whether the market in question is facing quantity constraints. We are not the first to test this hypothesis, most notably Glaeser and Gyourko [2002] find that several cities across the nation have substantial gaps between the value of land at the extensive and intensive margin. However, this paper has four distinct advantages over this previous work. Most notably, we were able to incorporate the tools of geographic information systems (GIS), allowing analysis at a finer geographic scale. Secondly, our price data are from arms length sales transactions, whereas Gyuorko and Glaeser rely on survey data of homeowners opinions of their home s value. Thirdly, our dataset is composed entirely of newly constructed homes. This has the advantage of representing recent density production decisions, and also allows us to ignore issues of depreciation. Lastly, we differentiate among housing construction qualities, rather than assuming all houses are of uniform quality. We believe the conclusions of this paper have an important voice in the discussion of the efficacy of regulations facing housing markets. Evidence that the value of land is significantly higher at the extensive margin suggests that the housing

3 REGULATION AND THE SHADOW PRICE OF HOUSING 3 market is already facing binding quantity constraints. In such circumstances, additional quantity restrictions will have significantly higher welfare impacts than in unconstrained markets. Conversely, policies that increase the costs of housing construction are not likely to affect the market equilibrium: since the price of housing is already greater than the marginal construction cost, policies such as increased development fees or permitting requirements will largely act as pure wealth transfers. These results have large repercussions in the cost-benefit analysis of new policies and serve to underscore the importance of understanding the fundamental nature of the regulated market. The next section offers a simple model of housing production and consumption, and formulates our testable hypothesis. Section 3 describes the data in detail and provides summary statistics for the study region. This is followed by a detailed description of the process used to calculate the extensive margin value of land. Section 5 explains the hedonic regression analysis used to estimate the value of land at the intensive margin. Section 6 then tests the hypothesis that the value of land at the intensive value is at least as great as the value at the extensive margin. We find that in most of the study region we can reject this null hypothesis, and conclude that the extensive margin value of land is substantially higher than the intensive margin value of land. We show that in some areas, this gap routinely accounts for 15% or more of the price of a home. The paper concludes with a further discussion of the policy implications of our results. 2. The Model Imagine a competitive firm in a neighborhood building houses in a profit maximizing manner. The firm must choose the number of houses to build and the amount of land associated with each house. Let H be the total number of houses produced (ignore, for now, issues of housing quality, and only consider issues of quantity). Let L be the quantity of land used per house. Let P ( ) represent the price facing the producer for each house produced as a function of the amount of land associated with it and the neighborhood specific index of amenities (α). If r is the per unit price of land, and k( ) is the cost of constructing H units of housing,

4 4 AARON SWOBODA then the profit maximization problem is (1) max L,H π = P (L, α)h rlh k(h) The profit maximizing First Order Conditions are given in equations (2) and (3) (2) π H = P rl k H = 0 (3) π L = P L H rh = 0 Equation (3) simplifies to P L = r, which when substituted into (2) yields... (4) P L = P k H L P L is the intensive margin value of land, the value of adding one more unit of land to a house. The RHS of (4) is the extensive margin value of land, the rents earned from the production of housing. Let ω denote the value of land at the extensive margin. In the unconstrained equilibrium these will be equal and the producer is indifferent between using the marginal unit of land as yard space or construction of a house. However, imagine that outside forces constrain the maximum number of houses produced in a neighborhood to be H. The maximization problem is then (5) max L,H,λ L = P (L, α)h rlh k(h) + λ(h H) The First Order Conditions are now equations (6), (7), and (3). (6) L H = P rl k H λ = 0 (7) L λ = H H = 0 Under this regulated equilibrium, instead of (4), we find (8) P L = P k H λ L Equation (8) shows that in constrained markets, the standard urban economics result no longer holds; land will be valued higher at the extensive margin than the intensive margin. With the proper data, we can estimate λ and make inferences about underlying equilibrium. Calculating the extensive margin value of land is

5 REGULATION AND THE SHADOW PRICE OF HOUSING 5 simple arithmetic, once the construction cost data are obtained. The intensive margin value of land cannot be calculated explicitly; it is the increase in the price of the house from an extra unit of land. As such, it must be inferred using hedonic regression techniques to estimate the marginal contribution of land to the price of the overall bundle of housing purchased. It is important to note that whereas we can obtain point estimates of the extensive margin value of land for each house in our dataset, estimates of the intensive margin value of land can only be obtained by comparing houses with different prices and land quantities. As a result, we must aggregate the data to a region level, and estimate the intensive margin value of land for a region. 3. The Data The data for this analysis were originally purchased from DataQuick 3, a company that aggregates and distributes real estate data. In this case, the data are the sales of roughly 18,000 newly built homes in Southern California between 1993 and There are several advantages to restricting the analysis to newly built homes. First, they represent recent density production decisions, which should give a more accurate estimate of the value of land. Secondly, the homes are of a common vintage, which reduces confounding issues of quality. For each home sale, the dataset contains information on the transaction, including the physical attributes of the home and basic characteristics of the sale. Most importantly, the dataset contains the geographic location each transaction, which allows the use of geographic information systems technology to help in the analysis. Specifically, Figure 1 maps the study region including parts of Los Angeles, San Bernardino and Riverside Counties, to the near east of the city of Los Angeles. It is a region characterized by high rates of recent growth, and represents a substantial portion of the Los Angeles exurban growth. It is also beginning to feel growth pressure from the south as San Diego continues to expand. To the north lie the San Gabriel 3

