AGEC 603 Urban Density and Structures Stylized Cited Assumptions q = a constant density the same throughout c = constant structures the same throughout Reality housing is very heterogeneous Lot size varies Structural characteristics vary Location matters Urban Density Density usually varies within a metropolitan area Factor substitution substitution of structure capital for land resource Denser residential development at more central locations Land is more expensive Key relationship between density and land value moves in both directions higher value not only encourages greater density but greater density increase the value of he land 1
People per sqaure mile (in People per sqaure mile (in Population (in 40 Population by Distance From City Hall 2010 Average of all U.S. metro areas 2000 Average of all U.S. metro areas 35 30 25 20 15 10 5 0 0 5 10 15 20 25 30 35 40 45 50 55 Distance from city hall (in miles) 2010 Average of all U.S. metro areas 2000 Average of all U.S. metro areas 7.0 6.0 5.0 4.0 3.0 2.0 1.0 2010 Houston-Sugar Land-Baytown, TX Metro Area 2000 Houston-Sugar Land-Baytown, TX Metro Area 2 18.0 16.0 14.0 12.0 1 8.0 6.0 4.0 2.0 2
People per sqaure mile (in People per sqaure mile (in People per sqaure mile (in 2010 Dallas-Fort Worth-Arlington, TX Metro Area 2000 Dallas-Fort Worth-Arlington, TX Metro Area 9.0 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 2010 Chicago-Joliet-Naperville, IL-IN-WI Metro Area 2000 Chicago-Joliet-Naperville, IL-IN-WI Metro Area 35.0 3 25.0 2 15.0 1 5.0 2010 San Antonio-New Braunfels, TX Metro Area 2000 San Antonio-New Braunfels, TX Metro Area 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 3
People per sqaure mile (in People per sqaure mile (in People per sqaure mile (in 2010 New York-Northern New Jersey-Long Island, NY-NJ-PA Metro Area 2000 New York-Northern New Jersey-Long Island, NY-NJ-PA Metro Area 9 8 7 6 5 4 3 2 1 2010 College Station-Bryan, TX Metro Area 2000 College Station-Bryan, TX Metro Area 8.0 7.0 6.0 5.0 4.0 3.0 2.0 1.0 2010 Champaign-Urbana, IL Metro Area 2010 College Station-Bryan, TX Metro Area 25.0 2 15.0 Both approximately 230,000 people 1 5.0 0 5 10 15 20 25 30 35 40 45 50 55 Distance from city hall (in miles) Note: Population-weighted density expressed as the average number of people per square mile of land area. Distances are measured to the city hall or similar municipal building of the metro area's first-named principal city. Metropolitan statistical areas defined by the Office of Management and Budget as of December 2009. Source: U.S. Census Bureau, 2010 Census and Census 2000. 4
People per sqaure mile (in 2010 San Francisco-Oakland-Fremont, CA Metro Area 2010 Dallas-Fort Worth-Arlington, TX Metro Area 7 6 5 4 3 San Francisco 4.3 million people Bay not circular Dallas 6.3 million people More circular growth 2 1 0 5 10 15 20 25 30 35 40 45 50 55 Distance from city hall (in miles) Note: Population-weighted density expressed as the average number of people per square mile of land area. Distances are measured to the city hall or similar municipal building of the metro area's first-named principal city. Metropolitan statistical areas defined by the Office of Management and Budget as of December 2009. Source: U.S. Census Bureau, 2010 Census and Census 2000. Negative exponential Density Gradient D(d) = D 0 e -αd where D is density at distance d α is an parameter D 0 is density at city center Estimate D(d) = D 0 e -αd = 22.86.7 e -7d R 2 = 0.84 5
Density per Acre 18 16 14 12 Predicted Actual 10 8 6 4 2 0 0 50 100 150 200 250 300 350 Population ( Population Densities for Different Texas Cities Housing Attributes Most houses on the market at any point in time are existing units and not newly constructed units Rarely a specific custom package Choose a complete package Housing is heterogeneous How to value individual characteristics Market Price True market price Housing market price - expenditures Select a house - selecting many characteristics Principle of Compensation Previous model equilibrium housing rents exactly compensate for commuting costs associated with different locations Expand this principle housing rents / prices will have to compensate for all of the desirable and undesirable features of each housing unit 6
Valuation Process Law of diminishing marginal utility additional utility (value) [marginal utility] of an additional consumption drops as more is consumed. housing rents / prices will have to compensate for all of the desirable and undesirable features of each housing unit Price $ House Price Incremental house price (marginal value) Size (sq ft) Equation P H = f(x 1, x 2, x 3,, x n ) Market price (P H ) is a function of the n attributes (x) How to get the equation? Functional Form Linear Log-Linear Hedonic Price Equation Obs. House Price Size (sq ft) Baths Noise (1 = yes, 0 = no 1 150,000 1,500 2 0 2 250,000 2,000 2 0 3 80,000 1,500 1 1 4 200,000 1,800 1 1 Linear Form and Interpretation P H = 47.1 + 82.3 * size + 13.9 * bath 36.1 * noise + e (6) (0) (0.15) (4) R 2 = 0.75 housing price in thousands size in thousands of sq ft Interpretation coefficients estimate of the implicit price that households are willing to pay for one more unit of each attribute sign magnitude significance R-bar squared 7
