Urban Economics. Yves Zenou Research Institute of Industrial Economics. July 17, 2006

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1 Urban Economics Yves Zenou Research Institute of Industrial Economics July 17, The basic model with identical agents We assume that the city is linear and monocentric. This means that the city is described by a line in which all jobs and all firms (which are assumed to be identical) are located in the Central Business District (CBD hereafter), which is normalized to zero for simplicity, and all workers/consumers endogenously decide their residential location between 0 and the city fringe f. Landlords allocate the land to the highest bids in the city. All workers/consumers are employed and are identical in all respect. There are eactly N identical workers. There are neither mobility costs within the city nor migration costs between outside the city and the city. However, individuals do incur commuting costs to go to work The individual location choice The question we have to solved is the following: What is the optimal residential choice of each individual? In fact, each individual is a worker and thus works every day in the CBD. He/she obtains a monthly wage of w L and incurs a monthly pecuniary commuting cost at distance from the CBD equal to τ, where τ is the commuting cost per unit of distance. Each individual is also a consumer. He/she consumes housing (or land) 1 and a non-spatial composite good (it is a miture of all goods that are not related to location). More precisely, he/she optimally determines the lot size of the house and the optimal consumption of the non-spatial composite good z L. 1 Land and housing is the same here as we do not modeled the optimal choice of land developers (see e.g. Brueckner, 1987, for such an analysis).

2 In order to solve this question we have to determine the preferences of this worker/consumer. For simplicity, we use preferences that are well-behaved, i.e. preferences are continuous and increasing at all > 0 and z L > 0 and all indifference curves are strictly conve and smooth and do not cut aes. The utility function (that represent these preferences) is given by: Γ(z L, ) (1.1) with Γ(z L, ) Γ(z L, ) > 0 > 0 z L and the function Γ(z L, ) is strictly quasi-concave. We can also write the budget constraint. Each worker has a net monthly wage of w L τ and spend money on the composite good z L plus housing R(), wherer() is the land price at a distance from the CBD. The budget constraint is thus given by: w L τ = R()+z L (1.2) Each individual chooses and z L that maimize U(z L, ) under the budget constraint (1.2). We need to calculate the bid rent of each individual, that is how much he/she is ready to pay for land at each location in order to reach a utility level Γ(z L, ). There are three different approaches that are all equivalent (Fujita, 1989, chap. 2) The indirect Marshallian approach In this approach, each individual solves the following program: ma Γ(z L, ) s.t. w L τ = R()+z L (1.3),z L Solving w L τ = R()+z L in z L,thisisequivalentto: ma Γ(w L τ R(), ) (1.4) where z L = w L τ R(). First and second order conditions of (1.4) give a unique Marshallian demand for housing (lot size) h M L (R(),w L ),which is implicitly defined as: Γ R()+ Γ =0 (1.5) z L By using the budget constraint, we obtain the Marshallian demand for the composite good: z M L (R(),w L )=w L τ h M L (, w L ) R() (1.6) 2

3 The indirect utility function can be written as: W (w L,τ,,R) (1.7) = Γ(w L τ h M L (R(),w L ) R(),h M L (R(),w L )) W L where W L is the equilibrium utility level in the city. Finally, we can solve equation (1.7) to obtain the so-called bid rent (see below for an eact definition) for specific preferences: Ψ L (, W L ) (1.8) Let us eplain the Marshallian approach using Figure 1. The aim of each individual is to solve (1.3), that is to find the highest utility level compatible with the budget constraint (1.2). Thus, by fiing the budget constraint as in Figure 1, we vary the utility functions W L and the solution of (1.3) is given by (z L,h L), which is precisely the highest utility level compatible with the budget constraint; in Figure 1, it corresponds to W L,2. Figure 1. The Marshallian Approach z L W > W > W L, 3 L,2 L,1 * z L W L,3 W L,2 W L,1 z = w τ R( ) h L L L 0 * We would like to know where this representative individual choose his/her residential location in the city. For that, he/she solves the following program: ma Γ(w L τ R(), ) 3

