Open Space Allocation and Travel Costs

Transcription

1 Open Spce Alloction nd Trvel Costs By Kent Kovcs Deprtment of Agriculturl nd Resource Economics University of Cliforni, Dvis Pper prepred for presenttion t the Americn Agriculturl Economics Assocition Annul Meeting, Montrel, Cnd, July 7-30, 003 Copyright 003 by Kent Kovcs. All rights reserved. Reders my mke verbtim copies of this document for non-commercil purposes by ny mens, provided tht this copyright notice ppers on ll such copies.

2 Abstrct Open spce lloction in city hs lrgely been ddressed by simultneously llocting lnd for open spce nd residentil housing in generl equilibrium model. Open spce competes with lot sizes of residentil housing to determine the optiml density nd lloction of open spce in city. Residents derive enjoyment from the open spce, but they never ctully visit it. In other words, these models ignore tht people must trnsport themselves to the open spce to enjoy it. Open spce is often defined s ny pristine nturl re, but the focus in this pper is lnd for prks. Rther thn ddressing whether prks tke too much lnd wy from residentil housing, this pper focuses on how prks should be llocted to mximize the net benefits from visittion to the prks. The net benefit from visit to prk is the vlue derived from the prk less the trvel costs to rech the prk. The plnning uthority is modeled to mke two decisions when llocting fixed mount of open spce in the city. The plnner decides how mny prks to hve nd where the prks should be locted in the city. When there re more prks, the trvel costs from visittion re reduced; however, when there re more prks, every prk is smller thn before, nd the vlue of the open spce is reduced. A condition is derived to inform the plnner how to optimlly llocte the open spce when both the number nd the loction of the prks re vrible. The model originlly hs trvel costs constnt throughout the city. However, lter in the pper, trvel costs re llowed to vry over spce. In prticulr, trvel costs re modeled to be more concentrted over the city center becuse there re demogrphic differences. The first order conditions for the optiml open spce lloction re determined nlyticlly, but the optiml plcements re computed numericlly. The

3 results indicte tht fewer number of prks re reltively more optiml when trvel costs re concentrted t the city center. Among the drwbcks to the model in this pper is tht the plnner lloctes fixed mount of lnd. More relisticlly, the plnner would simultneously determine how much lnd there should be for open spce, how to divide the lnd nd where to plce the lnd. Such n pproch hs been used in pst ppers except tht those ppers did not consider the trvel costs incurred by residents to rech the open spce. Another drwbck is tht the plnner lloctes ll the open spce t once. However, prks re usully creted in city when there is public demnd or money in the budget for them. There re mny possible extensions to this reserch. For instnce, some trvel costs re rodblocks so tht trvel costs spike t those res in the city. Further, the vlue derived from prk is probbly dependent on its nerness to other menities like n ocen or mountin. Clerly, the number of prks, their loction, nd their size hs gret influence on the net benefits residents derive from them. These chrcteristics of open spce require considertion before meningful model of competing lnd use between open spce nd residentil housing is developed. 3

4 Introduction Open spce in urbn res, such s prks, prkwys, greenbelts nd public squres provide numerous services to city residents. Clen ir, scenic vists nd recretionl opportunities re mong the benefits vilble to open spce visitors, nd city plnners utilize open spce to shpe nd contin urbn res (Fujit, 1997). Lrge proportions of some cities re occupied by open spce, but there hs been reltively little reserch done from n economic perspective to determine how this open spce is best distributed throughout city. There re mny considertions tht go into determining how to llocte prks in city. When the question is wht lloction of prks provides the gretest surplus to city residents, the economic content is the ingredients people use to decide if nd wht prks to visit. In prticulr, wht re the trvel costs to visit prk nd how do the benefits of prk depend on its chrcteristics. A city plnner s point of view is tken to conduct the nlysis of the open spce lloction. The city plnner is benevolent ruler tht ims to find the open spce lloction tht provides the city residents with the mximum surplus, i.e. the vlue of the visit less the trvel costs, from the open spce. The city plnner hs vrious options t its discretion to increse the surplus residents derive from open spce. A subset of these options is tht the city plnner cn mnipulte the sptil chrcteristics of the open spce. For exmple, the city plnner cn mold the open spce into ny shpe it wnts. Another option the city plnner hs is to divide into pieces fixed mount of lnd for prks. In other words, rther thn being forced to put single prk someplce in city, the city plnner hs the freedom to split 4