6 6 AARON SWOBODA +,-.0 $%&%'()* +,-./ +,-./ +, :; +,-123 +,- / +,-./ M >DR@IHB=BJDKL EBFGDHI?IF=BJDKL SC>DHI=BJDKL UTVWXYZ +,-/ T Figure 1. Overview of Study Region nopnqorstuvwxyzo{qs m [\]^_`]a]^bcdefag`h]ijgk^dcl\ < =>?@ABCD@> and San Bernardino Mountain Ranges, and the southwestern border of the study region is bordered by the Santa Ana Mountains. As stated earlier, an important step in my analysis is to estimate the hedonic price function for housing, in order to calculate the marginal contribution to price from land in a neighborhood. Most of this section is devoted to describing the variables in this function, the quantity of land, size of the house, as well as other variables that may affect the price of a house in Southern California. However, an important component of the analysis involved choosing an appropriate aggregation level at which to analyze the data. The smaller the individual neighborhood, such as census tract, the more plausible the assumption of constant land prices, but the fewer observations with which to estimate the price. Larger analysis regions, such as the county level, afforded a greater number of observations, but called into question the assumption of constant land prices. In the end, sub-county regions were used as the compromise aggregation level. These regions are defined by the U.S. Census Bureau, and are displayed in Figure 2.

7 REGULATION AND THE SHADOW PRICE OF HOUSING 7 $ %! "# % % &''&'(')*+,- % Figure 2. Map of Analysis Regions 3.1. The Price of Housing. Figure 3 shows the distribution of sales price over the entire study region. The distribution can be characterized by a majority of the observations at the lower end of the distribution (minimum sale price=$43,250), and a few observations at extremely high values skewing the distribution to the right (maximum sale price=$1,137,000). Table 1 shows the distribution of housing sale prices by subregion Lot Size. The variable of most interest in this study is the amount of land associated with each house. From equation (4) we know that profit maximizing house producers will continue to demand land as an input for yard space until the marginal revenue is driven to the marginal cost of its provision. It should come as no surprise that lots in the study region tend to be small (the mean lot size is essentially.2 acres). In the entire dataset, less than 5% of homes had lot sizes over one-half acre (approximately 22,000ft 2 ), in the final analysis, the data were censored to include only observations with lots smaller than an acre (this represented removing fewer than.9% of the observations). Figure 4 gives the reader a visual sense of

8 8 AARON SWOBODA Frequency Sale Amount (in $1,000s) Figure 3. The Distribution of Sale Price Table 1. Distribution of Sale Price, in $, by Region Sub-County Region Min Mean Max σ N Corona 71, , ,500 80,401 1,832 E. San Gabriel Valley 56, ,425 1,060, , Elsinore Valley 55, , ,000 80,359 1,597 Hemet-San Jacinto 47, , ,727 34, Jurupa 96, , ,773 54, Lake Mathews 95, , , , Murrieta 58, , ,000 68,382 3,946 Norco 200, , ,818 87, Ontario 105, , ,500 75,017 2,958 Perris Valley 43, , ,000 57,911 1,465 Riverside 54, ,297 1,137,000 95, San Bernardino 85, , ,000 43,230 2,672 San Gorgonio Pass 49, , ,000 53, Yucaipa 97, , ,000 80, Entire Study Area 43, ,996 1,137,000 88,300 18,227 the overall distribution of lots in the study area. Table 2 gives the reader a more detailed description of the distribution of lot size throughout the region.

9 REGULATION AND THE SHADOW PRICE OF HOUSING Frequency Lot Size (in sq ft) Figure 4. The Distribution of Lot Size Table 2. Distribution of Lot Size, in ft 2, by Region Region Min Mean Max σ N Corona 2,178 8,543 41,382 4, E. San Gabriel Valley 2,648 12,254 42,828 7, Elsinore Valley 4,356 8,842 37,462 3, Hemet-San Jacinto 3,049 7,235 23,522 2, Jurupa 4,356 7,635 39,204 4, Lake Mathews 6,970 23,778 43,560 12, Murrieta 2,614 7,725 43,124 3, Norco 16,988 24,869 43,124 5, Ontario 2,767 7,576 40,946 3, Perris Valley 3,049 7,119 43,560 3, Riverside 2,178 9,073 43,560 5, San Bernardino 3,150 7,392 39,000 3, San Gorgonio Pass 3,049 7,104 39,204 4, Yucaipa 5,000 11,325 43,560 6, Total 2,178 8,206 43,560 4, Other Variables. This section describes the available variables in the dataset. Although land is an important determinant of the price of a house, other characteristics of the homes are vital in any housing price hedonic function. This dataset

10 10 AARON SWOBODA Table 3. Descriptive Statistics of Control Variables Variable Min Mean Max σ Living Space (ft 2 ) , , Number of Bedrooms Number of Bathrooms contains the following elements variables the size of the house (living space) and the number of bedrooms and bathrooms in the house. Other important variables may be flags for the presence or absence of other potentially important housing amenities like a swimming pool or fireplace. The variables for the number of bedrooms and bathrooms in the house are treated as continuous variables since the values are in fact cardinal, and a difference of one represents a real difference between homes regardless of scale. Although it seems implausible to consider rooms in non-integer quantities, it is conceivable, and quite common, for the number of bathrooms to consist of a fraction of a bathroom (for example, a half bathroom is equivalent to a bathroom consisting of only a toilet and sink). Descriptive statistics for the square footage of living space, and number of bedrooms and bathrooms in each house in the dataset are given in Table 3.The binary variables representing the presence or absence of housing amenities are summarized in Table 4. Table 4. Description of Binary Variables Yes No Swimming Pool 7.2% 92.8% Fireplace 90.7% 9.3% Issues of Time. As can be seen from the Table 5, the home sales are well distributed throughout the study period of ten years. In order to avoid problems of improper comparisons, all analysis is in terms of 2003 dollars. Understanding that real estate prices change over time subject to macroeconomic fluctuations and other market changes, the house prices were transformed into year 2003 dollars using the Conventional Mortgage Home Price Index (CMHPI), which is published quarterly