4 It is easily checked that this leads to: τ + R 0 () =0 (1.9) which is the so-called Alonso-Muth condition. It says that, by optimally choosing his/her location, each individual face the following trade off. If the individual decides to reside closer to the city-center (here 0), then he/she pays a higher marginal land rent (R 0 () < 0) but incurs a lower marginal commuting cost (τ >0). On the contrary, if he/she locates closer to the periphery of the city, then the marginal land rent is lower but the marginal commuting cost is greater. Of course, in equilibrium, all (identical) individuals are indifferent between all location since they reach the same utility level W L (there is no mobility costs). Eample with a Cobb-Douglas utility function. Take Γ(z L, )= 1 2 log z L log (1.10) which is equivalent to: Γ(w L τ R(), )= 1 2 log (w L τ R()) log The Marshallian demand for housing and the composite good are respectively given by: h M L = w L τ 2R() z M L = 1 2 (w L τ) Plugging these values into the utility function leads to: W L = 1 µ 1 2 log 2 (w L τ) + 1 µ 1 2 log (w L τ) 2 R() and thus the bid rent function is equal to: (w L τ) Ψ L (, W L ) = R() = h ³ i 2ep 2W L log (wl τ) 2 = (w L τ)(w L τ) /2 2ep(2W L ) Ψ L (, W L )= (w L τ) 2 4ep(2W L ) Plugging back (1.11) in h M L,wefinally obtain: h M L =2 ep (2W L) w L τ (1.11) (1.12) 4

5 1.3. The direct approach Let us define more in a more direct way the concept of bid rent. This is referred to as the direct approach of calculating the bid rent. Definition 1. The bid rent Ψ L (, W L ) is the maimum rent per unit of land that an individual can pay for residing at a distance from the CBD while enjoying a fied utility level W L. Using the budget constraint (1.2), this bid rent can be written as: ½ ¾ wl τ z L Ψ L (, W L )=ma Γ(z L, )=W L z L, (1.13) where, for an individual residing at, w L τ z L is the money available for land rent and where (w L τ z L )/ represents the land rent per unit of land. By solving the equation Γ(z L, )=W L we obtain a composite good consumption which is a function of and W L,thatisZ L (,W L ) and thus we can write (1.13) as: ½ ¾ wl τ Z L (,W L ) Ψ L (, W L )=ma (1.14) Solving (1.14) leads to an the optimal lot size which is a function of and W L, that is h d L (, W L), which is implicitly defined by: Z L (h d L(, W L ),W L ) + w L τ Z L (h d L(, W L ),W L )=0 (1.15) Plugging h d L(, W L ) back to (1.14), we obtain: Ψ L (, W L )= w L τ Z L (h d L(, W L ),W L ) h d L (, W L) (1.16) We can illustrate the direct approach using Figure 2. To solve program (1.13), one has to fi theutilityleveltow L and then find the highest bid rent Ψ L (, W L ) compatible with W L. Since in the plane (, z L ), Ψ L (, W L )=R() is the slope of the budget constraint, i.e. z L = w L τ R(),then graphically solving program (1.13) means to find the optimal slope of the budget constraint that is compatible with W L.InFigure2,itisΨ L,2 (, W L ). 5

6 Figure 2. The Direct Approach z L w L τ * z L W L Ψ, W ) L, 1 ( L Ψ L, 2 (, W2 ) Ψ, W ) L, 3 ( L 0 w L,1 τ R ) 1 ( * w L,2 τ R ) 2 ( w L,3 τ R ) 2 ( We can again calculate the optimal location for each individual. By differentiating (1.16) with respect to, we obtain: Ψ L (, W L ) ( h = 1 h 2 L ( = 1 h 2 L i h L τ Z L(,W L ) [w L τ Z L (,W L )] τ " ) Z L (,W L ) +[w L τ Z L (,W L )] where h d L(, W L ). Now observing from (1.15) that Z L(,W L ) [w L τ Z L (,W L )] = 0, we obtain (this is known as the Envelope Theorem): Ψ L (, W L ) τ = h d L (, W L) In equilibrium (see below), Ψ L (, W L )=R() andthusthisequationiseactly equivalent to (1.9). #) 6