5 the prk into numerous pieces nd distribute those pieces nywhere it likes in the city. The city plnner s use of this option to llocte prks in city is the focus of this pper. Before going into further detil bout the model, n exmintion into how this nlysis reltes to the pst literture identifies the contribution mde here. Open spce hs received ttention recently becuse economists hve tken n interest in urbn sprwl. The reserch on urbn sprwl exmines the fctors tht cuse open spce outside city to be developed into residentil housing or frmlnd. In Wu (001), lepfrog development is explined by communities interest in locting ner nturl menity such s river or scenic hill. While Wu investigtes how n urbn re expnds to fill round nturl menities, this pper exmines how nturl menities re optimlly plced inside n lredy existing city. Yng (1990) looked t the provision of centrl prk, nd Fujit nd Lee (1997) exmined the efficient configurtion of greenbelt menity. A more generl pproch by Fujit nd Yng (1983) llowed the menity density distribution to vry cross the city. In prticulr, the finding in tht pper ws tht if households hve log-liner or Cobb- Dougls utility function, the efficient density of the menity is uniform cross the city. Unfortuntely, the form tht the uniformly dense menities hve throughout the city is left mystery. It could be tht there re few medium sized prks locted bout the city in uniform intervls, or there my be numerous smll prks distributed bout the city. In this pper, the model ddresses how much the open spce should be split prt. The purpose here is to emphsize tht there re detils to the lloction of open spce never explicitly considered in more complex models tht hve n importnt impct on the totl surplus derived by city residents. 5

6 Multiple prks The investigtion of the optiml lloction of city prks is simplified by considering city tht is represented by line. While this is not representtive of most cities, smll towns where min street is the center of ctivity re well represented by line. If the smll town explntion does not ppel to the reder, then consider tht city plnner is trying to decide how to plce prk long single street in mjor city. The generlity of the results reched re not ffected by using line, but there is likely more richness in the two dimension results tht is lost. Consider city plnner tht is required to plce open spce of length, l, in city. The city plnner is free to divide the prk into however mny pieces it desires, nd those pieces cn be plced wherever it wishes throughout the city. The length of the city not including the prk is normlized to one. People in this city py constnt cost of k per unit of length trveled. The city plnner is posed with the dilemm of whether to crete one prk or multiple eqully sized smller prks tht hve totl length equl to the single prk. Evluting these lterntives requires knowledge bout the vlue people derive from prk, nd the trvel costs people incur to obtin tht vlue. Suppose tht people in city visit prk only once, nd the only vlue people derive from the prk is from the visit. There is no vlue derived from the trvel to the prk. All people derive n identicl vlue from prk visit, but the vlue derived is dependent on the size of the prk. The only cost incurred from prk visit is the trvel cost. In other words, there is no fixed fee for entrnce to the prks. 6

7 In order to determine wht lloction of the open spce is best, the surplus (i.e. the totl vlue derived from the prk less the totl costs to society from visiting the prk) for n bstrct number of prks is computed. The totl cost of visiting the prks is determined in two steps. First, the optiml plcement of the prks to minimize trvel costs is found. Second, the totl cost to society ssocited with tht optiml prk plcement is found. The benefit from the prk system depends on the number of prks the city plnner chooses to crete. The benefit from hving r prks is U(,, s t z ) where s = ( A / r) is prk size, t = br is the number of trips to the prk, nd z is vector of other prk chrcteristics. U ( ) is qusiconcve nd monotoniclly incresing in its rguments. Since A is the fixed mount of lnd for prks the plnner hs to distribute, the prks get smller, i.e. s flls, when their number is incresed. Although it seems nturl to ssume diminishing returns to prk size, the extent of the incresed vlue from combining two prks into one is n empiricl question. The number of trips, t, to the prks increses with the squre of the number of prks becuse people visit prks more frequently when they re nerby. The choice to the mke trips incresing with the squre of the number of prks is d hoc; look t estimted trvel cost demnd curves for city prks should llow more relistic derivtion of the reltionship between trips to the prk nd the number of prks. The digrm below shows the ppernce of city when the city plnner decides to crete r prks. The prks re the drkened portions of the line. The res where the resident live nd commute to the prks re x1, x,..., x n. These res sum to one so tht prks re in fct only points on the line. x 1 x 3 x. n 1 x x n 7