11 REGULATION AND THE SHADOW PRICE OF HOUSING 11 Table 5. Temporal Distribution of Observations Year Built N by Freddie Mac 4. This index uses repeat home sales to establish a price index for Metropolitan Statistical Areas (defined by the Office of Management and Budget) over time in order to compare home prices across time. For this analysis, I used the index for Los Angeles, and San Bernardino/Riverside. The sales prices of homes, adjusted by index, are shown in Figure The Extensive Margin As discussed and defined earlier, the value of a unit of land at the extensive margin, ω, is the difference between the price of the home and the construction costs of the home divided by the amount of land associated with the home. The Dataquick dataset contains the price and quantity of land, but no information on the construction costs of the home. We use the Residential Cost Handbook by Marshall and Swift [2002] to obtain estimates of the construction costs for each house in the dataset. The extensive margin value of land is then calculated for each of the regions Construction Costs. The Handbook provides a means of estimating the construction costs of homes based on their location, size, quality, and construction materials used. The Dataquick dataset contains the absolute location and size of each house, two of the most important determinants of housing construction costs, 4 Index data are available at the Freddie Mac website,

12 12 AARON SWOBODA 2000 Frequency Inflated Sale Amount (in $1,000s) Figure 5. The Distribution of Inflated Sale Price but does not explicitly reveal the quality of the home or construction materials used. Therefore, we must make some assumptions in order to proceed. First, we assume that all houses are constructed of the same material, wood framed stucco. This construction material has the advantage of being common, and is neither the most, or least, expensive material detailed in the Handbook. This reduces the problem to determining the quality of the house. Rather than assuming all of the houses in our study are of one quality, we categorize houses as one of four quality levels to allow for different construction costs, average quality for homes less than $350, 000, good quality $350, 000 $750, 000, very good quality for homes $750, 000 $100, 000, and excellent quality for homes greater than one million dollars. The largest fear of a less-than perfect quality determination is that it leads to over rejection of the null hypothesis, which occurs if the estimated construction costs are biased downward. We use the sale price of the home as a guide to the construction quality of the home even though there is clearly not a one-to-one mapping from the price of the house to its construction

13 REGULATION AND THE SHADOW PRICE OF HOUSING 13 quality. This rule will bias our results against the null hypothesis for houses with low prices but high construction quality of, an unlikely situation. Additionally, while the Handbook details the construction costs of houses for six levels, from low to excellent, we assume that the houses in the dataset are solely of the top four quality levels. The decision rule is shown in Table??. (It should be noted that this rule results in 14% of the homes having estimated construction costs greater than 90% of their sale price.) With an estimate of the quality, and location and size data, the total production costs can be calculated. Aside from the physical construction costs, we incorporate the costs of the design process, site preparation, and marketing the homes for sale. Just as costs of construction vary across quality levels, these additional costs vary across quality levels. These additional costs can be considered either hard costs, costs associated with converting land to a suitable construction site, and soft costs, fees and other costs associated with design, marketing, and obtaining the necessary permits for construction. We use estimates of $35 per square foot for hard and soft costs for average quality homes, and $55 for higher quality houses 5. From equation (4), the value of land at the extensive margin, ω, is equal to the price of the house minus the construction costs of the house divided by the quantity of land associated with the house. Table 6 shows the mean, and weighted mean extensive margin value of land by region. The right columns represent a weighted average, where each observation is weighted by its relative lot size within the region. It is less subject to be influenced by values associated with small lots. Each area weighted estimate is lower than the simple arithmetic mean, and will be used throughout the rest of the analysis. Figure 6 shows the spatial distribution of the weighted mean extensive margin value of land across the study region. 5. The Intensive Margin Value of Land This section displays and describes the results from the regression analysis used to estimate the intensive margin value of land. As is stated earlier, the primary 5 Personal communication with the Newhall Construction Company

14 ! "# $ % 14 AARON SWOBODA Table 6. The Value of Land, Extensive Margin Region Non-Weighted ω σ N Area-Weighted ω σ N Corona $ $ E. San Gabriel Valley $ $ Elsinore Valley $ $ Hemet-San Jacinto $ $ Jurupa $ $ Lake Mathews $ $ Murrieta $ $ Norco $ $ Ontario $ $ Perris Valley $ $ Riverside $ $ San Bernardino $ $ San Gorgonio Pass $ $ Yucaipa $ $ &'()*+,-)./01,*2/34)567/*8 :;<=>?@ABCD ;<=E:F<=>?@ABCD F<=E:G<=>?@ABCD I 9 : :G<=>?@ABCD JK J L JMNOPQ H Figure 6. The Spatial Distribution of the Extensive Margin Value of Land goal of this section is to estimate the marginal contribution of land to the price of a house. Given a hedonic function, we can then estimate the intensive margin value of land by estimating the change in price of a house from a change in the amount

