FACULTY WORKING PAPER NO UW*V. Growth Controls and Land Values. /an K Brueckner

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3 " BEBR FACULTY WORKING PAPER NO Growth Controls and Land Values in an Open City Th8 UW*V < «* OCT \ \ *,e*v rf WW* *8 /an K Bruekner WORKING PAPER SERIES ON THE POLITICAL ECONOMY OF INSTITUTIONS NO. 31 College of Commere and Business Administration Bureau of Eonomi and Business Researh University of Illinois Urbana-Champaign

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5 BEBR FACULTY WORKING PAPER NO College of Commere and Business Administration University of Illinois at Urbana- Champaign August 1989 Growth Controls and Land Values in an Open City Jan K. Bruekner Department of Eonomis For presentation at the TRED Conferene on Growth Management and Land Use Controls. Cambridge, Massahusetts, Otober 6-7, 1989.

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7 ABSTRACT This paper analyses the impat of a growth ontrol law on land values using a variant of the open-ity model of Capozza and Helsley (1988). Sine population growth in the model generates a negative externality, a growth ontrol regulation raises urban land rents by improving the ity's quality of life. The ontrol also delays development, however, and both these effets must be taken into aount in determining land value hanges.

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9 . Growth Controls and Land Values In an Open City by Jan K. Bruekner* 1. Introdution In the fae of rapid regional population growth, many loalities in the U.S. have turned to growth ontrols in an attempt to divert unwanted extra residents to other ommunities. These ontrols take a variety of different forms, inluding redutions in allowable development densities, inreases in development fees paid by builders, annual building permit limitations, timing ordinanes designed to delay development, and various other regulations. Rosen and Katz (1981) provide a exellent survey of regulations adopted by ommunities in the San Franiso Bay Area, where growth ontrols are ommonplae There is now a large empirial literature doumenting the effets of growth ontrols on housing and land markets. The evidene to date onlusively establishes that growth ontrols raise housing pries in ommunities where they are imposed (see Elliot (1981), Shwartz, Hansen, and Green (1981), Dowall and Landis (1982), Shwartz, Zorn, and Hansen (1986), and Katz and Rosen (1987)). Additional evidene suggests that by delaying or banning eventual development, imposition of growth ontrols lowers the value of agriultural land near the ity (see Gleeson (1979), Blak and Hoben (1985), Knapp (1985), Vaillanourt and Monty (1985), and Nelson 1 (1988) J. The literature identifies two fores that aount for the positive impat of growth ontrols on housing pries. First, by restriting the supply of housing in the fae of population pressure, ontrols are thought to reate exess demand, whih in turn leads to higher pries. Seond, by preserving a

10 ommunity's "quality of life," ontrols reate an amenity whose value is then 2 apitalized into housing pries. Unfortunately, despite muh ogent disussion of these fores, the literature does not offer a formal dynami model that illustrates their operation. The purpose of the present paper is to offer a first step toward suh a model. Building on the framework of Capozza and Helsley (1988), the model fouses on the land development deision (onversion from rural to urban use) of a landowner operating with perfet foresight in an dynami open-ity environment. The time path of urban land rents in the model in part reflets the presene of a negative population externality (a large population lowers the ity's quality of life and redues the rent that urban land ommands). After deriving the optimal date of rural-urban onversion (the date that maximizes land value), the analysis onsiders the effet of a growth ontrol regulation, whih delays onversion at eah loation. The model's population externality is, of ourse, the key fator in the analysis. Given the externality, a slowing of population growth due to the ontrol raises land's rent in urban use at every date and loation as onsumers pay a premium to live in a smaller ity. For land that is already developed, imposition of the ontrol raises all future rents and therefore inreases the value of the land. This orresponds to the amenity effet of growth ontrols that has been identified in the literature. The ontrol's impat on the value of undeveloped land is, however, not as straightforward as the literature would suggest. The impat is the net effet of two hanges: first, the ontrol delays the date at whih urban rents an be earned, whih lowers value; seond, the ontrol yields a lower population path (and hene higher urban rents) after development, whih raises value. Sine the seond of these effets may dominate, growth ontrols an raise the value

11 . of undeveloped land In some loations, In ontrast to the literature's impliit assumption to the ontrary. The analysis attempts to pinpoint the loations of undeveloped land that benefit from the imposition of a ontrol. In addition, the paper derives the form of the optimal growth ontrol poliy, whih maximizes the total value of land in the ity. This poliy an be used as a benhmark in evaluating other growth ontrol programs. As a final exerise, the paper explores an example based on a speifi utility funtion, where various urban growth paths an be omputed expliitly. It is important to realize that, beause an open-ity model is used in the analysis, exess demand for housing (as disussed above) plays no role in determining the market impat of growth ontrols. Consumers denied residene by the presene of the ontrol simply reloate to other ommunities. Sine it eliminates population pressure as a market fore (fousing instead on the ontrol's amenity effet), the analysis may not offer an entirely aurate piture of the operation of atual growth ontrols. However, it is useful to gain an understanding of the working of an amenity-based model. One the analysis of the model is omplete, the paper skethes a losed-ity model of growth ontrols (where population pressure is a fator), and points out some problems that arise under suh an approah (politial issues are also disussed) 3 2. The Model and the Unontrolled Eoulllbriua In standard fashion, the ity is assumed to be radially symmetri, with all employment loated at the CBD. Radial distane to the CBD is represented by x, and ommuting ost from a residene at distane x equals kx, where k is a positive parameter that is onstant over time. Urban residents, all of whom are idential, earn inome y(t) at time t. Preferenes are given by the wellbehaved utility funtion U(g,,P), where i is onsumption of land, g is