8 Since trvel costs re constnt k per mile, the societl cost of visiting prk for people tht live long line of length is: kx dx= k x = k. Since the totl cost of the prk is only the trvel costs, the cost minimiztion problem where there re n res tht the residents live nd r = ( n 1) prks is: n 1 n 1 1 min kx ( 1 + i n ).. i 1 1,..., n x + x st x= x x i= i= 1 The first order conditions re tht 1 = n nd tht x1 = x i i =,.., n 1. In other x x words, the prks re plced symmetriclly long the line so tht no resident trvels x x further thn 1 = n distnce to rech prk. Solving these first order conditions, it is x x = x = found tht 1 = n (1 / r) nd tht i = (1 / r) i,.., n 1. When these res re plugged into the objection function, the totl trvel costs when r = ( n 1) prks re optimlly plced in city is found to be ( k / 4 r ). There is no indiction yet wht the optiml number of prks is since the benefits from creting prks hve not been introduced. Below net benefits re mximized to yield first order condition for the number of prks, nd there re some comprtive sttics for chnges in the prmeters. mx r k NB() r = U(,, s t z) 4r The first order condition is: dnb () r Gr () U A = = + U br+ k dr s r t 4r 8

9 The comprtive sttics re: r G / k (1 / 4 r ) ( ) = = = k G / r ( / + ) ( U A U A + U U br+ b k ) r s r s 3 r r t t 3 4r r G / b r( U / t) ( ) = = = b G / r G / r ( + / ) U 1 r G / A s r ( + ) = = = A G / r G / r ( + / ) The rgument in the benefits function, tht looks like utility function, for prk trips gives the comprtive sttics their mbiguous signs. If the prk trips rgument is dropped nd only prk size is considered, then the comprtive sttics hve expected signs. In prticulr, r / k is positive, nd this is expected since plnner would wnt to crete more prks when ech mile to prk is more costly to trvel. Also, r / A is negtive. Here it is difficult to sy wht the expected sign is since more lnd might men you wnt to combine it with n originl prk or crete new prk in the city. The single first order condition result in this model is due to the simplicity of the model, but there is more hppening here thn the choice bout wht prk configurtion yields the highest surplus. There re clerly equity issues rised from this nlysis. In prticulr, suppose tht the higher surplus prk configurtion is the one where there is single centrl prk. The mjority of the surplus goes to the people living close to the lrge prk while the people on the fringe of the city receive fr less surplus thn if the prk were split into two. When the prk is split into pieces, the trvel costs incurred by the city residents re much more similr thn when there is single centrl prk. Equivlently, the surplus from the prk menity is much more equitbly distributed if 9

10 prk is split into pieces. Accordingly, if equity is n issue, the city plnner might consider foregoing higher surplus centrl prk option for more equitble two prks option. Additionlly, besides using the first order condition to illuminte the nlysis bout how to optimlly distribute prks in city, this condition cn be used to infer the beliefs city plnners hve bout the vlues of different sized prks to visitors. The first order condition sys to equte the chnge in benefits from n dditionl prk with the chnge in trvel costs. An empiriclly determined chnge in totl trvel costs llows for n estimte on the chnge in vlue the plnner hs for tht configurtion. In prticulr, GIS dt llows resercher to determine the totl trvel costs ssocited with n lterntive numbers of prks. Next, by noting wht prk configurtion the city plnner ctully chose, the resercher retrieves some informtion bout the difference in vlues the city plnner believes the residents hve for the different sized prks. For instnce, if centrl prk is chosen for city nd the difference in totl trvel costs from the two prks option is some mount x, the resercher would conclude tht the city plnner believes the vlue from visit to lrge prk less the vlue from visit to prk hlf its size is greter thn x. Hedonic studies on housing prices hve estblished the vlues people plce on nerby open spce. If the hedonic studies hve estblished vlues for prks of different sizes, the difference in vlues in those studies cn be compred to the inferred beliefs tht the city plnner hs bout peoples vlues for different sized prks. Noting whether those vlue differences devite significntly from ech other provides check on whether city plnners re ppropritely llocting open spce in city. 10