7 Eample with a Cobb-Douglas utility function. function (1.10), i.e. Take the utility Γ(z L, )= 1 2 log z L log The bid rent function can be written as: ½ wl τ z L Ψ L (, W L )=ma 1 z L, 2 log z L + 1 ¾ 2 log = W L By solving 1 2 log z L log = W L, we obtain: and thus h 2 L Ψ L (, W L )=ma z L = ep(2w L log ) = ep (2W L) n o Solving ma wl τ ep(2w L )/ hl leads to: µ 1 ep (2WL ) h 2 L ½ ¾ wl τ ep (2W L ) / w L τ ep (2W L) =0 2ep(2W L ) (w L τ) =0 h d L = 2ep(2W L) w L τ Plugging this value back to the bid rent function yields: ½ ¾ wl τ ep (2W L ) / Ψ L (, W L ) = ma (1.17) = w L τ ep (2W L ) /h d L h d L = w L τ w L τ 2 2ep(2W L ) w L τ Ψ L (, W L )= (w L τ) 2 4ep(2W L ) (1.18) 1.4. The indirect Hicksian approach The last approach consists in using the Hicksian compensated demand for land. In that case, the consumer minimizes his/her epenses in order to reach 7

8 a certain level of utility (this is referred to as the ependiture-minimization problem), that is: E(, W L ) min z L, {z L + R()+τ Γ(z L, )=W L } (1.19) where E(, W L ) is the ependiture function. Again solving Γ(z L, )=W L, we obtain Z L (,W L ).Thusthisprogramcanbewrittenas: E(, W L ) min {Z L (,W L )+ R()+τ} which defines implicitly h H L (R(),W L ), the (compensated) Hicksian demand for land, as follows: Z L (h H L (R(),W L ),W L ) + R() =0 (1.20) Plugging h H L (R(),W L ) back to the ependiture function, we easily obtain: E(, W L ) Z L (h H L (R(),W L ),W L )+h H L (R(),W L ) R()+τ (1.21) From the budget constraint (1.2), we have which using (1.21) is equivalent to: E(, W L )=w L Z L (h H L (R(),W L ),W L )+h H L (R(),W L ) R() =w L τ (1.22) Inverting this equation and observing that Ψ L (, W L )=R(), we obtain the bid rent function. Now, by differentiating equation (1.22), we obtain: ( h = 1 h 2 L Ψ L (, W L ) τ Z L(,W L ) i h L [w L τ Z L (,W L )] ) ( " = 1 τ h h 2 L L Z L (,W L ) +w L τ Z L (,W L ) where h H L (R(),W L ). Using the budget constraint (1.2), we have w L τ Z L (,W L )= Ψ L (, W L ) 8 #)

9 and thus = Z L (,W L ) +[w L τ Z L (,W L )] h L ZL (,W L ) + Ψ L (, W L ) Observing that in equilibrium Ψ L (, W L )=R() and using (1.21), it is easy to see that (Envelope Theorem): Z L (,W L ) + w L τ Z L (,W L )=0 and thus Ψ L (, W L ) τ = h H L (, W L) < 0 Again, we can eplain the Hicksian approach using Figure 3. It is easy to see that this approach is the dual of the Marshallian approach since we are fiing the utility function to some level, here W L,andwevarythebudget constraint. Indeed, in order to solve program (1.19), one has to find the lowest budget constraint compatible with utility W L and in Figure 3 it is BC 2. w L,3 τ z L Figure 3. The Hicksian Approach w L,2 τ w L,1 τ * z L W L BC 3 BC 2 BC 1 0 * w L,1 τ R( ) w L,2 τ R( ) w L,3 τ R( ) 9