12 4 onsumption of a numeraire nonland good, and P is urban population. The marginal utility of population U is nonpositive, with population beoming a disamenity (U < 0) when P is suffiiently large. This externality presumably arises from traffi ongestion, air pollution, rime, and other phenomena assoiated with a large population. For simpliity, these underlying fores are not modelled in detail. To further simplify the analysis, individual land onsumption is fixed at one unit per person (this assumption is inessential, serving only to simplify notation). The budget onstraint then beomes g + r + kx = y(t), where r is land rent per are. Land rent is determined via the open-ity assumption, under whih the time path of utility is given by an exogenous funtion u(t). Substituting for using the budget onstraint, urban residents ahieve utility u(t) when r satisfies the equation U[y(t)-r-kx, 1, P] = u(t). (1) This equation impliitly defines the urban land rent funtion r = r(t,x,p), with r = -k < 0, r n = U_/U < 0, and r A = y'(t) - u'(t)/u. Land x P P g t g rent is a dereasing funtion of distane from the CBD, and for given t and x, a higher population redues rent via the disamenity effet (for low P's, this effet is absent). It is assumed that r > 0, so that holding x and P fixed, rent is inreasing over time. This requires that inome is inreasing suffiiently rapidly (or falling suffiiently slowly) relative to utility. Land is owned by absentee landlords who deide on the time pattern of its use (agriultural vs. urban) to maximize the present value of rents. Land earns a rent of r per are when in agriultural use, and onversion to urban a 4 use entails a ost of D per are. Consider the optimization problem at time zero of a landlord with holdings at loation x. Letting P(t) denote the

13 (equilibrium) population growth path of the ity and assuming that the landlord has perfet foresight, his goal is to hoose the onversion date T to maximize 5 rr re dt + o a 00 r(t,x,p(t))e _lt dt - De" lt (2) where i is the onstant disount rate. For future referene, expression (2) (whih is land value per are) will be denoted V(T,x,0 P). V gives the value at time zero of land at loation x as a funtion of the onversion date T, onditional on the population growth path P. The first-order ondition for hoie of T is r(t,x,p(t)) = r + id, (3) a whih shows that the land should be onverted when urban rent equals agriultural rent plus the flow ost of onversion. The seond-order ondition requires that the total derivative of r with respet to time (dr/dt = r + r_p') is positive at the optimal T. Given that r < 0, it then follows that T P x is an inreasing funtion of x, indiating that the ity grows outward over time (dt/dx = -r /(dr/dt) > 0). Aside from the population externality, this A model is idential to that of Capozza and Helsley (1988). The population growth path P(t) is in fat determined by the onversion deisions of landlords, and this must be reognized in solving for the equilibrium of the model. The first step in doing so is to note that (3) an reinterpreted as giving the x value where development is ouring at a given time. That is, rewriting (3) as r(t,x,p(t)) = r + ID, the equation determines a the loation x of land being onverted at time t. But sine the ity grows outward, this x value (all it x) is in fat the boundary of the ity at time t. Then, realling that individual land onsumption is fixed at one unit,

14 -2 population P an be written kx. Substituting this expression in plae of P(t) in the above equation yields r(t,x,kx 2 ) = r + id. (4) a This equation determines the time path x(t) of the urban boundary along with 2 the equilibrium population growth path P(t) = ^x(t). The inverse of the funtion x(t), written T(x), gives the onversion date T at loation x. Totally differentiating (4), it is easily seen that x'(t) = -r /(r + L A 2rxr ). A neessary and suffiient ondition for the assumed outward growth of the ity is therefore r. > (reall r, r^ < 0). This ondition also t x P guarantees satisfation of the developer's seond-order ondition. Substituting T(x) and the equilibrium population path into (2), the value of _2 land in equilibrium is written V*(x,0) = V(T(x),x,0 jttx ). Before proeeding to the disussion of growth ontrols, one final assumption is useful. The assumption is that x(0) = 0, whih means that the ity starts out as a point at time zero. This, of ourse, is simply a matter of speifying the time origin Growth Controls and Land Values In the early stages of urban growth, population size is a matter of indifferene to onsumers, with U and r both equal to zero. At some point, however, the population externality omes into play, so that U and r beome negative. Suppose that along the ity's equilibrium growth path, r is zero for all x when t < s and negative for all x when t > s (in other words, ' - 2 r (t,x,7rx(t) ) = (<) as t < (>) s). Although it an be shown that r must have the same sign for all x, r is not guaranteed to remain negative after it first falls below zero (it ould oneivably beome zero again at some future

