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Intermediate goods and the spatial integration of land uses
Regional Sck
to be 2 mdbted
1.
and Ur
Economics 6 ( f376) 1Y- 145. ~9 North-Holland
one &MI end one intermediate. is set OUT !:I n spatial contest. nd must be transported at positive costs between an? cd WE in which the finat 2nd intermediate goods hich final and intermediate goods occupy distinct ning which pttcm will be &ticient is obtained and the result rnpk are rebted to atirvatzons reported on the history of urban areas. two distinct equitibria is fok nd in a model which permits another factor
for kml in prixhtion.
OD
In this FYWr. attention is fwused on the role of intermediate goods in models Mills (1970) pointed ou! tnat in a Von Thiinen model with two prgduccri gsgds - one final and one intermcdiatc - two distinct land USCpatterns are feasible solutions. In one sofution, called krt~gratcvl, the linal good and its intermediate good share the same site, so to speak, whereas in the other solution, ads ali occupy one zone or distinct area, and the other distinct area. The details of Mills’ model and the condition which determirles which land ore precisely. Subsequently it is shown that both “CWpatterns can obtain in d model in which stituting another factor for land as the price of land vatic~
*My
C~~~~j~
t~~~~ An
with computations in the numerical to Phihp C. ~,~~~w~~~ foor ~,:,&wx cnrlkr venion of this paper teas presented at the Econometric Society, P 1974. Rob Eergk, Cohn ~~~~~r~~)~ 3rd Pravin Varaiya provided helpful
128
JM. Hartwick, Intrmediate goods in land.-usemodels
‘I:: U850,Boston was a tig’htty p acked seaport, by 1901:it sprawled over a ten mile radius and contained th;r:; ;K cities and towns . . . The old settll:ment oi 1850 became by 1900 the prin$+J zone 3f work - the industrial, commercial, and coiumunications center c,‘l‘the metropcIitan region. Beyond the inner concentrated section there .:~r r an equally novel environment, the enormous outer ring of new commuters“ . -‘uses . . . the wide extent of settlement in ?h* outer residential zone was :na.~‘cpossible by the elaboration of the new street railway transportation system 2 :ti a parallel extension of city services.* Warner (1961, pp. l-3) Adna Weber calls the transformation 1900, *city-building’.
undergone by Boston between ISSO-
‘The original settlement becomes the brlsiness center and for some time continues to grow rapidly. But if the city prol;pers, the time will come when this old center is more and more needed for strictly business purposcJ; k~ses disappear before the march oi office buildings, government buildings, banks, etc., until the only residents left are the janitors and porters, the keepers of the great buildings. With continued growth, the business center extends itself and steadily pushes the dwellings toward &c circumference, until at length the municipal limits are reached and passed.’ Weber (1899, p. 467) Observe that Weber considers that the business center ‘pushes’ residential areas bcjond the urban periphery of a previous pericd. For Weber, trolley cars did not cause suburbanization or ‘pull the city out’ as Warner suggests but this transportation mode was developed in response to the ‘American penchant for dwelling in cottage homes instead of blusiness blocks after the fashion of Europe’ [N’eber (1899, p. 469)]. For Weber it was obvious that Philadelphia was the ‘city of homes’ long before rapid transit. A glance at the record of the average number of rides annually on street railways in the 11th Cef~cs Report on ih~~portatbn by Lartd reveals that American cities consistently ranked higher (more trips per person per year) than Euiropean cities. In co&nental Europe, Weber considered it the practice for artis:ms and shopkeepers to live in rooms connected with their workshop or store, whereas in the United States, presumably these same types of people would commute to their place of work each day. In the 11th Cmsus, Social Statistics of Cities, Weber (1899, p. 468) raoted that the Urlited States had about 15.2 urban dwellers per acre compared with 38.3 for England and 25.9 for Germany. In summary (p. 469), ‘the comparative figures of milea and number of rides per inhabitant of American and European cities are indii:d-!+rl of low or high density ~)op~~Iatio~,whit regarded as the catial: of street railway building’. Weber then implies that density patterns, determined oppot;ed to eco ic:forces, led to the development of differ t~a~s~~Ftat~~~~
1s vkw
contrasts
with
that
of F
J.&i. Hartwick,
InrPrmediate goods in land-use models
129
?3ne can only guess just how large metropolitan Boston worrld have grown ha*1there been no invention of new communication &vices. If the spread of the city had begun to exceed the distance a man might walk in about an hour, say a three-mile radius, the shops and offices of the metropolis would have fallen out of easy daily communication with each other. The result would have been the destruction of a single .unit*ed communication network and the development of semi-autonomous subcities which would h..ite had to duplicate many services and facilities offered in other parts of the c&r.’ Warner (1962, p. 16) In section 2, we will identify precisely tbc conditions under which the spatial integration or segregation of work-place and residence-place occurs. We will be able to identify the fclrces which lead a city to change from one form to another. We will be able to isolate the cases alluded to by Warner and Weber. Thus a technological change in transportation such as the electric trolley will be shown to make possibie the spatial segregation of work-place and residenceplace as Warner suggests. Also an exogenously given taste for relatively less land per iamily will lead to a spatial integration of work-place and rsidenceplace as Weber suggests.’ We shall examine these issues in the familiar model which postulates that there exists a geographic irregularity such as a deep water port which makes shipments to and from a city rcratively cheap. Thus activity agglomerates about this node and outp:Pt of ,.on-residential activtty is shipped from the city through this point or node. There will t : three sectors: production activity, residential activity, and transportation activ;‘.S In our first model there will be fixed production coeflicients and transportarion will !X assumed to use no land. Production activity will require land and workers, who iive in the residential sector. Workers spend all their income on commuting (if the allocation has spatially segregated sectors) and on this residence which requires only land as an input.’ ‘In the general area cl urban model building. it is interesting to note that since modern cities appear to display segregation of residence-place and work-place, model-builders take this property as given whereas they should really make the spatial integration or segtcgation of land uses a variable in tf e model. Solow (1973), for example. avoids the problem hy postulating enough increasing I :t;r11s to scale in production to lead to production being geographically concentrated but no! ~tfftcicnt increasing returns to scale to make necessary the USCof a theory of pricing for all firclot-s different from the marginal productivity approach. Outside of the spatial context, Grct P (1974) has investigated circumstances in whiih an output requiring an intermediate inpuf . I !w produced in a single vcttically irttegr;,ttcti firm or in two d,stinLt ‘upstream’ and Ldcwr~~tre;m-t”firms. Green’s results turn on a parlicular ‘nind of tran:..lction cost being inrcru,~liz A; by vertical intcgratior). In the snatial n-&cl, transpo~tatron coht\ CZI! be profitably interurli~ed t,nder the ertplicit conditions de~elopeti below in ihe *~:-a I. 2Thls model 15 vuiua;ty ideutlcal to that in ?\liih (1970) and (IW?, ch. 5). Howcvcr, Mill&) erred in solving his model JIX.! the basic condition on detcrmirrirrg whether spatial integration or’ segregation of activity WCuld result is not valid. Milis also did not attempt to rela?e his model to the historical dcAopment of cities ;is we are emphaGzing. Clouevcr. Mrlls pond the integration or segregaimportant question cleirlj.: What cflnclrtions determined whet er spati tion of activity res At:; I &en one good is an interme
li30
J.M. Hilrtwick, Znterrnedrate xaods in land-use models
A model with an activity analya.15 techno!ogy is then set out and examples are reported which display the switch from a spatially integrated allocation of actitities to a spatially segregated ;~!locati,on. In this last model, transportation a!so uses land and capitai. 2. The tied cwfficiemt model
, ,_
11: the town or center of our economy, there is an infinitely elastic demand for ,t!~efmal good at price p per unit. The solution to the problem will be an allocation which maximizes t!le value of output delivered. to the center. Since p is fixed, this is equivalent to maximizing the amount of the final good produced and d.e!ivered to the center. The final good, good 1, r;;qtires 1 unit of gocd 2, the mtermediate good, for each unit of good 1 produced. ‘u’ units of good 2 are produced from 1 unit of land and 1 Il!lit of good 1 requires 1 unit of land in productifon. The salient qu,r of Iii:ermediat;~?~ess in this mode! is that a ur.it of the intermediate good net ’ I’ ;;I ’ ,e +attsgorted to the site of the final good and not to the center point of our disc-shapeg.. economy. Transportation costs per unit distance arc i, and t2 Jollars for goods 1 and 2 respectively. Good 1 is rhe final good and good 2 is housing. Associated with each house is one worker, assumed for concreteness. There are two possible land use allocations in this mode!: integrated solution or the case in which intermediate goods are produced adjacent to the final goods and we observe a h ..nogcneous distribution of activity over space. Segregated solution or the case in which the h-la1 good is produce+ in one zone and the intermediate good ic produced in a separate distinct .-one. We shall see that wllich solution obtains depends on the parameters of the problem. A central planner would allocate activities to sites so as to maximize the amount of delivered output of good 1 (or the value of good 1 delivered net of transportation costs). Under the integrated strategy, he would select the boundary rr: so as to maximize
R’ = p,Q',
-T,',
where
pr = value of a unit of good 1 in town, and a/(! ta) its interinediate good adjacent) per unit of land.
= output c f :ood
I
(given
’
131
Thl; problem is solved when -&(pI
-t,lr\)
= 0,
when land rent falls to zero; c)r when
14;= Pllh * When solved, ignoring the subscript on p, Q’,
,
Under the segregated strategy, the planner would select boundaries ui and us so as to minimize RS = plQ; -T;-T,“.
subject to
where
We form the Lagrnngian 14 = pQt-7’q--T,$-R[QI;-&“I],
132
(3) From (3) we obtain ti2 = kui, wherek = [(I -t-a)/olk > 1 since a > 0. From (2) we obtain aA = t,a(u2- ill), where aA is the price of land at distance Ur from the center. A is the value in dollars of rent of one additional unit of the intermediate good or equivakently dT,/dQz.. Substituting, we get A = r2~r(k - 1). From (I) we obtain
Note that A.--(fZfi,j2) is the price of one unit of the intermediate good.3 Recall aA is the price of land at the margin between land devoted to the production of good 2. Now
p-[i.-q! - t1u 1
is the land rent function for good 1, where C1is the equilibrium value of U, and u iis a variable indicating distance from the center. From (2)
is the land rent function for good 2, and again u is a variable indicating distance from the center. Observe that at the margin between land uses, u = 6 and land rent is as we noted above al. Fig. 1 illustrates the two land reut functions. We require that the price of good 2 to be positive; that is t&-I
2-21’O34inceoi. is the pricepa unit of Innd at the spatial mnrgin between goods I and I?. one nlikI:t assume that this was the cost of a unit off good 2 since land is the only input in its production. This would illdeed be the case if good 2 was being dehvcred to i.t stationary poiut in the plane. (In the famihar case of 2 commodities competing for spncu for production, both IIFCclcl1~w.9 to he center of the economy and this center remains stationztry.) lloccc\cr, in our case a hkt I‘cbr unit of intcrmcdiste good 1 a’t the sparial mar!~~nc;~uscs the margin to III~WC nwy frcm tnc center or reduces the distance which an average unit of good 2 must travel. What are the ~nxctnences of the margin being driven from the center? Saving (cs~~~,Q)x (1/ri,) appca s or tbc transportation cost per unit distance where the cost is that of delivering all of good 2 potr,ntialiy produced between 0 and 12~. Thus thk: 3djllsted price of land at the mar od.-itar.rlli2) and the price of a unit of:go:ood2 is ..I--jf2~lp,i2).
SubstINting for L from (3) yields rztil[k--$] thatk > ~01=(8+a)/a>~ora<0.8. So!Ang (l), (2) and (3) yields
required
to be positive. This implie
i=i1 =p/(t,+at,jk-l]+r,[k-2]),
and e2 =
b/h
+nf,[k-
lJ+tJk--$1).