10 Eample with a Cobb-Douglas utility function. Take again (1.10), i.e. Γ(z L, )= 1 2 log z L log Then, E(, W L ) min {z L + R()+τ Γ(z L, )=W L } z L, = min ½z L + R()+τ 12 log z L + 12 ¾ log = W L z L, By solving 1 2 log z L log = W L, we obtain: and thus z L = ep(2w L log ) = ep (2W L) ½ ¾ ep (2WL ) E(, W L ) min + R()+τ Solving this minimization problem yields: ep (2W L) h 2 L + R() =0 Thus ep (2W L) h H L = h 2 L ½ ep (2WL ) E(, W L ) min = R() s ep (2W L ) R() ¾ + R()+τ s ep (2W L ) R()+τ R() = ep (2W L) p R() p + ep (2WL ) = p R()ep(2W L )+ p R()ep(2W L )+τ = 2 p R()ep(2W L )+τ Finally, to obtain the bid rent function, we have to solve: E(, W L )=w L 2 p R()ep(2W L )=w L τ 10

11 4R()ep(2W L )=(w L τ) 2 Ψ L (, W L )=R() = (w L τ) 2 By plugging back (1.23) into h H L, we obtain: h H L =2 ep (2W L) w L τ 4ep(2W L ) (1.23) (1.24) Finally, if we compare the different approaches, it is easy to verify that: h L(, W L ) h M L (R(),w L )=h d L(, W L )=h H L (R(),W L ) (1.25) when h M L (R(),w L ) and h H L (R(),W L ) are evaluated at R() =Ψ L (, W L ). Therefore the three approaches are totally equivalent. By taking our Cobb-Douglas eample where the utility function is given by (1.10), then it is easy to see that the three approaches are equivalent because they all lead to the same bid rent function (see (1.11), (1.18), (1.23)) and the same housing consumption (see (1.12), (1.17), and (1.24)). By differentiating equation (1.25), it is easy to show that: h L (, W L) > 0, h L (, W L) W L > 0, h L (, W L) w L < 0 Indeed, people living further away from the CBD will consume more land because it is cheaper there (remember that Ψ L(,W L ) < 0). When utility W L increases, bid rent decreases and thus individuals increase their housing consumption because land is cheaper. The same reasoning applies to w L since when it increases, they are able to pay more for land and thus reduce their housing consumption. Also, in all these approaches, it is easy to show that: Ψ L (, W L ) w L, Ψ L (, W L ) W L, Ψ L (, W L ) τ Indeed, the bid rent increases (decreases) with the wage w L (the commuting cost τ) because, individuals being richer (poorer), they are able to offer more (less) for land at each location. On the other hand, the bid rent decreases with utility W L because the higher the utility level to be reached the lower the bid rent individuals can afford. 11

12 1.5. The urban-land use equilibrium We have now to determine the equilibrium of this city. We will have different cases depending on whether the city is closed or open and the landlords are absent or not The closed-city with absentee landlords We have now to determine the equilibrium of this city. The fact that the city is closed means that the utility level W L is endogenous while the size of the population N is given. Also, landlords do not reside in the city and thus the revenue for land does not appear in the income of any resident. We have the following definition: Definition 2. A closed-city urban equilibrium where landlords are absent and individuals are all employed is a vector (W L, f,r()) such that: ( Z f 0 Ψ L ( f,w L )=R A (1.26) 1 h L (, W d = N (1.27) L) R ma {Ψ L ( f,w L ),R A } for f () = (1.28) 0 for > f where f is the city fringe and R A agricultural land rent outside the city and h L(, W L ) is defined by (1.25). Equation (1.28) means that the land is offered to the highest bids in the city. In equation (1.26), the bid rent at the city fringe is equal to agricultural rent (normalized to zero for simplicity). Equation (1.27) is the population constraint condition. By solving the two last equations, we obtain the equilibrium values of the two unknowns WL and f as functions of the eogenous variables w L, τ. Of course these conditions are written for a linear city. In order to better understand them, let us eplain the case of a circular city. Let there be n()d individuals residing in a circular ring whose inner radius is and outer radius is + d. Their total demand for land in that ring is thus h L (, W L)n()d. On the other hand, the total supply for land in that ring is 2πd. Then in equilibrium, we must have supply equals demand for land, that is: h L(, W L )n()d =2πd 12