15 date). To avoid inessential ompliations, however, this is assumed to not happen. Suppose that in response to the population disamenity, the ity imposes a growth ontrol law at time s. This law takes the form of a restrition on the future growth of the urban boundary. Formally, the law speifies a new time path x (t) for the boundary beyond s, with x (t) < x(t) holding for t > s (see Figure 1 for an example). Sine the law delays development, the onversion date funtion T(x) is also replaed by a new funtion T (x), whih C satisfies T (x) > T{x) for x values beyond x(s) (T (x) is the inverse of x (t)). Of ourse, the law ould be written as a "growth management timing ordinane" (Rosen and Katz (1981)), in whih ase the law would diretly speify T (x). An important assumption is that imposition of the growth ontrol is unantiipated by developers. Without this assumption, development ativity might aelerate in antiipation of the ontrol With x(t) replaed by x (t), the population growth path of the ity U beyond s is lowered from P(t) = rx(t) to P (t) = kx (t). Consumers denied residene in the ity loate elsewhere in the eonomy. It is important to note that this rediretion of population has no effet on the time path of utility in the eonomy (reall that the funtion u(t) is exogenous). For suh an effet to be absent, the ity imposing the ontrol must be small relative to the rest of the eonomy. The onsequenes of relaxing this impliit assumption are disussed below. Sine the lower population growth path improves the ity's quality of life relative to the equilibrium path, urrent and future urban land rents rise. Conversion of undeveloped land is also postponed by the ontrol, and together, these two effets lead to windfall hanges in land values throughout the ity. Consider first the hange in the value of developed land. Sine the

16 onversion ost D has already been inurred for suh land, value is simply the present value of the flow of future (urban) rents. Therefore, prior to the imposition of the growth ontrol at date s, the value of a developed are at loation x is given by 8 V* u (x,s) s /4. ~/*^ 2 \ -i(t-s),. r(t,x,ftx(t) )e dt (5) (the supersript denotes developed land). After imposition of the ontrol, the value of the same are of land is V*"(x,s) = C r(t,x,nx (t) )e dt C (6) The rent expressions in (5) and (6) differ beause the growth ontrol hanges the ity's population path. With population growth slowed under the ontrol, rent is higher and land value greater. In other words, with x (t) < x(t) 2 for t > s and r (t,x,7tx(t) ) < holding by assumption beyond s, it follows that (6) exeeds (5) and that the ontrol raises the value of developed land. Consider now the growth ontrol's effet on the value of land that remains undeveloped at time s. Prior to imposition of the ontrol, the value of an undeveloped are at loation x is V*(x,s) ft(x) re -Kt-s) Ht dt + a s r(t,x,rx(t) 2 )e 1(t s) dt T(x) -i(t(x)-s) - De (7) After imposition of the ontrol, value equals

17 s V*(x,s) T (x).,., -i(t-s) J-t s re dt + a /* ~ /4.\ 2 \ -i(t-s). r(t,x,kx (t) )e 'dt T (x) - De- i(t (x»- s) (8) To ompare these expressions, it is useful to rewrite (8) as V*(x,s) ft tx),,+, v -i(t-s),. re dt + a s T r(t,x,rx(t) )e dt (x) - De- I(T (x) - S» [r(t,x,7tx (t) ) T (x) r(t,x,*x(t) 2 )]e 1(t s) dt (9) Realling that V*(x,s) is equal to V(T(x),x, -2 \tx ) (see (2)), it follows that the differene between pre- and post-ontrol land values an be written V*(x,s) - V*(x,s) = V(T(x),x,s rx 2 - ) V(T (x),x,s 2 Irx ) [r(t,x,7tx (t) 2 - ) r(t,x,*x(t) 2 )]e 1(t s) dt C JT J (x) (10) -2 Note that V(T (x),x,s kx ) is equal to the first three terms in (9). By repeating the argument used above, it follows that the integral in (10) is positive (r.. is negative along the the equilibrium path, and x (t) < x(t)). P To sign the differene between the first two terms, note that by definition, -2 T(x) maximizes V(T,x,0 7tx ) (T(x) is the optimal onversion date under the equilibrium population path). As a result, T(x) also maximizes _2 3 V(T,x,s *x ). Sine onversion date T (x) is, by ontrast, nonoptimal under 2-2 population path rx(t), it follows that V(T(x),x,s rx ) >