_I? 5
u2 radiel distance
Fig. 1
Since k > 3, at,[k- I] and r,[k- $1 are positive, we get fil < r/r1 and ii2 <: p/t, and iI $j p/t, according as f, $ [NI- ~.jt,, where 1’ = (k--$)/(/c-- 1) > 0. Recall that p/t, was the fsonticr of the gcngraphic margin of land UC under integrated production. We now consider our principal rc
J.M. Hartwick, Intsmediate gook in land-use models
134
Rs -=:
h2u; Py-3.
2
=
=
h2u,3p ht, T_T{_F+T
ph"u;
hut2
I-- h3 [-p-j_!$!!(~)[!g-J
h2ujp ht, hat,, I -h2 -I-_.__{.F+__[7;j--]_~!]y}
ph’u;
R’ FM, 2n
Thu: when ii, = p/t,, then Is’ = R’, Small increases in t2, from the value oft, obtaining when U2 = p/t,, result in u2 < y/r, and R” < R’. Small increases in a lead to the same rest&. Decreases in t, and/or a resuit in R” > R’. Hence the theorem. One might ask if the above planning solutions will be replicated by decentralized market processes. Although we cannot precisely define the process of transition from one solution to the other, we can co&u& ihnt decentralized market decisions will lead to Ibe cflicient result. For example, consider the silua:ion when R’ -=cR’ and the segregated land use solution is in effect. Tn !hir: case, afi entrepreneur can move to thn i‘mn+:-_UBlrlr.~,lllarginally beyo,nd 1C2and start production of good 1 in an integrated fashion, pay’zero land rents, and make positic;e profits by seliing his output of good 1 at the town (center). ‘Integration will continue until the system of’ land use is completel> integrated. Thus market. processes will lead to the planning solution obtaining. Observe that at the point in lxhich both integrated and segregated solutions yIe!d she :;ame total rent, the slope of the r-e ..IAt ~~i~i;uuiCiii’~~~.~~lfor the inte.gratlsd soIu?;on wili be in value between the slopes of‘ the two segments of the rent schedule for the segregated solution. The followin,o numerical exnmple illustrates the transition from cme efficient solution to the ather as f2 is varied; tote that JI/C~ = 18.75: 1’ -1 f5, N = 0.2, 1, =4: Let us review how our re~‘t relates the observatioris 26‘Warner and Weher as presented in the introducGon. First we observe from our illustrative numerical example that for all values of ,t, greaier than 4.8, the integrated land LEE
J %f. Hart wick. Itz~erntediafe goods &Iland-use models
33s
Table 1 ‘r ranspdrt cost per unit distance
Distances
for intemxdiate good
Aggregate rents
-I--
u2
k
IE9
R’
t2
__-.15.81
15.03 .
7:73 7.54 . .
5:30
38.73 36.8CI
-.-
3 124.72 2822.08
732.42 732.42
. .
.
. .
Ii.94 18.47
747-A 7!0.34
73i42 732.42
.
12:97
. .
350:53
--
0.6 0.8
416 4.8 ,
.
732142
t;:2
prevailed and for ail values 1e:;s than 4.8 the spatially segregated land use pattern prevailed. Thus any technolo:icA change i F transportation which tuok the cost of commuting from above 4.8 to below 4.2 would release eco:,U-zrr,l, the land input cmfficient few production activity may be declining in recent decades. We did not isolate ,C:is parameter in the model. It was set at unity a priori. IIt v.wdd, of course, be straightforward to redevelop the model with this coefiicient identified hy a symbol such as b. The GaBidityof ahe tixxmm would remain.
136
./MS Har:wick, intermediate goodf in land-use models
It should be noted that thle bs ‘:c condition indicating whether an integrated or segregated land use solution oh: lined carries over to Mills’ formulation of the problem. hlihs let Q, be fixed. FT,+ ask oneself which solution results in the least costs being incurred. It is easy to ,!ee that the %,ar?eamount of land is used under both the integrated and segregaied strategies XC: Q, fixed. Hence we can set a priori E2 = pl/tl (p’ will equal i:re betivered ::nit cost of good 1, the final good, say cc’).Given ti2 fixed and the t&.: .;~r!ditions: l(i)that Q1 = Q2, and (ii) that the rental price on land must be e~uc at the margin between competing us& in equilibrium, then the delivered C-I:. cost of the final good cy can be determined for the segregated solution. F&Y-V :‘s$ ci according as tl p [a + o]tz , u and o as defined above. Clearly thF; iand bse pattern which prevails will be the one for which unit costs are lea.sr, and this outcome depends on the same parameters as vie had for the previous analysis. (Mills condition should have been the same as the above.) 3. Some numerical exarnL&s with inputs substitutable In Hartwick and Hartwick (1974) a rmodel of land use was set out which incorporated technology in its activity analysis characterization and considered land in discrete units. Transportation of goods required inputs of land and capitaLs Parcels of land were annuluses of equal width centered on a particular point. Let us consider a two-good version of this model with one good intermediate to the other and each good produced with land and capital available iri unlimited quantities at price r. We shall select techniques so that the resulting iscqzlants ore convex to the origm. We allow transportation services produced directly from land and capital and in so doing make the model slightly more realistic compared with the one above. To this point we have considered the maximization of the value of output given a fixed delivered price for the final good at the center. In this model we examine a dual problem, namely, the maximization of total factor returns for achieving a prescribed level of output of the final good is in the center. 6 There are three possible solutions one might expect for different parameters of demand, the technology of production and the technology of transportation : (i) an integrated land use allocation iu which houses (workers) and workplaces are both in annulus i in amounts such that no workers commute across any boundary of an annulus, (ii) a segregated solution in which no annulus has both houses and production activity lying with it except for a single annulus between two land areas specialized in different uses, (iii) a mixed land use allocation in which at least two boundaries have flows of workers moving across in cquiliVhis model was similar to that in Hartwick and Hartwisl, (!c17-92). !;i ilik euriier lnCk.el Iand a priori into a grid of square parcris, each parcel equal in area to another. Vhe model is set out in brief in the appendi/r to the paper. A detailed description of the general model is in Hartwick and Hartwick (1974b). was divided
J.&f. Hnr twick, Intermediate
goods iIf land-use models
137
briwm and also in the same equilibrium, at least one annulus, with positive levels