13 which can be written as 2π n()d = h L (, W L) As a result, the equilibrium condition for space is given by: N = Z f 0 n()d = Z f 0 2π h L (, W L) d whichiseactlythepopulationconstraint(1.27)forthecaseofacircularcity since 2π is the circumference of a circle of radius. Indeed, at a distance from the CBD, there is 1 piece of land available in a linear city and 2π in a circular city. The second equilibrium condition (1.26) is independent of the type of city (linear or circular) and is mainly the result of perfect competition in the land market The open-city with absentee landlords We can easily etend this model to the case of an open city. In that case, where mobility is free between cities, the utility obtained by city residents W L becomes eogenous (it is just their outside option) but the number of people N living in the city is endogenous. The definition of equilibrium is eactly as in definition 2 but one has to solve (1.26), (1.27) and (1.28) in terms of N, f and R(). It can be shown (see Fujita, 1989, Proposition 3.5, pages 62-63) that the closed- and open-city models are identical if one uses the value of equilibrium utility W L of the closed-city model in the open-city case The case of resident landlords In both closed and open cities, landlords can reside in the city. This is referred to as the public land-ownership model. To be more precise, the city residents are now assumed to form a government, which rents the land for the city from rural landlords at agricultural rent R A. The city government, in turn, subleases the land to city residents at the competitive rent R() at each location. We can define the total differential rent (TDR)fromthecityas: TDR = = Z f 0 Z f 0 [R() R A ] d (1.29) R()d R A f This will lead us to study, where urban land is rented from absentee landlords at a price equaling the agricultural rent (see, for eample, Pines and Sadka, 13

14 1986, for the fulled closed city model, i.e. a closed city model with public land-ownership). The analysis of this case in both closed and open cities is quite straightforward. The only difference is that the income of each individual is now given by w L +TDR/N.SinceTDR/N is taken as given by each individual, the analysis is straightforward and follows closely that of the benchmark case. Again, Fujita (1989, Proposition 3.5, pages 62-63) has shown that the absentee landlord and public land-ownership models are identical if they are evaluated with the same values. 2. The basic model with housing (the Muth model) The city is monocentric, closed and linear where the Central Business District (CBD hereafter) is located as the origin (zero). All land is own by absentee landlords Firms (land developers) There is a housing industry with has the following production function: 2 Q = H(L, K) (2.1) where L and K are respectively land and capital (or nonland input). This function is increasing and concave in each of its argument. It is assumed that the housing production function H(L, K) has constant returns to scale, which implies that the production function can be written as: h = h(s) (2.2) where S K/L represents the capital per acre of land or improvments per acre. S is also referred to as the structural density (Brueckner, 1987) and is an inde of the height of buildings. The function h(s) Q/L defined by (2.2) is the housing output per acre of land, with h 0 (S) > 0 and h 00 (S) < 0. When there are no taes, the profit isgivenby: Π = R H H(L, K) RL rk 2 Observe that the housing capital K is assumed to be perfectly malleable. This strongly simplifies the analysis since it implies that producers are able to costlessly adjust both their capital and land inputs, and, as a result, the issue of durability of structures is not analyzed here. 14