15 REGULATION AND THE SHADOW PRICE OF HOUSING 15 of land associated with the house. (9) House P rice = P (location, living space, lot size, amenities) We are interested in estimating P L, the change in the price of a home given an increase in the amount of land associated with the house. This is an estimate of the intensive margin value of land, and represents the opportunity cost of devoting land to the production of more housing. The first estimation method employed is Ordinary Least Squares. OLS is appealing for its simplicity and ease of use, but makes restrictive assumptions regarding the data and functional form of the model. As a result, other estimation strategies are used to check the robustness of the results, including Huber-Tukey robust estimation to allow for the presence of influential outliers. Other functional forms are also common in the housing literature, so the intensive margin value is also estimated using a log-log specification and the flexible Box-Cox Transformation Ordinary Least Squares. The simplest manner in which to estimate the intensive margin value of land is to assume that the marginal contribution is constant. If this is true, then the hedonic function in (9) can be rewritten as (10). The result of such a formulation is that the regression coefficient for the quantity of land is the estimate of the intensive margin value of land. (10) P = α + ˆβL + X ˆδ + ɛ Where α is a neighborhood specific constant, L is the amount size of the lot for the house, ˆβ is the intensive margin value of land, X is a vector of other house attributes, ˆδ is the vector of their marginal effects, and ɛ is the usual error term. Following this estimation strategy, the adjusted price of housing was regressed on the size of the lot, the size of the house, the number of bedrooms, the number of bathrooms present in the house, dummy variables indicating the presence or absence of swimming pools and fireplaces, the age of the house, and the year of the sale. (Including the year of the sale allows for regional differences in the trend of housing prices than those specified in the Freddie Mac real estate inflator.)

16 16 AARON SWOBODA The results of these neighborhood regressions are displayed in Table 7, with each column representing an independent regression for each area. Upon examining the table, it seems clear that the quantity of land present is an important determinant of the price of the house. Indeed, 11 of the 14 estimates are deemed significant at the 1% level (while two of the other three are significant at the 5% level). A careful examination reveals that the three regions with the least statistically significant lot size coefficients are also the three regions with the smallest samples in the area. Other coefficients are predominantly as one would expect. The coefficients for square footage of living space are positive and predominantly in the range of $70-$100. There are some surprising results in from these regressions. Most notably, the negative coefficient for square footage in column (6). With only 34 observations, this regression seems to contain little useful information. The reader also may be surprised by other coefficients of the wrong sign. A greater number of bedrooms present in the house seems to decrease the sale price. At first glance this may seem to turn convention on its head; one would generally assume that houses with more bedrooms would be worth more. This rule of thumb, however, may still be correct if we ponder the impact of a marginally increasing the number of bedrooms, keeping all other variables constant, especially the size of the house. Comparing houses similar in all other aspects, houses with fewer (and presumably larger) bedrooms are then valued in the market higher. There are other examples of coefficients being the wrong sign, bathrooms are supposedly bad in regressions (4), (5), (7), (10), and (11). Although luckily only statistically significantly so in two instances. The presence of swimming pools and fireplaces yield little descriptive power, and sometimes in a nonintuitive manner. While using the Freddie Mac real estate inflator presumably removed temporal influences, a glance at the coefficients for time (calculated as years past 1993) shows that time still retains descriptive power for the price of the houses.

17 REGULATION AND THE SHADOW PRICE OF HOUSING 17 Table 7. OLS Coefficients for All Regions (1) (2) (3) (4) (5) (6) (7) Lot size, ft (0.24)** (0.40)** (0.28)** (0.45)** (0.36)** (1.41)* (0.20)** Living Space, ft (3)** (5)** (3)** (3)** (4)** (41)* (2)** Bedrooms -12,710-20,991-12,070 2, ,221-5,964 (1,613)** (3,785)** (1,271)** (1,446) (2,572) (18,119) (869)** Bathrooms 20,684 12,065 22,091-7,971-11, , (2,870)** (4,382)** (2,866)** (3,283)* (5,096)* (39,373)** (1,740) Swimming Pool 4,028-3,574 3,239 5,804 15,393 6,830 (3,267) (7,841) (4,206) (4,425) (7,782)* (2,517)** Fireplace -24,499 43,662-3,946 18,914 8, ,800 2,110 (12,410)* (76,507) (4,750) (4,015)** (7,478) (45,226)* (4,123) Age of House 5,533 16,022 7,187 2,619 7,796-35,536 11,766 (1,749)** (3,632)** (1,810)** (1,301)* (2,998)** (21,475) (1,183)** time -4,566-9,291 1,599-1, ,370 2,747 (453)** (1,561)** (457)** (359)** (660) (7,760)* (276)** Constant 98,139 25,175 58,622 70, , ,919 86,555 (13,710)** (13,636) (7,089)** (6,924)** (11,459)** (89,592)** (5,093)** Observations R-squared (1)=Corona, (2)= E. San Gabriel Valley, (3)= Elsinore Valley, (4)=Hemet-San Jacinto, (5)=Jurupa, (6)=Lake Mathews, (7)=Murrieta, (8)=Norco, (9)=Ontario, (10)=Perris Valley, (11)=Riverside (12)=San Bernardino, (13)=San Gorgonio Pass, and (14)=Yucaipa (8) (9) (10) (11) (12) (13) (14) Lot size, ft (0.95)* (0.38)** (0.27)** (0.45)** (0.18)** (0.52)** (0.47) Living Space, ft (15) (3)** (3)** (6)** (2)** (6)** (8)** Bedrooms 1,440-2,550-15,108-8,799-2,353-20,972-4,019 (7,883) (2,043) (1,002)** (4,104)* (914)* (2,214)** (4,777) Bathrooms 60,826 41,096-2,322 26,520 1,551 20,460-14,699 (11,753)** (3,740)** (2,849) (7,350)** (1,595) (5,226)** (6,439)* Swimming Pool -8,041 4,358 11,092 14,301 11,882-28,052 1,506 (18,124) (4,768) (3,907)** (8,976) (2,712)** (9,082)** (9,799) Fireplace -105,771 18,046 28,709 28,287 6,407 15,099-16,223 (42,137)* (5,516)** (2,666)** (8,571)** (2,094)** (7,967) (23,438) Age of House 15,807-8,461 2,195 1,063 1,846 7,745 2,498 (9,524) (1,905)** (1,102)* (3,408) (757)* (2,530)** (2,827) time 2,222-4, ,632-6,131 3,538 (2,589) (586)** (372)* (936) (234)** (576)** (1,231)** Constant 228,231-7,294 80,678-24,107 58,038 40,189 93,112 (64,188)** (9,872) (5,554)** (16,081) (4,098)** (10,788)** (27,687)** Observations R-squared Figure 7 visually displays the spatial variation of the estimates of the intensive margin value of land in the study region. Table 7 shows that the in most circumstances the estimated intensive margin value of land is lower than the extensive margin values of land calculated in Table 6. Before undertaking the hypothesis test, the remainder of this section is devoted to estimating the intensive margin value of land using different techniques to check the robustness of the OLS results.