of output for the export good, has no workers entering or exiting from it in equilibrium. In fact, we can rule out case (iii) ba!cause of the linear homogeneity of the technology and demands for intermediate goods. in the examples below, we observe only two possible er.juilibria: (i) the one we called the integrated land use pitttern, and (ii) the segregated land use pattern. Tb: experiment reported aelow was one in which the physical transportation services demanded by each unit flow of housing or workers was decreased relative to the analogous magnitude per unit flow of the final good being exported. Thus in the 6 runs performed (only 5
PBl PB2 PB3 PB4 TB5 PB6
0.1 0.175 0.250 0.250 0.250 0.30
0.5 0.5 0.5 0.25 0.1 0.1
For the intermediate good, good 2, the demands for transportation fall through run 1 to 6 relative to the demands made b!r the final good, good 1. Our general finding is that iha integrated land use becomes segregated as the transportation costs on intermediate goods fall below a certain value relative to those transported across the border between :my two annuluses. IY PB5, laborhousing is located only in annuluses 10 and 1i, and flows to the ;o:alions of the production of good 1, annuluses 2 to 9. Annulus 10 transships lab.Jr to annulus 2-9. Residential activity has become suburbanized. PB6 is similar to PBS in the sense that all residential activity is suburbanized. However, in PB6, all residences are concentrated in a single annulus rather than being divided between two as was t’re case in PBS. Tile outputs are summarized in fig. 2. Each square represents an annulus. Along the horizontal axis we indicate the proportions of the area of an annulus occllpied by an activity. There are four activities possible: transpol tation, good 1, good 2, and agriculture. The vertical axis indicates the capital intensity of the technique used for the relevant activity. Transportation, good 1 and good 2 can be produced with three different techniques, each technique corresponding to a different ratio of capital to land in production. The highest ratio of capil:.d to land is indicated by the activity ‘occupying’ the complete vertical dimension for an annulus. Thus in P 3, in a.lnulus 1, transportation 15 produced with tlvn techniques; the maximum capital to land ti:chnique and the technique with m intermediate value for the ratio of capital to land. Again, for example, we see that in annulus P and in P 1, t~a~sportat~oi~ occu ies alil the Iani; availa
31725
P63
24138
i
1914
0.817
1267
0.506
0.771
0.236
11224
12.560
Il.200
12 240
i3600
12560
Fig. L
3 555
2;27
? ve
1.652
1.362
1154
0646
0594
fgjgp#q~Q~~
2 676
1173
i--
0103
0.!52
0348
0.0
0.0
0.0
0.0
J.M. Hartwick, Intermediate goods in land-use models
139
uses the techniques with a minimum possible capital to land ratio and the technique with an intermediate value for the ratio of capital to land. The numbers beneath each square representing an annulus are shadow prices on land or land rents net of the agricultural price for land. These rents generally decline with the distance the annulus lies frieci the center except for the case of annuluses, not at the certcer. which specialize in transportation. For example, annulus 3 in PB3 exhibis the anomalous rent value. The reason for these anomolies is that the resources used in transporting a unit of good across a boundary are divided into equal parts and are iequi .d?dsupplied by the two annuluses on either side of the boundary in the same factor proportions. Thus in annulus 3 of PB3, the resources used are entirely determined by the flows from and to the adjoining annuluses. This subarea in ‘t sense is not free to select its own activity in a way which minimizes t!-iecosts of production within the particular annulus. It should be emphasized that the se rinomolies in the rent values only occur in annuluses completely specialized m the transportation activity. What might we expect with more than one final and more than one intermediate good? There will now be the possibility of different intermediate goods ‘integrating’ with different final goods and depending on the technology of production and on transport costs, we can expect different patterns of integration and segregation of land t,ses in difIerent areas. Exogenous changes in parameters can integrate or segrer;ate two activities at the same site. These results are apparent in light of the result; for one final and one intermediate good and of the strticture of the activity analysis land use models with many commodities. 4. Conclusions Th: fuildamental characteristic of intermediate goods in location analysis is ihat they are 4 necessary element of a model which is to exhibit a spatial mixing of activities at various sites given constant returns to scale. When all commodities are final goods we observe the Von Thiinen pattern of segregated land use. Nowever, given that intermediate goods are in a model, not every allocation will display the mixing or spatial integration of land uses. In the fixed coefficient mode! we determined the precise conditions under which one allocation versus another would occur. The naidre of technology and relative transportation costs for tht: final and the intermediate good were crucial. In particular, we noted in all mcfdels that, given an efficient integrated land use patkrn, some decline; in the rela?ite cost of transporting the intermediate good would eventually lead to the segregated land use pattern being efficient. Such land use changes have been discussed by urban historians. They ide~t& 4 the intermediate good a’; labor required for rh-: production of the final good. We have analyzed formaliv what appear to be a fundamental transformation of cities in history - the changt from a spatially integrated land !~se pattern to a spatially segregated one. One might ask what are the reasons why contemporary North American citrss
J.&f. Hartwick, Intermediate good9 in land-use models Table 2
Total rents
Distances
0.158107E-k02 0.1502558402 0.143146E+02 0.136680Ef02 0.130773Et02 0.125355E-t02 0.1203688 + 02 0.115763E-k02 O.l11497E+O2 0.1C7534E+ 02 0.183843E+r)2 G.l00398E+02 0.971734E+01 0.941497EfOl 0.913084E+Ol G.886337E+ oi 0.861112E+Ol 0.837,828 +Ol 0.814737E+Oo1 0.793373E+Of 0.773102E+Ol O.75384OE+ 01 0.355515E+ 01 0.71806OE+Ol 0.701414E-t01 0.685522E + 01 0.67033.1E+01 0.6558058+ 01 0.64189tE f 01 0.628558E +01 0.615766E-701 0.6032828 + 31 0.5916838+ 01 0.580335E + 01 O.S69413E+Ol 0.558895E+Ol 0.548758E+Ol 0.588983E -1-01 0.52955OE-t 01,
Transport cost for intermediate goods
a2
R”
R’
t2
0.387281E+G2 0.368048E+O2 0.350636E-k 102 0.334797E-k02 0.320326E+02 0.307055E-k102 0,29484OE+ 02 0.2835598 + 02 0.273llGE+02 0.263404E + 02 0.25436448-I-02 0.245923E + 02 0.238025E-k 02 0.320619E+O2 0.223459E+ 02 0.2i7107EfO2 0.210928E+02 0.206091E+02 0.199569E f 02 O.l94336E+02 0.18937OE+ 02 O.l84652E+02 0.180164E+02 O.l75888E+02 O.l71811E+02 O.l67918E+02 0.16419bE+O2 0.160639Ef02 O.l572?1E+02 O.l53965E+O2 0.150831Et02 O.l47823E+02 O.l44932E+G2 O.l42152E+02 0.139477E + 02 h13690lEf02 0.134418E+02 0.1320238+@2 0.129713E+K!