18 _2 V(T (x),x,s ttx ). The differene between the first two terms in (10) is therefore positive. With the integral also positive, the sign of the entire 10 expression is indeterminate, indiating that the value of undeveloped land an rise or fall when the ontrol is imposed. As explained in the introdution, the reason for this indeterminay is that the growth ontrol has two opposing effets. The ontrol delays the date at whih urban land rents an be earned, whih tends to redue value, but it lowers the population growth path (and thus raises rents) after development, whih tends to inrease value. Despite this general indeterminay, the hange in land value an be signed under some irumstanes. Consider first the ase of a "marginal" ontrol, whih involves only a slight delay in development at eah loation. Under suh a ontrol (illustrated by the dotted line in Figure 1), T (x) for x > x(s) an be written T(x) + <5(x), where <5(x) > is infinitesimal. Similarly, x (t) = x(t) + e(t) for t > s, where e(t) < is again infinitesimal. The hange in the population path indued by the ontrol 2 an then be written nx (t) - nx(t) = 27tx(t)e(t). Under these C assumptions, the land value differene in (10) beomes V*(x,s) - V*(x,s) = V t (T(x),x,s kx 2 )<5(x) "oo r (t,x,rx(t) 2 )2*x(t)e(t)e" 1(t " s) dt. p T (x) (11) Sine T(x) is the optimal development date, it follows that the partial derivative V equals zero when evaluated at T(x). Eq. (11) then redues to the negative of the integral in the seond line, an expression whih is positive given that r < along the equilibrium path and e(t) < 0. Imposition of a marginal ontrol therefore Inreases the value of all undeveloped land (the

19 1 1 inrease, of ourse, will be small). The reason is that sine the ontrol is marginal and initial onversion dates are optimal, the loss of value from delayed development vanishes. The gain in value from a lower population growth path remains, however, so that the net effet is positive. Now onsider the ase of where the ontrol is not neessarily marginal but is "ontinuous" in the sense that the limit of T (x) as x approahes x(s) from above equals T(x(s))). This means that the ontrol does not interrupt the development proess when it is first imposed. Equivalently, ontinuity of the ontrol means that the funtion x (t) is inreasing near s, as in Figure 1. If x (t) were flat near s, then the urban boundary would initially be frozen by the ontrol, and T (x(s)) would exeed T(x(s)) (the ontrol would then be 9 disontinuous). Under a ontinuous ontrol, the land value differene in (10) an be signed at loations near the urban boundary. To see this, onsider the behavior of (10) as x falls toward x(s). Sine T (x) * T(x) as x + x(s) by ontinuity of the ontrol, it follows that the differene between the first two terms of (10) approahes zero as x * x(s). With the last term in (10) negative for all x, the entire expression therefore beomes negative as x approahes x(s). It follows that the imposition of the ontrol raises the value of undeveloped land adjaent to the urban boundary. To relate this result to the previous disussion, note that a ontinuous ontrol is neessarily marginal near the urban boundary. By the previous analysis, land value must rise in suh loations. If the growth ontrol is disontinuous, with T (x(s)) > T(x(s)), then the first part of (10) remains positive as x falls toward x(s), and the land value differene annot be signed. The differene is determinate, however, in one highly disontinuous ase: where the ontrol prohibits development beyond

20 12 x(s). With future development banned, T (x) is infinite for x > x(s), and the integral in (10) equals zero. Sine the rest of the expression is positive, it follows that land value falls when the ontrol is imposed (value at eah loation falls by the present value of the differene between agriultural and forgone urban rents). Returning to the ase of a ontinuous ontrol, it is natural to wonder whether more omplete results on the ontrol's spatial impat beyond x(s) an be derived. To address this question, an appropriate proedure is to ompute the derivative of the land value differene (10) with respet to x. If this derivative were positive, then (given that land value rises near x(s)) it would follow that value rises in response to the ontrol between x(s) and some x and falls beyond x (x ould be infinite). Unfortunately, the derivative in question is ambiguous in sign, so that this simple spatial pattern of value impats need not emerge. To see this, subtrat (8) from (7) and differentiate with respet to x. The result is C i(t S) 1(T (x) s) 2 ke dt + T'(x)e [r(t (x),x,kx - ) r - ID] TOO (12) (reall that r = -k). To sign the seond term in (12), note first that A 10 2 T'(x) > 0. Moreover, sine r(t(x),x,7tx ) = r + ID, r. > 0, and T (x) > a t T(x), it follows that the term in brakets is positive. With the entire seond term therefore positive and the integral negative, the sign of (12) is indeterminate. As a result, the impat of the growth ontrol on the value of undeveloped land may have a omplex spatial pattern. For example, after rising near x(s) when the ontrol is imposed, value may fall farther from the boundary only to rise again at still more distant loations.

21 13 The previous results stand in sharp ontrast to the usual laim that growth ontrols redue the value of undeveloped land. It has been shown that if a ity imposes a very mild growth ontrol (a marginal ontrol), then the value of all undeveloped land rises. If the ontrol is instead a stringent one that happens to be ontinuous, then the value of undeveloped land near the urban boundary rises, and more remote land may rise in value as well. As mentioned in the introdution, empirial evidene suggests that growth ontrols redue the value of undeveloped land, a finding that is not fully onsistent with the above results. There are a number of possible explanations for this inonsisteny. First, land value gains near the urban boundary may be hard to pik up empirially, espeially if the estimating equation does not allow for interation between loation and the effet of the ontrol. Seond, atual ontrols may be quite disontinous, in whih ase all undeveloped land may fall in value. Whatever the explanation, future empirial investigators should be aware that the impat of a ontrol on the value of undeveloped land need not follow onventional wisdom. 4. The Effiient Growth Control The preeding analysis foused on the effets of an arbitrary growth ontrol law. The purpose of this setion is to derive the form of the effiient growth ontrol. Sine onsumer utility is exogenous in the model, the planner's goal in hoosing an effiient ontrol is simply to maximize the total value of land in the ity (this maximizes returns aruing to landlords). To derive total land value, the value expression (2) is integrated aross all loations x in the planner's jurisdition. This yields a double integral involving land rents that is not onvenient to use. A more useful expression is gotten by reversing the order of integration, with integration ouring