15 or equivalently π Π/L = R H h(s) (R + rs) where π is the profit per acre of land, R H is the rental price per unit of housing service q, R istherentperunitofland(landcostperacre)andr the price of capital (or the cost per unit of S). Each profit-maimizing firm of the housing industry behaves as: ma {π = R H h(s) (R + rs)} at each [0, f ] (2.3) S 2.2. Consumers/Workers Each household contains one person. Each individual chooses z and q that maimize their utility function under their budget constraint, i.e. ma U(z, q) s.t. z + R H q = y t (2.4) z,q where isthedistancetothecbd,z and q are, respectively, the consumption of the composite good (which price is taken as the numeraire) and the lot size (or dwelling size), y, the common income, and t the pecuniary commuting cost per unit of distance. It is assumed that U(z.q) is well-behaved, i.e. it is increasing and strictly concave in each of its argument and smooth (differentiable). The program (2.4) is equivalent to ma U(y t R H q, q) q which leads to 3 U q (z,q) U z (z,q) = R H (2.5) where we use the following notations: U z U z, U q U q Equation (2.5) implicitly defines q = q(, y). Using the budget constraint,we obtain z(, y) =y t R H ()q(, y) 3 The second order condition is given by U zz R 2 H + U qq 2R H U zq and is assumed to be negative. A sufficient condition is that U zq > 0. 15

16 Plugging these two values into the utility function gives the following indirect utility function: U((y t R H ()q(, y),q(, y)) u (2.6) where u is the common utility level reached in equilibrium by all residents in the city. Finally, by taking the inverse of this function, we can determine the bid rent of all individuals as It is easy to show that R H (, u) R H = R H (, u) < 0, R H(, u) u Plugging this value R H (, u) in q = q(, y), which using (2.5) defines q = q(, u) by the following equation: U q (z, q) U z (z, q) = R H(, u) (2.7) Again, it is easy to show that 2.3. Equilibrium q(, u) > 0, q(, u) u > 0 < 0 Plugging (2.6) in (2.3), the land developer s program becomes ma {π = R H (, u)h(s) [R()+rS]} at each [0, f ] S First order condition yields: 4 R H (, u)h 0 (S) =r (2.8) which implicitly gives S = S(, u) Again, we have S = R H h 0 (S) R H h 00 (S) < 0, S u = R H u 4 The second order condition is always satisfied since R H (, u)h 00 (S) < 0 h 0 (S) R H h 00 (S) < 0 16

17 We can now define the population density as D h [S(, u)] /q(, u), which is the ratio between square feet of floor space per acre of land and square feet of floor space per dwelling (person). This is a different concept than the structural density or improvments defined by S(, u). The two concepts are important to understand the main results of Brueckner and Kim (2003). As noted above, the improvments (i.e. the intensity of land development) are a measure of building height so a higher S means that developers construct higher buildings, containing more housing floorspaceperacreofland. Onthe other hand, a higher population density means that either the housing floor space is higher or the dwelling size is lower. Let us now go back to the analysis. Since H(.) has constant returns to scale, in equilibrium, the housing industry is such that all firms make zero profit ateach, thatis which implies It is easy to show that R H (, u)h(s(, u)) [R()+rS(, u)] = 0 R(, u) =R H (, u)h [S(, u)] rs(, u) (2.9) R(, u) = R H R(, u) h<0, = R H u u h<0 We can now formally define the equilibrium. Definition 3. An urban land-use equilibrium in a linear and closed city with absentee landlords is a vector (u, f,r()) such that: Z f R( f,u)=r A (2.10) h [S (, u)] d = N (2.11) 0 q(, u) R() =ma{r( f,u),r A } at each 0, f (2.12) where R( f,u), S (, u), q(, u) are defined by (2.9), (2.8), and (2.7), respectively. Equation (2.10) says that the bid rent of the individuals must be equal to the agricultural land at the city fringe. Equations (2.11) gives the population constraint. Finally, equation (2.12) defines the equilibrium land rent as the upper envelope of the equilibrium bid rent curves of all workers types and the agricultural rent line. 17