0.3124728+04 0.2822038 + 04 0.251%36E-I-04 0.2335188+04 0.2137698 + 04 0.196J23E-t04 0.181105E-kG4 0.1675128+04 0.155394EfO4 O.l44545E+O4 O.l34793E+O4 0.125996E A-04 O.l18033E+O4 O.l10802E+-04 0.104215Ei G4 0.98199iE+O4 0.92689lE-k04 0.8763028 + 04 0.829745E+ 04 0.786801E+O4 0.747108E+O4 0.710343E+o4 0.676228E + 04 O.tA4512E+ 04 0.614976E$04 0.537425E + 04 0.561685E+O4 0.5376OOE-k04 0.515032E+04 0.4938568-t-04 0.473959E + 04 0.4552415~04 0.4376108 + 04 0.420984E + 04 0.4052888 f 04 0.390454E -I-04 0.376419E c 04 0.363328E+04 0.3505288+04
0.732422B + 03 0.732422E-k 03 0.7324228-k 03 0.732422E + 03 0.732422E $03 0.7324228+03 0,732422E+ 03 0.7325228 -t 03 0.7324228 + 03 0.732422E+ 03 0.7324228 i- 03 0.7324228 + 03 0.732422E~t03 0.732422L:+O3 0.7324228+03 0.7324228+03 0.7324228 + 03 0.732422E f 03 0.7324228 -I-03 0.732422E + 03 0.732422E+03 0.7324228 + 03 0.732422E.t 03 0.732422E+ 03 0.732422E-t- 03 0.732422E + 03 0.732422E + 03 0.732422E + 03 3.732422E+03 0.732422E + 03 0.732422E + 03 0.732422E+O3 0.7324228 -i-03 0.7324228 -I-03 0.732422E + 03 0.7324226 -403 0.732422E -I-03 0.732422E f 03 0.73242X? + 03
0.6 0.6 1.0 1.2 1.4
-_1__
:*: 210 2.2 2.4 2.6 z*: 3:2 :.t 3:8 44.0 :: 4:6 4.8 5.0 0.2 0.4 0.6 0.8 6.0 0.2 ::: :: 0:2 0.4 0.6 0.8 ::; -
do not display a clearly defined segregated land use pattern. I believe the two primary reasons are that in modern cities there are a multiplicity of final goods and intermediate goods with some sectors spatially integrated an segregated, and second! yn that people of the sarrle ;ncomc classes h income elasticities of demand 1’~ reGlentia1 spxe and ccpm!a~ufinr. element causes a city to display considerable heterogeneity of resi to give an appearance of integrated residential land use. For an example of ahe
J. M. Hurtwick,
htermeciiote
goods irt land-use m-de/s
!41
first reason in operation, one could scrutinize the example reported in Hartwick and Martwick (1974a). A wide vaiizty of land use pttterns are possible for difl’er- 1 ent specifications of the technology and transport costs. To this point. we have restricted our attention to patterns of land use in long-run equilibrium. Some might argue that the neglect of the durability of structures and the related disequilibrium paths of adjustment in urban growth represents a major omission. In the opinion of the writer, our propositions require inessential changes when this omission is considered in detail. Considerable debate would no doubt arise on the precise nattile of the path of adjustment from one equilibrium to another. Appendix
Land in this model is simply Van Thtinen’s homogeneous plain with a center. We divide a priori land into n annuluses of Pqual radial distance or width. Commodities must be shipped at rrositive transportation costs to where they are used (intermediate goods) cr where they are to be exported (final goods more to the center). Land must be bid away from agricultural use at rent R, per unit, and capital must be bid away from alternate use.; at r dollars per unit. The driving force causing the city to exist is the exugenou ;ly given demand for exports of the fina! good. The non-transportation activities must be allocated to each parcel 0,’ land agriculture, the final good {good 1) and housing (good 2), plus transportation. Each unit of goods 1 and 2 crossing the boundary between two annuluses requires land and capital to be expended. These are the costs of transport. The allocation of activities must be such as to minimize the value of land and capital used in the city net of the value of transportation costs. The dual problem is the maximization of the value of exported goods net the value of land used for urban (nqn-agricultura!) activity. The objective function t3 be minimized with respect to activity levels and intersite flows is
where the variables in the pri nal linear progr’uu are defined as follows: S
=
index ranging from I to produce commodity
to S, q,
indicating the technique or process used
J.M. Hartwick, Intermediate goods ir! hd-use
142
d R RA W %q.=
z,(i)
%i)
x,‘!+‘(i)
=
= =
= =
models
index ranging from 1 to 2 indicating the boundary of the ith aunulus over which the indicated flows are moving, exogencausly given rental price of a un.Aof capital, exogenously given rental price of a unit of agricultural land, exogcnously given wage rate for a man-week of labor, input uf commodity k required to produce a unit of commodity 5.with the 8th technique, k ranges from 1 to !’+ 3 where cornmoLt ICS l to 4 are not primary, input r+ 1 is land, input r+2 is capital and input r-t3 is labor; commodity r is housing and ar,4j is set equal to u~+~,~,~in order to identify one worker with one hoi!Gn;; unit; q ranges from 1 to r plus one additional commodity, transy!Jrtr ion indexed to b, coefficient indicating the relative economic attractiveness of annulus (i) for exporting commodity 6, relative to other transportation nodes, coefficient taking on values of 1 if annulus (i) is a transportation node and 0 if it is not a node. Only for i = i will 6 = 1 in this paper, the activity level of the transportation sector used to transport aggregated commodity flows ficross the (i, if 13boundary of annulus (i) using the sth transportation technique, flow of the qth commodity 3eaving annulus (i) across the dth boundary, coefficient indicating the amount of the transportation commodity Tequired to move a unit of commodity q from one annulus to 6 coniIguous annulus, amount of commodity q exported from annulus ii), qs activity level in annulus (i) or the gross output of commodity q when the sth technique is annulus (i) is operated.