22 first over distane and then over time. The present value of total land rent an then be written 14 t=0 (_ X (t) e - 2 2rxr ( t, x rx (t) )dx +, x=0 B 2nxr dx \ e dt a x (t) e (13) The first inside integral is total urban land rent at time t (the ity boundary at t under the effiient ontrol is denoted x (t), with total population 2 equal to rex (t) ). The seond inside integral is total agriultural rent at t. Note that the upper x-limit B represents the outer boundary of the planner's jurisdition (the jurisdition ould be visualized as a irular island with radius B). Total land rent (the sum of these two integrals) is then disounted and integrated from time zero onward. Total onversion ost in present value terms is found by multiplying the last term of (2) by 2;tx and integrating over x, whih yields _ -it (x)_, 2ftxDe e y 'dx, (14) x=0 where T (x) is the effiient onversion date for land at x. To make (14) e ommensurate with (13), a hange of variable from x to t is performed and the resulting expression is integrated by parts. This yields the following equivalent expression for total onversion osts: kx (t) 2 ide lt: dt t=o e (15) The effiient growth ontrol is found by hoosing x (t) to maximize the differene between (13) and (15), whih equals total land value. Note that sine the planner's jurisdition stops at x = B, the optimal boundary path must

23 15 satisfy x (t) < B. Differentiating the total value expression inside the time integral, the first-order ondition for hoie of x (t) is e r(t,x (t),nx (t) 2 ) = r + id - e e a X (t) e - 2 2nxr (t,x,x (t) )dx * e (16) This ondition differs from the previous first-order ondition (3) by the presene of the integral, whih is nonpositive given r < 0. The interpretation of the differene is straightforward. The negative of the integral represents an additional ost of onverting land from agriultural to urban use, over and above the foregone agriultural rent and the opportunity ost of the funds spent on onversion. This ost is the redution in the rent on previously onverted land that omes from the population growth aused by further onversion. When this ost is taken into aount, the spatial growth of the ity is slowed relative to the equilibrium path. No effet ours, however, before time s (x(t) = x (t) for t < s). This an be seen by noting that sine r equals zero along the equilibrium path before s, the first-order ondition (16) is satisfied by x(t) in this range. Beyond s, the fat that r < holds along the equilibrium path means that the RHS of (16) exeeds the LHS along 12 this path. From the seond-order ondition, it follows that the boundary must be ontrated relative to the equilibrium path to satisfy (16). As a result, x (t) < x(t) holds for t > s. Under suitable smoothness assumptions on the land rent funtion r, x (t) will diverge from x(t) in a smooth manner at t = s. This means that x (t) must be inreasing in t immediately after s, whih in turn implies that 14 the effiient ontrol is ontinuous. The previous setion's results on the land value impats of a ontinuous ontrol then apply. In partiular, land

24 ) 16 near the urban boundary rises in value when the effiient ontrol is imposed. Other undeveloped land may fall in value, but given the effiieny of the ontrol, total land value rises. Of ourse, imposition of an arbitrary ontrol, ontinuous or otherwise, may not have this effet (the ontrol ould redue total land value). 5. An Bxaaple To generate a simple example, suppose that the utility funtion U(g,l,P) 1/2 is given by g - ap, where a > (land onsumption is suppressed sine = 1). Suppose also that inome and utility vary linearly with t, with y(t) = y + 1/2 #t and u(t) = r + pt. Then, using (1 ), r = r? + 0t - kx - ap, where 57 = j - x > and 6 = <}) - p > 0. Eq. (4) then beomes t - kx - a(7tx 2 ) = r + id, (17) a and solving for x yields x(t) = (a + 0t)/0, (18) 1/2 where a = rj - r - id and $ = k + an. For x(0) = to hold as assumed, a a must equal zero, whih then makes x(t) equal to (0/j3)t. Sine the population externality is present under the given utility funtion for all values of P, the ritial date s in the preeding analysis is equal to zero (growth ontrols are imposed immediately). Consider the optimal -1/2 growth ontrol first. To find its form ± note that sine r = -(a/2)p, the x (t integral in (16) is (a/2)(*x (t) 2 )" 1/2 e f 2nxdx = (a/2)x 1/2 (t)r. Adding this expression to the RHS of (17) and solving for x yields x (t) = (9/B)t (19) e e