11 The next section exmines more rigorously how city plnner should best distribute prks when trvel costs re bunched up over prticulr re. The opportunity costs of time incurred by prk visitors re considered. A closer look t trvel costs Suppose tht there is uniform difficulty trveling throughout the city, i.e. no rodblocks, but people in the city hve vrying opportunity costs of time. On the outskirts of the city the opportunity cost of time is low while t the city center the opportunity cost of time is the highest. This representtion of the vrying opportunity costs of time is bsed upon the ssumption tht the high pying jobs re locted t the city center, nd people visit the prk during the dy. At ech spot in the city there is different opportunity cost of time, i.e. wge, nd for the person emnting from tht spot tht cost is constnt per mile mount wherever the person goes in the city. All people still py the constnt k per mile relted to the costs ssocited with their vehicle. The curve cx( x 1) describes the vrying opportunity cost of time where x is the loction long the line nd c controls the mgnitude of this opportunity cost. Note, tht t x = 0 nd x = 1, the opportunity cost of time is zero no mtter how lrge c is mde. At the fringes of the city, no people work, nd their opportunity cost of time is zero. The curve reches its mximum where x = 0.5 t the city center. Presumbly, the CEOs working downtown hve the highest opportunity costs of time. These two costs combined describe the trvel costs fced by the city residents. Since trvel costs hve been mde more complex, this nlysis simply compres the one versus two prks lterntives to keep the mth from obscuring the informtion provided by the new economic content. The surplus from ech option is sought to 11

12 determine the best lterntive. The benefits side remins simple since the vlues derived from visits to different sized prks re given. However, the prk plcements to minimize trvel costs hve become more involved. The minimiztion problem for the one prk scenrio is: min ( ) φ = ( k + cx(1 x)( x) dx+ ( k+ cx(1 x)( x ) 0 1 dx Leibniz s Rule is pplied to obtin the necessry condition to this problem: φ = ( k + cx(1 x) ) dx ( k + cx(1 x) ) dx = = c c c c c kx x + x kx x + x = k c + c k = The only solution to the necessry condition tht lies within the intervl [0,1] is = 0.5. In other words, for ll k nd c, the optiml plcement of the prk is t the center of the city. Recll tht in the first model where k is the only trvel cost component, the optiml plcement for the single prk ws lso t the center of the city. Since the opportunity costs of time component to trvel costs reches its mximum t the center of the city, the conclusion is not surprising tht trvel costs re minimized when the prk is plced t the center of the city. At tht plcement, the CEOs locted downtown, those with the highest opportunity cost of time, trvel the lest. Since the prk is lwys plced t the center of the city, the totl trvel costs re 1 8k + c esily determined in terms of the prmeters k nd c. In prticulr, φ =, 3 8k + c nd the surplus from this lterntive is v1. 3 The minimiztion problem for the two prks scenrio is: 1

13 min b, ( ) 0 ( b+ ) φ b, = ( k+ cx(1 x)( x) dx+ ( k+ cx(1 x)( x ) dx b + ( k + cx(1 x)( b x) dx+ ( k + cx(1 x)( x b) dx ( b+ ) 1 b Agin, Leibniz s Rule is pplied to get the necessry conditions: 0 ( b+ ) φ = ( k + cx(1 x) ) dx ( k + cx(1 x) ) dx= 0 1 ((( ) 3 3( ) b b ) c 1( b ) k 8 ((3 ) c 6 k )) 0 = = 4 b φ b = + ( k + cx(1 x) ) dx ( k + cx(1 x) ) dx = 0 1 ( b ) b 1 ( ( 3 3 ( 1) 3 ( ) 3 b b b b (7 5 b ) 4) c 1 k ( b ) 4 8 bk ) 0 = = 4 Now the optiml nd b vry bsed upon the vlues tht k nd c hve. Accordingly, explicit solutions for nd b in terms of k nd c would be useful. However, the first order conditions re too intrctble to find nd b other thn numericlly. Below the tble shows the optiml prk plcement(s) for both the one prk nd two prk scenrios for different vlues of k nd c. Further, the tble shows the totl trvel costs for both scenrios nd the rtio of those totl trvel costs. k / c One prk vs. Two prks Alterntives for Different k nd c Single Prk Prk Loction Totl Trvel Cost: TC1 Prk 1 Loction Two Prks Prk Loction Totl Trvel Cost: TC Rtio of Totl Trvel Costs: TC/TC1 500/ / / /