If no urban activities are assigned to an area in the system, then that arca remains in agricultural WC. Some peripheral annuluses may not bid away from agriculture for use in the city. Agricultural activity can be considered as all othx activities operated at zero activity levels. Now each annulus must be in flow equilibrium for each of the 2 non-transportation and non-agricultural activities. That is, we have the: following constraints:
where the first two terms indicate flows of commodity q to ammlus (1’)from each ofthe two continguous annuluxs, the third term is the gross output of comrrodity y lin annulus (i), the fourth term is total intermediate use of commodiry q by
J.M. Hartwick, Intermediatf
3
goods iE land-use models
143
other activities in an&us (i), the fifth term iii&“,_ -dates total flows of commodity q from annulus (i) to its neighboring annuluses, and the sixth term indicates the export or import of cohmodity q from annulus (ij for the case in which (i) is a transportation node. The exports of each commodity must at least satisfy the final demands whiayh have been directed to the urban are?. for the commodity in question. The resulting constraints are c &-)x,(i) 3 Yq, 4 = 1,2, i
j
z
(A.3)
1, - - l, n.
The 2nd commodity, hcusing is treated as purely intermediate to the urban economy since we equate labor with housing and assume that associated with each worker fs one hotlse. and that ail workers reside in the urban area. The I* i (3rd) sector i:$the transportation sector denoted by the subscript t, The demand for transportation services derives from the flows of commrditics (including housing/labor) moving between annuluses. The production of transportation services requin:s inputs of capital and land, transformable from processes in a finite array. More capital-intensive transportation producticn corresponds to, say, the tise of subways rather than automobiles. The process = tIra,+l,r,,,a,.i2,,,~ ) indicates the requirement of land (ar+l ,t,,) and ca&al ar+ ? 2,r,s) for producing one unit of transportation services with technique S. On the demand side, since, 1, is the amount of transportation good required in one annulus to move a unit of commodity q between two adjacent annuluses, then t,#(i)+I@i+ I)) is the amount of transportation good required produced in annulus (i) in order to transport the floes ofq across the (i, i+ 1) boundary of annulus (i). An identical amount of transportation good must be produced in annulus (i+ 1) in order to accomodate the incoming flow to annulus i+ 1 across the i, i+ 1 frontier. We thus get the following additional ctinstraints:
1 xj:i+‘ii)s
f
t,(Ti(i)+T,2(i+ 1))
2 0,
i = 1, . . ., M-
4=1
1.
(A.4)
Finally, all land in each annulus must be exhausted in use by the production ot commoditie:; I and 2, clnd/or transportation, and!or bsriculture. The land use the constraints
for 0 fixed and 0 < 0 5 I, where in (A.5), the firsi Lerm indicates land syortation, the second term production other than agricult
J.iW. Hartwick, Interm~Late goods in land-use models
144
land use in transportation, and the inequality indicates the land can be left in agr,-iculture (when the relevant slack variable is not zero).
ilatafor probienzs
in text
Max. number of rings : 12 (if 12 rings do not appear, then the remaining rings are filled with agricultural activity). Number of goods: 2 (1 export good and housing). Number ofcapital intensities: 3. Product ion technology ----~ Good I
Good 2
Land
Capital
0.25 &O
0.6 0.66
2.5 2.5
Height =: 1 output
1
0.0
2
0.0
.--~--iieight = 2 o!atput 1 2 --- _I_-__ Height = 3 Ourpui 1 2
--0.25 0.0
0.0
0.0
0.48 0.2
3.2 4.0 -
0.0 0.0
0.25 0.0
0.4 0.1
4.0 6.0
Final demand for export good: 100 units. Transportation
production technology
Congestion kvel _-I_-
Land
Capital
1 2 3
0.3 X::
0.2 2.4 4.0
Commodity transportation
3.2 3.3
II I’ 0.1 rl ,= 0.175 tt = 0.250
12 = 0.5 t2 = 0.5 tz = 0.5
PI3 3.4
?, = 0.250
II = 0.25
PB PB PI;
3.1
weights
This problem in spite of these differences in b:sic coeficients has an identical solution to th?t of PB 3.3. PB
3.5
Cl = 0.25
tr .:=0.1
PB
3.6
t1 = 0.30
bz = 0.1
References Green, J.R., 1974, Vertical integration and ~:ssurance of markets, Discussion Paper no. 383 (Harvard Institute of Economic Research, Cambridge, MA). Hartwick, P.G. and JX Hartwick, 197&r, EZicient resource allocation in a multinucleated city with intermediate goods, Quarterly Journ, ofEconomics, May, 340-352. Hartwick, J.M. and P.G. Hartwick, 1974b, The activity analysis approach to nrban model I building, Papers Regional Science Association, Nov., 75-85. Mills, E.S., 1370, Theefficiency of spatial competition, Papers Regional Science Association 75. 71-82. Mills, E.S., 1973, Studies in the structure of the urban economy !Re=ources for the Future, Washington, DC). Solow. R.M., 1973, On equitibrium models of urban locatio “, in: J.M. Parkin, ed., Essays in modem econtimics (Longs ans, London). Warner, S.B., 1962, Streetcar suburbs: The process of growth ix Boston 189t.LI%0 iHarv3rd University and MIT Press. Cambridge, MA). %‘e~,,-r, A., 1899, The growth of cities in the nineteent:] century (Cornell University Pre’ss, Ithaca, NY).
to be 2 mdbted
1.
and Ur
Economics 6 ( f376) 1Y- 145. ~9 North-Holland
one &MI end one intermediate. is set OUT !:I n spatial contest. nd must be transported at positive costs between an? cd WE in which the finat 2nd intermediate goods hich final and intermediate goods occupy distinct ning which pttcm will be &ticient is obtained and the result rnpk are rebted to atirvatzons reported on the history of urban areas. two distinct equitibria is fok nd in a model which permits another factor
for kml in prixhtion.