25 1/2 1/2 where = k + (3a/2)r (a = is used). Sine 3 > S = k + an, it 17 follows that x (t) < x(t) for t > e With the distane to the urban boundary proportional to t under both the equilibrium and effiient growth paths, it is interesting to onsider the entire lass of linear paths, where the boundary distane at t is equal to Xt for some X > 0. Sine it an be shown that total property value is a singlepeaked funtion of X, a piture suh as Figure 2 applies. Total property value inreases as X falls from 0/(9, reahing a maximum at X = 0/3. Further redutions in X redue total value, with value falling to an expression equal to the present value of agriultural rent as X approahes zero (X = orresponds to a total development ban). Figure 2 shows that while a moderate ontrol an raise total property value above the equilibrium level, a stringent ontrol leads to a redution in total value. 1 fi 6. Politial Considerations and Closed-City Analysis Up until now, no mention has been made of the politial fores leading to the imposition of growth ontrols. Note first that sine utility is fixed in the analysis, onsumers are indifferent to the presene of a ontrol (qualityof-life gains are dissipated in higher land rents). Landlords, however, have a strong interest in the nature of the ontrol law. Imposition of a partiular growth ontrol will be supported by landlords who stand to reap windfall gains under the law and opposed by landlords who expet windfall losses. For a partiular ontrol to be politially viable, the gainers must have more politial lout than the losers. Sine gainers and losers under many growth-ontrol proposals will orrespond roughly to the owners of developed and undeveloped land, a politial struggle between these groups is likely. Interestingly, in one ase where this mathup is exat (the ase of a omplete development ban after date s), owners

26 18 of developed land reap the largest possible gains. If this group is suffiiently powerful relative to owners of undeveloped land, a development ban might well be imposed. By ontrast, in a Coasian world where transations osts are absent, any growth ontrol (development ban or otherwise) that lowers total land value is not politially viable. The reason is that potential losers are better off paying gainers to vote against it. In suh a world, adoption of an effiient growth ontrol is in fat a likely outome. 17 With this brief disussion of politis in mind, a natural next step is to investigate a growth-ontrol model where onsumers are not indifferent to the presene of the ontrol. The obvious way to onstrut suh a model is to assume that the ity is losed rather than open, with a population growth path 18 that is fixed exogenously. One the losed-ity assumption is imposed, however, the growth ontrol an no longer have an effet on the ity's quality of life (the ontrol restrits spatial growth, but this simply paks the 19 population into a smaller area without affeting its size). As a onsequene, land value hanges indued by the ontrol have no amenity omponent. Inreases in the value of developed land are purely the result of a supply restrition in the fae of a growing population. Similarly, the ontrol's only impat on undeveloped land is to delay onversion, whih unambiguously redues its value in the absene of an amenity effet. Identifiation of gainers and losers among landlords is easy in the losed-ity model sine these groups orrespond exatly to the owners of developed and undeveloped land. However, there is a new group of losers in suh a model, namely onsumers, whose utility is lowered at all dates following Imposition of the ontrol. This is a onsequene of the inrease in urban land rents that follows from spatial onstrition of the ity.

27 With a new group of losers present, it appears that the politial basis for growth ontrols is muh weaker in the losed-ity model than in the amenity-based open-ity framework. Indeed, as desribed, the losed-ity model seems unsatisfatory as a framework for the analysis of growth ontrols. This verdit would hange, however, if the assumption of absentee landownership were altered. If landlords were instead to live in the ity, then the share of the renter lass in the urban population would be redued and the politial opposition to growth ontrols diluted. With the ost of oupany (i.e., mortgage payments) fixed for the owners of developed land but with urrent land rents (and hene values) inreasing under the growth ontrol, the landowner 20 portion of the urban population would benefit from suh a law. If it were large enough, this group ould enfore its will at the ballot box against the opposition of renters and the owners of undeveloped land. Although the amenity aspets of growth ontrols are absent from this model, it might yield further 19 useful insights. 7. Conlusion Interest in the impat of growth ontrols on land values has generated a host of empirial studies. The literature, however, offers no formal analysis of this issue. To remedy this omission, the present paper has analysed the impat of growth ontrols in an open-ity model similar to that of Capozza and Helsley (1988). The paper offers a number of insights, the most important of whih is that growth ontrols in an amenity-based model may raise rather than lower the value of undeveloped land in some loations. More generally, the paper shows how to onstrut a simple yet realisti framework for the analysis of growth ontrols. Tasks for future researh ould inlude analysis of the modified losedity model disussed above. In addition, it might be useful to explore a

28 20 variant of the open-ity model in whih the ity imposing the growth ontrol is "large" relative to the rest of the eonomy. In this situation, the diversion of population aused by the ontrol would be great enough to depress the utility level in other ities. This welfare loss (whih would also our within the ontrolled ity) would ount as an additional ost of the ontrol. This type of model might be espeially relevant for regions like the San Franiso Bay Area where the widespread use of growth ontrols undoubtedly leads to a general equilibrium impat on onsumer welfare.