18 18 AARON SWOBODA *+),! -. ()! " # $%&' / :;9<=>;?@AB=9C@DE;FGH@9IJKHL NOPQRSTUV OWOXYNZWXPQRSTUV M N ZWXYN[PQRSTUV N[PQRSTUV \ Figure 7. Spatial Distribution of OLS Estimates 5.2. M-Estimation. Ordinary Least Squares is a member of the class of estimators known as M-Estimators, so called because they are the result of a min(max)imization. The general M-Estimator objective function can be written as (11) min n ρ(e i ) where e i = y i X i ˆβ i=1 That is, choose ˆβ in order to minimize a function of the residuals. OLS sets ρ(e i ) = e 2 i (minimizing the sum of the squared residuals). This makes the OLS estimator especially sensitive to outliers. One solution in such circumstances is to weight these outlying observations less in the minimization process, by choosing a ρ( ) that does not increase as rapidly as e 2 i, which reduces the influence of extremely high residuals. One common technique was developed by Huber [1964], and is commonly implemented in concert with a system of bi-weighting developed by Beaton and Tukey [1974]. Throughout the rest of the paper, the results of this estimation process will be referred to as Huber Tukey (H-T) estimates. Table 8 shows the estimated intensive margin values of land using the Huber Tukey methodology. The results show no strong indication of influential outliers pulling the estimates of

19 REGULATION AND THE SHADOW PRICE OF HOUSING 19 the intensive margin value of land towards zero. In fact, rather than increasing the estimates, the estimated marginal effects decrease in more regions than it increases Log-Log Formulation. Both of the previous estimation strategies assumed a constant marginal effect of yard space on the price of a house. The next two parts estimate the intensive margin value of land allowing for non-constant marginal effects from lot size. The first formulation is the log-log specification, which is commonly used in the literature regarding housing, including Glaeser and Gyourko [2002]. Under a log-log specification, equation (12) is estimated. (12) ln(p ) = ˆα + ˆβln(L) + ɛ Given the formulation in equation (12), the intensive margin value of land is no longer equal to the coefficient associated with the size of the lot of the house; it varies with the lot size and predicted price. Equation (13) shows how the intensive margin value of land is calculated in a log-log specification. (13) PL = ˆP L ˆβ Under a log-log specification, the standard error of P L will be equal to (14) σ PL = σ ˆβ ˆP L The calculated intensive margin values of land for each region are shown in Table 8, and are calculated at the median values for the independent variables. The estimates of the intensive margin value of land increase relative to the OLS estimates in only 3 of the 14 regions Box-Cox Transformation. The previous estimates of the intensive margin value of land were calculated while assuming that the marginal value of land was constant across lot sizes (or, in the case of the log-log function, a constant elasticity). This section calculates the intensive margin value of land for the study region using the Box-Cox transformation, a flexible functional form that allows for non-constant returns to land. The Box-Cox transformation allows for varying functional forms,

20 20 AARON SWOBODA from the pure linear formulation (θ = 1), to log-linear (θ = 0), to even a model that is linear in reciprocals (θ = 1). The Box-Cox estimates (15) y = ˆα + ˆβ xˆθ 1 ˆθ From examining equation 15, the reader will notice that the estimate of β is no longer a measure of the marginal contribution to price from the regressor (the exception, of course, is if θ = 1 in which case we are in the linear case again). Under the Box-Cox transformation, the estimate of the intensive margin value of land is shown in equation (16). Following convention, the marginal effects of lot + ɛ size are estimated at the median of all the regressors. (16) PL = ˆβLˆθ 1 Since the estimate of P L is a function of both ˆβ and ˆθ, we must use the delta method as shown in equation (17) to calculate the standard errors of P L (see Chapter 10: Nonlinear Regression Models in Greene [2000] for a description of obtaining the variance-covariance matrix). [ (17) σ pl = P L ˆβ P L ˆθ ] V ar[ ˆβ] Cov[ ˆβ, ˆθ] Cov[ ˆβ, ˆθ] V ar[ˆθ] P L ˆβ P L ˆθ Table 8. The Intensive Margin Value of Land Region OLS H-T Log-Log Box-Cox P L ˆσ PL ˆσ PL ˆσ PL ˆσ Corona $5.41 (0.244) $3.94 (0.158) $3.53 (0.298) $4.23 (2.249) East San Gabriel Valley $2.81 (0.407) $2.77 (0.343) $3.60 (0.494) $3.55 (2.474) Elsinore Valley $0.71 (0.280) $0.39 (0.221) $0.15 (0.303) $0.19 (0.175) Hemet-San Jacinto $1.71 (0.447) $1.74 (0.290) $1.00 (0.347) $1.20 (2.389) Jurupa $1.25 (0.366) $1.26 (0.342) $1.46 (0.523) $1.31 (2.298) Lake Mathews $3.97 (1.458) $2.93 (1.220) $1.83 (1.410) $0.46 (13.287) Murrieta $4.87 (0.198) $0.70 (0.125) $1.57 (0.197) $3.09 (1.410) Norco $2.18 (0.951) $0.62 (0.707) $1.86 (0.814) $0.51 (25.985) Ontario $4.38 (0.383) $3.91 (0.349) $3.78 (0.327) $1.40 (1.663) Perris Valley $1.13 (0.274) $1.75 (0.226) $1.79 (0.299) $1.54 (0.974) Riverside $4.36 (0.452) $1.37 (0.146) $3.61 (0.384) $0.48 (0.906) San Bernardino $4.57 (0.182) $3.51 (0.125) $2.97 (0.135) $3.84 (2.064) San Gorgonio Pass $2.91 (0.524) $3.09 (0.453) $1.21 (0.653) $1.57 (2.123) Yucaipa $0.58 (0.453) $0.65 (0.389) $0.65 (0.559) $0.63 (1.225)