Linear Form Use - Examples P H = 47.1 + 82.3 * size + 13.9 * bath 36.1 * noise + e What is the expected price of a 2,000 sq ft house with three baths and can hear the noise from the factory? P H = 47.1 + 82.3 * 2 + 13.9 * 3 36.1 * 1 P H = $217.3 thousand or $217,300 What is the expected price of the same house but can not hear the noise? P H = 47.1 + 82.3 * 2 + 13.9 * 3 36.1 * 0 P H = $253.4 thousand or $253,400 WTP $36.1 thousand to avoid the noise Problem with Linear Form Square feet first 1,000 sq ft adds $82.3 thousand to the value second 1,000 sq ft adds $82.3 thousand to the value third 1,000 sq ft adds $82.3 thousand to the value Problem no diminishing marginal utility or value Overcome this limitation use other functional forms Log-Log Form P = β 0 x 1 β 1 x 2 β 2 x 3 β 3 x n β n e Transform by taking natural logarithm ln P t = lnβ o + β 1 ln x 1 + β 2 ln x 2 + β 3 ln x 3 + β n ln x n + e t Interpretation of coefficients the coefficients represent the elasticity of price with respect to increases in the attribute [except for 0-1 variables] percentage change in price that results from a 1% change in the independent variable 8
Log-Log Example ln P t = 10.3 + 0.23 ln size + 0.62 ln bath f (0) (5) (0) +0.18 Mass + e t (0.11) R 2 = 0.54 housing price in dollars size in sq ft Mass = 1 near mass transit and = 0 not near mass transit Interpretation of coefficients continuous variables elasticities 0-1 qualitative variables are not elasticities just the change in ln (price) for living close to mass transit over the base on not close to mass transit Log-Log Example cont. ln P t = 10.3 + 0.23 ln size + 0.62 ln bath + 0.18 Mass + e t What is the expected price of 1,000, 2,000, and 3,000 sq ft houses with two baths and not near mass transit? ln P t = 10.3 + 0.23 ln 1000 + 0.62 ln 2 + 0.18 0 ln P = 12.31853 P = e 12.31853 = $223,806 ln P(2000) = 12.47796 => P = $262,488 P(3000) = $288,144 9
Optimal Characteristics - New Single attribute size Tangent Total cost Price $ House Price Maximum distance S* Size (sq ft) Density Residential density more complicated than previous slide Two Key solutions Increased density => decreased unit value Lower density better Increased density => increased land rent Higher density better Price equation P = α βf FAR = F floor area ratio ratio of total housing to total land area Cost equation C = μ + τ F Net Returns = P C per sq ft floor space Net Returns = F(P C) per land unit Density $/sq ft housing α μ F 0 C = μ + τ F P = α βf FAR 10
Density - Example What are net returns if F = 1,000 sq ft / acre P = 1000-0.5 F = 1000-0.5 * 1000 = 500 C = 200 + 0.2F = 200 + 0.2 * 1000 = 400 Net returns per sq ft of floor space = P - C P- C = 500 400 = $100 / sq ft Net returns per land unit = F(P C) = 1000 (100) = $100,000 / acre Land price = F(P C) => substitute in P and C p = F(P - C) = F[(α βf) (μ + τ F)] = Fα βf 2 Fμ -τ F 2 p = F(α μ) F 2 (β + τ) => quadratic equation $/sq ft housing α Optimal FAR Construction costs = C = μ + τ F $/sq ft land μ House price = P = α βf FAR: F Land price = F(P-C) Note F = 0 Note P - C = 0 F* FAR: F Optimal FAR and Land Price p = F(α μ) F 2 (β + τ) Take partial w.r.t. F and set equal to zero=> quadratic equation p = α μ 2F β + τ = 0 F Solve for F F α μ = 2(β + τ) Substitute F* into land price eqn. p α μ = α μ 2 β + τ (α μ)2 = 4(β + τ) α μ 2 β + τ 2 β + τ 11
Location and Residential Density $/sq ft housing C α Increase in α P $/sq ft land μ FAR: F new p* original p* Land price = F(P-C) F* F* FAR: F Different Densities Why do we see different densities at similar (same) locations within a city? Key redevelopment Initial development at optimal FAR at that time City grows optimal FAR changes to higher density Redeveloped at the new FAR Timing issues Not all of the city is developed at the same time Not all redevelopment occurs at the same time Summary The price of a housing unit can be decomposed into implicit prices for each attribute that makes up the unit, such as the presence of a garage, the amount of square feet, and the total density. There are distinct patterns to residential density in cities throughout the world. At higher densities otherwise identical housing units tend to have lower prices. The cost of constructing units, however, tends to increase with density of development. Thus, the profit per housing unit tends to be lower at greater density although more units can be placed on a given parcel of land. Determining the most profitable density of development requires trading off these two opposing considerations. 12
Summary At more desirable locations, this tradeoff leads to a highest and best use that involves greater density. In effect, higher prices for housing units lead to greater land values, and thus land use is used more sparingly per housing unit. Most cities have density gradients, with denser residential uses located near transportation centers, highways, parks, waterfront, or other amenities. As cities grow in population, new development initially tends to occur horizontally at the expanding border of the city. Thus, at any time existing units reflect the history of development, with the oldest units at the center and newest at the fringe. Summary At critical times in the development of a city, it can become profitable to demolish existing housing units built years earlier, particularly at more desirable locations. At such times, the net return to just the land from new development can exceed the total price for existing housing. Normally, this occurs only when new development calls for much greater density than exists with the current housing. Combining the above, we see a decreasing density gradient from the city center to the fringe but also see different densities at similar distances from the city center. 13