29 X Fig. 1 total land value Fig. 2 6/6 e/s

30 . 21 Referenes Blak, J. Thomas and Hoben, James "Land Prie Inflation and Affordable Housing: Causes and Impats." Urban Geography 6 (Jan. -Mar.): Capozza, Dennis and Helsley, Robert "Fundamentals of Land Pries and Urban Growth." Unpublished paper (forthoming Journal of Urban Eonomis ) Cooley, Thomas F. and LaCivita, C.J "A Theory of Growth Controls." Journal of Urban Eonomis 12 (Sept.): Dowall, David and Landis, John D "Land Use Controls and Housing Costs: An Examination of San Franiso Bay Area Communities." Amerian Real Estate and Urban Eonomis Assoiation Journal 10 (Spring): Elliot, Mihael "The Impat of Growth Control Regulations on Housing Pries in California." Amerian Real Estate and Urban Eonomis Assoiation Journal 9 (Summer): Fishel, William "Do Growth Controls Matter: A Review of Empirial Evidene on the Effetiveness and Effiieny of Loal Government Land Use Regulation." Unpublished paper, Dartmouth College. Gleeson, Mihael E "Effets of an Urban Growth Management System on Land Values." Land Eonomis 55 (August): Katz, Lawrene and Rosen, Kenneth T "The Interjurisditional Effets of Growth Controls on Housing Pries." Journal of Law and Eonomis 30 (April): Kim, Chung-Ho, 1989, "Urban Spatial Growth and Land Pries." Unpublished paper, Korea Loal Administration Institute. Knapp, Gerrit J "The Prie Effets of Urban Growth Boundaries in Metropolitan Portland, Oregon." Land Eonomis 61 (Feb.): Nelson, Arthur C "An Empirial Note on How Regional Urban Containment Poliy Influenes an Interation Between Greenbelt and Exurban Land Markets." Journal of the Amerian Planning Assoiation 54 (Spring): Rosen, Kenneth T. and Katz, Lawrene "Growth Management and Land Use Controls: The San Franiso Bay Area Experiene." Amerian Real Estate and Urban Eonomis Assoiation Journal 9 (Winter): Shwartz, Seymour I.; Hansen, David E.; and Green, Rihard "Suburban Growth Controls and the Prie of New Housing." Journal of Environmental Eonomis and Management 8 (De): Shwartz, Seymour I; Zorn, Peter M. ; and Hansen, David E "Researh Design Issues and Pitfalls in Growth Control Studies." Land Eonomis 62 (August):

31 22 Sheppard, Stephen "The Qualitative Eonomis of Development Control." Journal of Urban Eonomis 24 (Nov.): Vaillanourt, Franois and Monty, Lu "The Effet of Agriultural Zoning on Land Pries, Quebe, " Land Eonomis 61 (Feb.): 36-42

32 . 23 Footnotes *I wish to thank Perry Shapiro and Jon Sonstelie for helpful disussions of some of the issues onsidered in this paper. Kangoh Lee also provided helpful omments. Errors or shortomings, however, are my responsibility. For an exhaustive and engaging survey of the empirial literature on growth ontrols and zoning, see Fishel (1989). 2 Higher development fees an also raise pries as they are passed on to onsumers 3 There have been few previous attempts in the literature to model growth ontrols. Cooley and LaCivita's analysis (1982) depits the hoie of optimal ity size in a stati model. Sheppard (1988) analyses the effet of restriting the land area available to various lasses of onsumers in a stati multi-lass ity. By onduting stati analyses, both of these papers omit key dynami elements of the growth ontrol problem. 4 Like the ommuting ost parameter k, r and D are assumed to be onstant over a time. All of these parameters ould be made funtions of time without affeting the onlusions of the analysis. 5 From above, this requires that dr(t,x,p(t) )/dt = r + r P'(t) > holds at t = T(x). Notingthat P'(t) = 2*x(t)x'(t), and substituting the above expression for x'(t), the seond-order ondition redues to r r /(r + 2axr p ) > 0, whih holds as long as r > 0. The first laim follows beause r_ = r _ = 3(-k)/3P = 0. The temporal Px xp behavior of r is unertain beause the total derivative dr (t,x,p(t))/dt is ambiguous in sign. 7 Note that r may fall to zero at a given t and x as population delines from P(t) to P (t). Sine r n starts out negative, however, it must be the P ase that (6) exeeds (5). 8-2,4$ -it -2 V(T,x,s 7rx ) is gotten by subtrating /re dt from V(T,x,0 kx ) and u a is multiplying by e

33 24 g Note that this definition does not rule out disontinuities in T (x) away from x(s) (or equivalently, flat ranges in x (t) away from t = s). The presene of suh disontinuities does not affet the results derived below. Note that for the purposes of this alulation, the funtion T (x) is assumed to be differentiable. With a hange of variable from x to t, (14) beomes 00 2ttx e o (t)x' (t)de _lt dt e Integrating the above expression by parts assuming x (0) = yields (15) 12 The seond-order ondition requires that r + 4rx rn + x e P X e 2-4k xx r dx < e PP This ondition is assumed to hold (note that satisfation of the ondition is guaranteed if r < 0, indiating that rent dereases at an inreasing rate with population) 13 - Stritly speaking, this inequality holds at values of t where x (t) < B. For larger t's, equality holds. 14 If r is twie ontinuously differentiable, then x (t) must be differentiable. Given that x '(t) = x'(t) > for t < s, x '(t) annot be zero immediately after s without violating differentiability. 15 The qualifiation stated in footnote 13 applies here. 16 An attempt was made to investigate the spatial pattern of land value impats under a linear ontrol using the above example. Unfortunately, muh of the ambiguity enountered in the general ase remained.