21 ! "# $ % REGULATION AND THE SHADOW PRICE OF HOUSING 21 &'()'*+,)-./0+'1.23) C ;<=>;<=>?;@<=>;@<=>?;A<=>B;A<=> DE D F D : Figure 8. Spatial Distribution of Box-Cox Estimates 6. Hypothesis Testing In this section we test the null hypothesis that the value of land at the extensive margin is less than or equal to the value at the intensive margin. The intensive margin values from the Box-Cox estimation are used, as well as the area-weighted estimates of the extensive margin value. Recall that ω is the area-weighted mean calculated extensive margin value of land in a region, σ ω is the standard error of the mean extensive margin value, P L is the estimate of the intensive margin value, and ˆσ pl is the standard error of the intensive margin value. The null hypothesis is stated in (18), while the alternative is in (19). (18) H o : ω P L 0 (19) H a : ω P L > 0 Assuming that both ω and ˆ P L are distributed around the true values, the null hypothesis can be tested using a one-sided t-test. If the null hypothesis is true,

22 22 AARON SWOBODA Table 9. Calculating the One-Sided t Region ω P L t-statistic Corona $ E. San Gabriel Valley $ *** Elsinore Valley $ *** Hemet-San Jacinto $ Jurupa $ *** Lake Mathews $ Murrieta $ *** Norco $ Ontario $ *** Perris Valley $ *** Riverside $ *** San Bernardino $ ** San Gorgonio Pass $ *** Yucaipa $ *** **=significant at 5% level ***=significant at 1% level then (20) ω P L ˆσ 2 ω + ˆσ 2 pl t Table 9 displays the estimated gap between the extensive margin value and intensive margin value of land for the 14 regions in the study area. It also shows the calculated t-statistic for the null hypothesis that the intensive margin value is at least as great as the extensive margin value. In only 4 regions do we fail to reject the null hypothesis at the 5% level, and in fact we can reject the null hypothesis at the 1% level for 9 of the regions. Table 10 displays the impact of the estimated gaps on the median houses in each region. The center column represents the dollar magnitude of the gap for the median sized lot. This is then translated to the percentage of the median priced home in the region to give a sense of the relative magnitude in the region. Table 9 states that the estimated gap per square foot of lot is $4.73, which, at the median lot size, represents $35, 056, or 12.6% of the median priced home in the region. Rearranging equation (8) shows that the middle column is the shadow price of housing in each region.

23 REGULATION AND THE SHADOW PRICE OF HOUSING 23! " # $ % & ' ( &)*+,-./ :;<53=:>?2@AB:3C DEFGHEFGHIEJGHEJGHIEKGHLEKGH Figure 9. Distribution of Estimated Gaps Table 10. The Impacts of Gap Gap at Median Percentage of Region Lot Size Median Price Corona $24, % E. San Gabriel Valley $120, % Elsinore Valley $45, % Hemet-San Jacinto $14, % Jurupa $44, % Lake Mathews $16, % Murrieta $35, % Norco $58, % Ontario $77, % Perris Valley $32, % Riverside $43, % San Bernardino $29, % San Gorgonio Pass $36, % Yucaipa $57, % 7. Other Possibilities, and Future Research The previous section tested the hypothesis that the intensive margin value of land is at least as great as the extensive margin value of land. We rejected this null hypothesis at the 1% level for 9 of the 14 regions. This section explores other