18 3. The basic model with housing (the Muth model) with apropertyta Each profit- We use the same model but with θ, the property ta rate. maimizing firm of the housing industry behaves as: 5 ma {π = R H h(s) (1 + θ)(r + rs)} at each [0, f ] (3.1) S Consumers/Workers Each household contains one person. Each individual chooses z and q that maimize their utility function under their budget constraint, i.e. ma U(z, q) s.t. z + R H q = y t (3.2) z,q This is eactly as before. The bid rent of all individuals as R H = R H (, u) Equilibrium The land developer s program is ma {π = R H (, u)h(s) (1 + θ)[r()+rs]} at each [0, f ] S First order condition yields: R H (, u)h 0 (S) =(1+θ) r (3.3) which implicitly gives S = S(, u, θ) Again, we have S = R H h 0 (S) R H h 00 (S) < 0, S u = R H u S θ = r R H h 00 (S) < 0 h 0 (S) R H h 00 (S) < 0 We can now define the population density as D h [S(, u, θ)] /q(, u), which is the ratio between square feet of floor space per acre of land and square feet of floor space per dwelling (person). This is a different concept 5 Observe that it does not matter whether the developer or the urban resident pays the property ta θ. The same results would emerge if the residents pay at a rate θ, sothat the gross-of-ta rent price is written R H (1 + θ). Then, the developer profit will just be R H h(s) (R + rs), with no ta term showing up. 18

19 than the structural density or improvments defined by S(, u, θ). Thetwo concepts are important to understand the main results of Brueckner and Kim (2003). Let us now go back to the analysis. Since H(.) has constant returns to scale, in equilibrium, the housing industry is such that all firms make zero profit ateach, thatis which implies It is easy to show that R H (, u)h(s(, u, θ)) (1 + θ)[r()+rs(, u, θ)] = 0 R(, u, θ) = R H(, u) h [S(, u, θ)] rs(, u, θ) (3.4) (1 + θ) R(, u, θ) = R H h R(, u, θ) < 0, = R H (1 + θ) u u h (1 + θ) < 0 R(, u, θ) θ We can now formally define the equilibrium. = R H h (1 + θ) 2 < 0 (3.5) Definition 4. An urban land-use equilibrium in a linear and closed city with absentee landlords is a vector (u, f,r()) such that: Z f R( f,u,θ)=r A (3.6) h [S (, u, θ)] d = N (3.7) 0 q(, u) R() =ma{r( f,u,θ),r A } at each 0, f (3.8) Brueckner and Kim (2003) find the following result: Proposition 1. The comparative statics of the equilibrium is as follows: f θ R 0 However, for CES preferences, 6 if the elasticity of substitution σ 1, then f / θ < 0, while if σ<1, the sign of f / θ is still ambiguous. 6 That is U(z, q) = αz β +(1 α) q β 1/β where 1 β<+ and with σ =1/(1 + β) giving the elasticity of substitution, where 0 σ<+. 19

20 By remembering our discussion about structural versus population density, the intuition of this result is easy to understand. There are two countervailing effects of an increase of a property ta θ on urban sprawl f. On the one hand, an increase in θ has a direct negative effect on the profit of developers, which accordingly reduce the level of improvements (or structural density). As a result, for a given size of dwellings, buildings are shorter and thus the population density is lower. Because population is fied (closed city), it has to be that the city increases in size (this is referred to as the building height effect). On the other hand, an increase in θ has an indirect negative effect on households housing consumption because the ta on land and improvments is partly shifted forward to consumers, which yields a higher price of housing and thus a lower dwelling size. Since smaller dwellings imply an increase in population density D and thus more urban sprawl (this is referred to as the dwelling size effect). The net effect is thus ambiguous in the general case. In the CES case, when the consumptions of z (composite good) and q (housing) are highly substituable (σ 1), the dwelling-size effect becomes more important and the net effect is such that an increase in θ decreases urban sprawl. 20