34 As usual, this outome requires a prior assignment of property rights. The natural assignment gives landowners the right to develop their land unless persuaded to do otherwise. 18 See Kim (1989) for an analysis of the losed-ity version of the Capozza- Helsley model used in this paper. 19 Land onsumption must be endogenous in this model rather than being fixed at unity. 20 Note that a ompliation in analysing this model is that the inome of urban residents is no longer endogenous (landowners' inome depends on endogenous urban rents)

35 " Papers in the Politial Eonomy of Institutions Series No. 1 Susan I. Cohen. "Pareto Optimality and Bidding for Contrats." Working Paper # 1A11 No. 2 Jan K. Bruekner and Kangoh Lee. "Spatially-Limited Altruism, Mixed Clubs, and Loal Inome Redistribution." Working Paper #1406 No. 3 George E. Monahan and Vijay K. Vemuri. "Monotoniity of Seond-Best Optimal Contrats." Working Paper #1417 No. 4 No. 5 No. 6 No. 7 No. 3 Charles D. Kolstad, Gary V. Johnson, and Thomas S. Ulen. "Ex Post Liability for Harm vs. Ex Ante Safety Regulation: Substitutes or Complements?" Working Paper #1419 Lanny Arvan and Hadi S. Esfahani. "A Model of Effiieny Wages as a Signal of Firm Value." Working Paper #1424 Kalyan Chatter jee and Larry Samuelson. "Perfet Equilibria in Simultaneous-Offers Bargaining." Working Paper #1425 Jan K. Bruekner and Kangoh Lee. "Eonomies of Sope and Multiprodut Clubs." Working Paper #1428 Pablo T. Spiller. "Politiians, Interest Groups, and Regulators: A Multiple-Prinipals Ageny Theory of Regulation (or "Let Them Be Bribed." Working Paper #1436 No. 9 Bhaskar Chakravorti. "Asymmetri Information, 'Interim' Equilibrium and Mehanism Design." Working Paper #1437 No. 10 Bhaskar Chakravorti. "Mehanisms with No Regret: Welfare Eonomis and Information Reonsidered." Working Paper #1438 No. 11 Bhaskar Chakravorti. "Communiation Requirements and Strategi Mehanisms for Market Organization." Working Paper #1439 No. 12 Susan I. Cohen and Martin Loeb. "On the Optimality of Inentive Contrats in the Presene of Joint Costs." Working Paper #1445 No. 13 Susan I. Cohen and Martin Loeb. "The Demand for Cost Alloations: The Case of Inentive Contrats Versus Fixed-Prie Contrats." Working Paper #1455 No. 14 Jan K. Bruekner and Kevin M. O'Brien. "Modeling Government Behavior in Colletive Bargaining: A Test for Self-interested Bureaurats." Working Paper #1481 No. 15 Jan K. Bruekner. "Estimating a Bargaining Contrat Curve: Prior Restritions and Methodology. Working Paper #1490 No. 16 Peter C. Reiss and Pablo T. Spiller. "Competiton and Entry in Small Airline Markets." Working Paper #1497 No. 17 No. 18 No. 19 No. 20 No. 21 Pablo T. Spiller. "A Note on Priing of Hub-and-Spoke Networks." Working Paper #1498 Larry DeBrok. "Joint Marketing Efforts and Priing Behavior." Working Paper #1500 Frank A. Wolak and Charles D. Kolstad. "A Model of Homogenous Input Demand Under Prie Unertainty." Working Paper #1502 Susan I. Cohen. "Reputation, Intertemporal Inentives and Contrating." Working Paper #1511 Lanny Arvan and Antonio Leite. "A Sequential Equilibrium Model of Cost Overruns in Long Term Projets." Working paper #1514 No. 22 Jan K. Bruekner and Pablo T. Spiller. "Competiton and Mergers in Airline Networks." Working Paper #1523 No. 23 Hadi S. Esfahani. "Reputation, Produt Quality, and Prodution Tehnology in LDC Markets." Working Paper # No. 24 Hadi S. Esfahani. "Moral Hazard, Limited Entry Costs, and 'Introdutory Offers'." Working Paper # No. 25 Bhaskar Chakravorti. "Mehanisms with No Regret: Welfare Eonomis and Information Reonsidered." Working Paper # No. 26 Susan I. Cohen. "Impliit Cost Alloation and Bidding for Contrats." Working Paper # No. 27 Rafael Gely and Pablo T. Spiller. "A Rational Choie Theory of the Supreme Court." Working Paper # No. 28 Rafael Gely and Pablo T. Spiller. "An Eonomi Theory of Supreme Court Statutory Deisions The State Farm and Grove City Cases." Working Paper # No, 29 Rafael Gely and Pablo T. Spiller. "The Politial Eonomy of Supreme Court Constitutional Deisions: The Case of Roosevelt's Court Paking Plan." Working Paper #1561 No. 30 Hadi S. Esfahani. "Reputation and Produt Quality Revisited." Working Paper #

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