14 When the problem is where to plce single prk, the conclusion is to put it t the center of the city for ll k nd c. However, when the problem is where to plce two prks, the optiml plcements vry bsed upon k nd c ; in prticulr, the rtio k / c rther thn the bsolute vlues of k nd c determine the optiml plcements. When k is lrge reltive to c, the constnt cross spce cost per mile component to totl trvel costs domintes, nd the optiml plcements re equivlent to those found when k is the only trvel cost component. When c is lrge reltive to k, the optiml plcements re squeezed closer to the center of the city. Since the opportunity cost of time component to trvel costs is the most significnt, trvel costs re concentrted over spce t the city center. The best wy to llevite the high trvel costs t the city center is to bring the prks closer to it. In this wy, those people tht hve the highest opportunity costs of time re those tht trvel the lest. Further, the tble shows how the totl trvel costs of ech lterntive vry with k nd c. Note tht for every k / c rtio, the totl trvel costs re lower in the two prk lterntive since the totl distnce people hve to trvel is reduced. Also, if k or c increse, the cost of every mile trveled hs risen so tht nturlly the totl trvel costs in both lterntives increse. Accordingly, the pproprite wy to compre the totl trvel costs is to clculte the rtio between them. In this wy, the gin observed from the reduced trvel costs in the two prks scenrio does not depend on the levels tht k or c hve. Observe tht the totl trvels costs rtio rises s the k / c rtio flls. When the opportunity cost of time is the dominnt trvel cost component, the gin from choosing the two prks lterntive is less. 14

15 The rtio, TC/TC1=0.5, when k is the only trvel cost is cler becuse the two prks lterntive mens the distnce trveled by people is cut in two. Tht is, the two prks lterntive exctly hlves the totl trvel costs. However, the conclusion tht the two prks lterntive is even less beneficil when the opportunity cost of time component is the most significnt is less cler. When the time cost component domintes, the distnce trveled by people ner the city center is cut by more thn two, but the totl trvel cost is cut by less thn two. When costs re the sme for ll people in city, the optiml plcements for the two prks option re those tht cut the distnce trveled by people in hlf. When trvel costs re greter for those t the city center, the intuitive response is to bring the prks closer to those with the high trvel costs. Those with the highest trvel costs re mde to trvel less since ech mile less they trvel contributes more to cost svings thn mile less for someone with smll trvel costs. However, the optiml plcements moved closer to the city center men tht the distnce trveled is cut by less thn two. Since residents re trveling further when the optiml plcements re closer to the city center, the totl trvel costs of the two prks option is cut by less thn two. When trvel costs re heterogeneous becuse of demogrphic differences in some city regions, the two prks option is less effective t reducing totl trvel costs. A policy prescription is tht homogeneous mix of people with different opportunity costs of time should be encourged in city. The extension here with the opportunity costs of time concentrted over n re in spce is wy to consider vrying demogrphics of ll kinds over spce. For instnce, the chief demogrphic feture of city center might not be tht it is where people with 15

16 high opportunity costs of time reside during the dy. Rther, city center might be the plce where the poor live who hve mlfunctioning utomobiles or rely on public trnsporttion. At the outskirts of town people hve expensive utomobiles to tke them esily nywhere throughout the city. In this instnce, the stereotypicl demogrphics of city center complement ech other to suggest tht trvel costs increse for people round city center so tht the two prks option is the most beneficil. Informtion bout the true demogrphics in city nd cler understnding bout wht those demogrphics men for trvel costs for those people is crucil to determining how trvel costs re concentrted over spce. Conclusion The plcement nd number of prks throughout city hs strong influence on the surplus residents derive from those prks. In prticulr, splitting open spce into pieces is powerful option the city plnner hs to increse surplus for residents. By incresing the number of prks, trvel costs re lwys reduced, but the size of the reduction depends on the kinds of trvel costs in the city. If trvel costs re concentrted in regions of the city, creting two prks is reltively more costly for residents lthough it is more equitble. 16

17 References Correll, M., J. Lillydhl, nd L. Singell. "Effects of greenbelts on residentil property vlues-some finding on the politicl-economy of open spce." Lnd Economics 54, no. (1978): Hotelling, H. "Stbility in Competition." (199). Johnson, M. "Environmentl impcts of urbn sprwl: survey of the literture nd proposed reserch gend." Environment nd Plnning A 33, no. 4(001): Lee, C. M., nd M. Fujit. "Efficient Configurtion of greenbelt: theoreticl modelling of greenbelt menity." Environment nd Plnning A 9(1997): Rollins, K., nd A. Lyke. "The Cse for Diminishing Mrginl Existence Vlues." Journl of Enviromentl Economics nd Mngement 36, no. 3(1998): Wile, J. "Open spces, Revenue shring, nd urbn structure." Journl of Urbn Economics 5, no. 1(1978): Wu, J. "Environmentl menities nd the sptil pttern of urbn sprwl." Americn Journl of Agriculturl Economics 83, no. 3(001): Yng, C. H. "The optiml provision of centrl prk in city." Journl of Regionl Science 30, no. 1(1990): Yng, C. H., nd M. Fujit. "Urbn sptil structure with open spce." Environment nd Plnning A 15(1983):