24 24 AARON SWOBODA Table 11. Extensive & Intensive Margin Comparison for Murrieta, year 2001 Region Extensive OLS H-T Box-Cox ˆω ˆσ ω PL ˆσ PL ˆσ PL ˆσ Murrieta $6.93 (0.09) $3.98 (0.38) $0.35 (0.25) $2.16 (2.17) possible explanations for rejecting this hypothesis, including issues of time, spatial autocorrelation, and spatial aggregation Time. One possible reason for the low marginal contributions of land to the price of a house is that we are improperly constraining the hedonic to be constant across time. One way to check this result is to rerun our regressions allowing for different valuations of land across time. However, one problem with this approach is that there may not be enough observations in each region in each year in order to obtain accurate estimates. In 2001 there were 907 sales of homes in Murrieta that were built in the same year. The analysis for this area in order to shed light on the hypothesis that issues of time are causing the difference between the extensive and intensive value of land. Table 11 shows the analysis after restricting the data to Murrieta in the year 2001 using OLS, Huber-Tukey and a Box-Cox transformation. The calculated marginal effect of an increase in lot size after the Box-Cox transformation (calculated at the mean of all variables) is 2.16, with a standard error of This standard error is large enough that the upper limit of the 95% confidence interval for the marginal effect is However, a t-test (t=2.17) still allows us to reject the null hypothesis that the intensive margin value is at least as great as the extensive margin value. While this is not conclusive evidence that there are no temporal problems with the analysis, it provides the best opportunity to check since other time period/location combinations contain sparser data. One other possible test is to re-estimate the intensive margin values of land by region including dummy variables for the year of the sale. While not allowing for different valuations across time, it does allow for fundamental differences in the prices of housing across time. Comparing the results of Table 12 to those of Table 8, yields insignificant increases in the estimated intensive margin values of land in the regions. Perhaps more data in each

25 REGULATION AND THE SHADOW PRICE OF HOUSING 25 Table 12. Extensive & Intensive Margin Comparison with Time Dummies Region Extensive OLS H-T Log-Log ˆω ˆσ ω PL ˆσ PL ˆσ PL ˆσ Corona $7.49 (0.14) $5.14 (0.25) $3.64 (0.16) $3.84 (0.26) E. San Gabriel Valley $16.52 (0.23) $2.24 (0.44) $2.39 (0.36) $1.65 (0.55) Elsinore Valley $6.05 (0.11) $0.75 (0.28) $0.39 (0.21) $0.58 (0.21) Hemet-San Jacinto $3.16 (0.13) $1.68 (0.44) $1.68 (0.28) $0.99 (0.27) Jurupa $7.70 (0.22) $1.16 (0.35) $1.06 (0.24) $1.47 (0.38) Lake Mathews $1.22 (0.50) $4.55 (1.77) $1.79 (0.91) $2.41 (1.51) Murrieta $7.83 (0.09) $4.84 (0.20) $0.62 (0.12) $1.39 (0.15) Norco $3.03 (0.21) $2.85 (0.89) $0.80 (0.63) $1.77 (0.65) Ontario $13.22 (0.13) $4.28 (0.38) $3.66 (0.35) $3.89 (0.32) Perris Valley $6.23 (0.14) $1.13 (0.28) $1.72 (0.23) $1.15 (0.23) Riverside $6.37 (0.19) $4.59 (0.46) $1.48 (0.14) $3.17 (0.33) San Bernardino $7.99 (0.07) $4.43 (0.18) $3.26 (0.11) $2.71 (0.11) San Gorgonio Pass $7.61 (0.21) $3.16 (0.50) $3.20 (0.44) $1.95 (0.56) Yucaipa $6.83 (0.23) $0.77 (0.49) $0.59 (0.41) $1.16 (0.48) year would allow for a more in depth analysis across the region like that done for Murrieta in The current extent of the data capabilities, however, shows no strong influence of time on the marginal price effect of land Spatial Data Issues Choice of region. One possible explanation for the prevalence of rejected hypotheses is that my choice of region is too large, and the assumption of constant land prices does not hold. This may be more of a data problem than anything, as it seems that there is an insufficient quantity of data to define the regions much smaller. Perhaps a test could be conducted at the census tract level to see if the difference between extensive and intensive value of land diminishes. Table 13 presents the results of the same analysis done by census tract (but only using OLS rather than the robustness checks run earlier). It shows that a majority of the results hold, we have no choice but to reject the null hypothesis that the intensive margin value of land is at least as great as the extensive margin value in a majority of the analysis regions Spatial Autocorrelation. One of the crucial assumptions necessary for an unbiased estimator is that the error terms are uncorrelated across space. Given the

26 26 AARON SWOBODA Table 13. Analysis by Census Tract Region Extensive Intensive-OLS Gap t-stat N Tract Region Name ω ˆσ ω PL ˆσ ω P L Corona $7.91 (0.27) $5.17 (0.58) $ Corona $7.23 (0.21) $6.04 (0.37) $ Corona $12.18 (0.83) $1.32 (1.30) $ E. San Gabriel Valley $15.03 (0.37) $8.25 (1.33) $ E. San Gabriel Valley $11.72 (0.41) -$0.07 (2.57) $ E. San Gabriel Valley $21.39 (0.38) $3.11 (0.91) $ Elsinore Valley $6.23 (0.11) $0.32 (0.26) $ Hemet-San Jacinto $5.37 (0.44) $3.89 (1.98) $ Hemet-San Jacinto $3.62 (0.22) $2.92 (0.77) $ Hemet-San Jacinto $3.16 (0.14) $3.13 (0.51) $ Hemet-San Jacinto $0.54 (0.29) $0.93 (0.65) -$ Jurupa $6.89 (0.36) $2.20 (0.39) $ Jurupa $7.12 (0.39) -$0.99 (1.41) $ Jurupa $8.77 (0.21) $0.68 (0.47) $ Jurupa $7.59 (0.62) -$3.37 (2.87) $ Murrieta $18.12 (0.27) $4.60 (0.66) $ Murrieta $19.48 (0.24) $2.99 (0.67) $ Murrieta $11.06 (1.00) $3.51 (2.24) $ Murrieta $10.30 (0.35) $3.45 (0.95) $ Murrieta $9.40 (0.44) $2.35 (1.17) $ Murrieta $7.89 (0.16) $8.33 (0.34) -$ Murrieta $23.35 (0.64) $4.59 (2.88) $ Murrieta $9.62 (0.21) $1.96 (0.56) $ Murrieta $5.25 (0.27) $4.65 (0.69) $ Murrieta $7.23 (0.35) $0.62 (1.16) $ Murrieta $8.56 (0.12) $3.78 (0.25) $ Murrieta $6.00 (0.29) $8.83 (0.49) -$ Perris Valley $4.64 (0.36) $0.12 (0.51) $ Perris Valley $6.14 (0.19) $1.16 (0.93) $ Perris Valley $4.32 (0.20) $0.89 (0.78) $ Perris Valley $6.39 (0.25) $4.19 (1.12) $ Perris Valley -$0.95 (0.58) $1.24 (0.65) -$ Perris Valley $10.68 (0.28) $0.01 (1.56) $ Riverside $7.19 (0.28) $4.62 (0.64) $ Riverside $7.12 (0.46) $3.73 (1.02) $ Riverside $6.32 (0.16) $0.67 (0.32) $ Riverside $3.20 (0.31) -$1.90 (1.51) $ Riverside $4.32 (0.23) -$3.64 (1.52) $ San Bernardino $7.36 (0.14) $3.74 (0.52) $ San Bernardino $9.46 (0.16) $3.22 (0.38) $ San Bernardino $8.18 (0.12) $1.35 (0.49) $ San Bernardino $5.62 (0.29) $6.40 (1.10) -$ San Bernardino $9.51 (0.23) $2.04 (1.07) $ San Bernardino $10.27 (0.29) $2.19 (0.57) $ San Bernardino $5.66 (0.56) $1.47 (1.12) $ San Bernardino $9.46 (0.18) -$0.52 (1.02) $ San Bernardino $8.03 (0.15) $0.69 (0.45) $ San Gorgonio Pass $2.94 (0.29) $5.02 (0.74) -$ San Gorgonio Pass $13.07 (0.35) -$3.42 (1.55) $ San Gorgonio Pass $3.96 (0.15) -$0.21 (0.41) $ Yucaipa $7.93 (0.34) $2.10 (1.01) $ Yucaipa $5.76 (0.31) $1.28 (0.52) $

27 REGULATION AND THE SHADOW PRICE OF HOUSING 27 nature of the data, it seems reasonable, in fact almost certain, that there are spatial aspects that are not incorporated in our data. Future research could take advantage of recent computing advances, including the creation of spatial analysis software. This would include the use of Kriging techniques to create a structural spatial autocorrelation matrix. Table 13 shows that there is still a large degree of heterogeneity within the sub-county regions, as evidenced by the results by census tract. As such, this information could manifest itself in the positive correlation of error terms of nearby housing observations Over Restrictive Assumptions. All of this analysis assumes that the housing production sector is a perfectly competitive industry. Can the model be expanded to include a non-competitive market structure? Would the results hold for a monopolistic market? What assumptions regarding the demand for housing are needed? What about the cost structure of the industry? Herein we have assumed CRTS technology. The real world maybe be more aptly described by Increasing Returns to Scale on a housing project scale. How would this affect the model s predictions? 8. Conclusion Previous research hypothesized that the value of land must be equal at both the intensive and extensive margins. This equilibrium concept is appealing for its simplicity and intuition: the price of a good should reflect its opportunity cost. If this result did not hold, producers of housing could increase their profits by reallocating land towards the higher value use. However, this paper has developed a model showing that market regulations may create a permanent wedge between the marginal value of land for different uses. Our results indicate that the value of land at the extensive margin can be much higher than at the intensive margin. While different econometric techniques may diminish this gap, current research shows that this gap between the extensive and intensive margin values of land can easily account for 15-20% of the price of an average home in this study region. These results have multiple implications for policy makers. First, it reinforces the importance of understanding the underlying market conditions when assessing

28 28 AARON SWOBODA the impacts of a potential policy intervention. Policies will have large impacts if they restrict the quantity of housing if they occur in an already quantity constrained market. The test implemented in this market (where the value of land is higher at the extensive margin than at the intensive margin). This means that policy makers may wish to implement the same test that this paper, whereas, small adjustments in the costs of construction (i.e. through taxes or development fees) will have little impact in the market, other than acting as transfer payments from producers to regulatory agencies. This is in contrast to standard market analysis which states that an increase in the costs of production will result in reduced market output. We have shown that in a majority of the regions in our study area, the costs of construction are not the limiting determinant of the ultimate price of housing. Our results suggest that policy makers may have another tool to combat the high prices of housing in some areas. In markets with large gaps between the extensive and intensive margin values of land, policies that increase the supply of housing are likely to improve the affordability of housing in an area. In such markets, the high prices of housing are not due to the scarcity of land, as much of the urban economics literature suggests. References William Alonso. Location and Land Use. Harvard University Press, A. E. Beaton and J. W. Tukey. The fitting of power series, meaning polynomials, ilustrated on band-spectroscopic data. Technometrics, 16:146 85, Jan K. Brueckner. The economics of urban yard space: An implicit market model for housing attributes. Journal of Urban Economics, 13:216 34, Jan K. Brueckner. Handbook of Regional and Urban Economics, volume 2, chapter The Structure of Urban Equilibria: A Unified Treatment of the Muth-Mills Model, pages North-Holland, FannieMae. Statistical fact sheet: America s housing affordability crisis. Technical report, Fannie Mae, Edward L. Glaeser and Joseph Gyourko. The impact of zoning on housing affordability. Harvard Institute of Economic Research Discussion Paper Number 1948,