OD
In this FYWr. attention is fwused on the role of intermediate goods in models Mills (1970) pointed ou! tnat in a Von Thiinen model with two prgduccri gsgds - one final and one intermcdiatc - two distinct land USCpatterns are feasible solutions. In one sofution, called krt~gratcvl, the linal good and its intermediate good share the same site, so to speak, whereas in the other solution, ads ali occupy one zone or distinct area, and the other distinct area. The details of Mills’ model and the condition which determirles which land ore precisely. Subsequently it is shown that both “CWpatterns can obtain in d model in which stituting another factor for land as the price of land vatic~
*My
C~~~~j~
t~~~~ An
with computations in the numerical to Phihp C. ~,~~~w~~~ foor ~,:,&wx cnrlkr venion of this paper teas presented at the Econometric Society, P 1974. Rob Eergk, Cohn ~~~~~r~~)~ 3rd Pravin Varaiya provided helpful
128
JM. Hartwick, Intrmediate goods in land.-usemodels
‘I:: U850,Boston was a tig’htty p acked seaport, by 1901:it sprawled over a ten mile radius and contained th;r:; ;K cities and towns . . . The old settll:ment oi 1850 became by 1900 the prin$+J zone 3f work - the industrial, commercial, and coiumunications center c,‘l‘the metropcIitan region. Beyond the inner concentrated section there .:~r r an equally novel environment, the enormous outer ring of new commuters“ . -‘uses . . . the wide extent of settlement in ?h* outer residential zone was :na.~‘cpossible by the elaboration of the new street railway transportation system 2 :ti a parallel extension of city services.* Warner (1961, pp. l-3) Adna Weber calls the transformation 1900, *city-building’.
undergone by Boston between ISSO-
‘The original settlement becomes the brlsiness center and for some time continues to grow rapidly. But if the city prol;pers, the time will come when this old center is more and more needed for strictly business purposcJ; k~ses disappear before the march oi office buildings, government buildings, banks, etc., until the only residents left are the janitors and porters, the keepers of the great buildings. With continued growth, the business center extends itself and steadily pushes the dwellings toward &c circumference, until at length the municipal limits are reached and passed.’ Weber (1899, p. 467) Observe that Weber considers that the business center ‘pushes’ residential areas bcjond the urban periphery of a previous pericd. For Weber, trolley cars did not cause suburbanization or ‘pull the city out’ as Warner suggests but this transportation mode was developed in response to the ‘American penchant for dwelling in cottage homes instead of blusiness blocks after the fashion of Europe’ [N’eber (1899, p. 469)]. For Weber it was obvious that Philadelphia was the ‘city of homes’ long before rapid transit. A glance at the record of the average number of rides annually on street railways in the 11th Cef~cs Report on ih~~portatbn by Lartd reveals that American cities consistently ranked higher (more trips per person per year) than Euiropean cities. In co&nental Europe, Weber considered it the practice for artis:ms and shopkeepers to live in rooms connected with their workshop or store, whereas in the United States, presumably these same types of people would commute to their place of work each day. In the 11th Cmsus, Social Statistics of Cities, Weber (1899, p. 468) raoted that the Urlited States had about 15.2 urban dwellers per acre compared with 38.3 for England and 25.9 for Germany. In summary (p. 469), ‘the comparative figures of milea and number of rides per inhabitant of American and European cities are indii:d-!+rl of low or high density ~)op~~Iatio~,whit regarded as the catial: of street railway building’. Weber then implies that density patterns, determined oppot;ed to eco ic:forces, led to the development of differ t~a~s~~Ftat~~~~
1s vkw
contrasts
with
that
of F
J.&i. Hartwick,
InrPrmediate goods in land-use models
129
?3ne can only guess just how large metropolitan Boston worrld have grown ha*1there been no invention of new communication &vices. If the spread of the city had begun to exceed the distance a man might walk in about an hour, say a three-mile radius, the shops and offices of the metropolis would have fallen out of easy daily communication with each other. The result would have been the destruction of a single .unit*ed communication network and the development of semi-autonomous subcities which would h..ite had to duplicate many services and facilities offered in other parts of the c&r.’ Warner (1962, p. 16) In section 2, we will identify precisely tbc conditions under which the spatial integration or segregation of work-place and residence-place occurs. We will be able to identify the fclrces which lead a city to change from one form to another. We will be able to isolate the cases alluded to by Warner and Weber. Thus a technological change in transportation such as the electric trolley will be shown to make possibie the spatial segregation of work-place and residenceplace as Warner suggests. Also an exogenously given taste for relatively less land per iamily will lead to a spatial integration of work-place and rsidenceplace as Weber suggests.’ We shall examine these issues in the familiar model which postulates that there exists a geographic irregularity such as a deep water port which makes shipments to and from a city rcratively cheap. Thus activity agglomerates about this node and outp:Pt of ,.on-residential activtty is shipped from the city through this point or node. There will t : three sectors: production activity, residential activity, and transportation activ;‘.S In our first model there will be fixed production coeflicients and transportarion will !X assumed to use no land. Production activity will require land and workers, who iive in the residential sector. Workers spend all their income on commuting (if the allocation has spatially segregated sectors) and on this residence which requires only land as an input.’ ‘In the general area cl urban model building. it is interesting to note that since modern cities appear to display segregation of residence-place and work-place, model-builders take this property as given whereas they should really make the spatial integration or segtcgation of land uses a variable in tf e model. Solow (1973), for example. avoids the problem hy postulating enough increasing I :t;r11s to scale in production to lead to production being geographically concentrated but no! ~tfftcicnt increasing returns to scale to make necessary the USCof a theory of pricing for all firclot-s different from the marginal productivity approach. Outside of the spatial context, Grct P (1974) has investigated circumstances in whiih an output requiring an intermediate inpuf . I !w produced in a single vcttically irttegr;,ttcti firm or in two d,stinLt ‘upstream’ and Ldcwr~~tre;m-t”firms. Green’s results turn on a parlicular ‘nind of tran:..lction cost being inrcru,~liz A; by vertical intcgratior). In the snatial n-&cl, transpo~tatron coht\ CZI! be profitably interurli~ed t,nder the ertplicit conditions de~elopeti below in ihe *~:-a I. 2Thls model 15 vuiua;ty ideutlcal to that in ?\liih (1970) and (IW?, ch. 5). Howcvcr, Mill&) erred in solving his model JIX.! the basic condition on detcrmirrirrg whether spatial integration or’ segregation of activity WCuld result is not valid. Milis also did not attempt to rela?e his model to the historical dcAopment of cities ;is we are emphaGzing. Clouevcr. Mrlls pond the integration or segregaimportant question cleirlj.: What cflnclrtions determined whet er spati tion of activity res At:; I &en one good is an interme
li30
J.M. Hilrtwick, Znterrnedrate xaods in land-use models
A model with an activity analya.15 techno!ogy is then set out and examples are reported which display the switch from a spatially integrated allocation of actitities to a spatially segregated ;~!locati,on. In this last model, transportation a!so uses land and capitai. 2. The tied cwfficiemt model
, ,_
11: the town or center of our economy, there is an infinitely elastic demand for ,t!~efmal good at price p per unit. The solution to the problem will be an allocation which maximizes t!le value of output delivered. to the center. Since p is fixed, this is equivalent to maximizing the amount of the final good produced and d.e!ivered to the center. The final good, good 1, r;;qtires 1 unit of gocd 2, the mtermediate good, for each unit of good 1 produced. ‘u’ units of good 2 are produced from 1 unit of land and 1 Il!lit of good 1 requires 1 unit of land in productifon. The salient qu,r of Iii:ermediat;~?~ess in this mode! is that a ur.it of the intermediate good net ’ I’ ;;I ’ ,e +attsgorted to the site of the final good and not to the center point of our disc-shapeg.. economy. Transportation costs per unit distance arc i, and t2 Jollars for goods 1 and 2 respectively. Good 1 is rhe final good and good 2 is housing. Associated with each house is one worker, assumed for concreteness. There are two possible land use allocations in this mode!: integrated solution or the case in which intermediate goods are produced adjacent to the final goods and we observe a h ..nogcneous distribution of activity over space. Segregated solution or the case in which the h-la1 good is produce+ in one zone and the intermediate good ic produced in a separate distinct .-one. We shall see that wllich solution obtains depends on the parameters of the problem. A central planner would allocate activities to sites so as to maximize the amount of delivered output of good 1 (or the value of good 1 delivered net of transportation costs). Under the integrated strategy, he would select the boundary rr: so as to maximize
R’ = p,Q',
-T,',
where
pr = value of a unit of good 1 in town, and a/(! ta) its interinediate good adjacent) per unit of land.
= output c f :ood
I
(given
’
131
Thl; problem is solved when -&(pI
-t,lr\)
= 0,
when land rent falls to zero; c)r when
14;= Pllh * When solved, ignoring the subscript on p, Q’,
,
Under the segregated strategy, the planner would select boundaries ui and us so as to minimize RS = plQ; -T;-T,“.
subject to
where
We form the Lagrnngian 14 = pQt-7’q--T,$-R[QI;-&“I],
132
(3) From (3) we obtain ti2 = kui, wherek = [(I -t-a)/olk > 1 since a > 0. From (2) we obtain aA = t,a(u2- ill), where aA is the price of land at distance Ur from the center. A is the value in dollars of rent of one additional unit of the intermediate good or equivakently dT,/dQz.. Substituting, we get A = r2~r(k - 1). From (I) we obtain
Note that A.--(fZfi,j2) is the price of one unit of the intermediate good.3 Recall aA is the price of land at the margin between land devoted to the production of good 2. Now
p-[i.-q! - t1u 1
is the land rent function for good 1, where C1is the equilibrium value of U, and u iis a variable indicating distance from the center. From (2)
is the land rent function for good 2, and again u is a variable indicating distance from the center. Observe that at the margin between land uses, u = 6 and land rent is as we noted above al. Fig. 1 illustrates the two land reut functions. We require that the price of good 2 to be positive; that is t&-I
2-21’O34inceoi. is the pricepa unit of Innd at the spatial mnrgin between goods I and I?. one nlikI:t assume that this was the cost of a unit off good 2 since land is the only input in its production. This would illdeed be the case if good 2 was being dehvcred to i.t stationary poiut in the plane. (In the famihar case of 2 commodities competing for spncu for production, both IIFCclcl1~w.9 to he center of the economy and this center remains stationztry.) lloccc\cr, in our case a hkt I‘cbr unit of intcrmcdiste good 1 a’t the sparial mar!~~nc;~uscs the margin to III~WC nwy frcm tnc center or reduces the distance which an average unit of good 2 must travel. What are the ~nxctnences of the margin being driven from the center? Saving (cs~~~,Q)x (1/ri,) appca s or tbc transportation cost per unit distance where the cost is that of delivering all of good 2 potr,ntialiy produced between 0 and 12~. Thus thk: 3djllsted price of land at the mar od.-itar.rlli2) and the price of a unit of:go:ood2 is ..I--jf2~lp,i2).
SubstINting for L from (3) yields rztil[k--$] thatk > ~01=(8+a)/a>~ora<0.8. So!Ang (l), (2) and (3) yields
required
to be positive. This implie
i=i1 =p/(t,+at,jk-l]+r,[k-2]),
and e2 =
b/h
+nf,[k-
lJ+tJk--$1).
_I? 5
u2 radiel distance
Fig. 1
Since k > 3, at,[k- I] and r,[k- $1 are positive, we get fil < r/r1 and ii2 <: p/t, and iI $j p/t, according as f, $ [NI- ~.jt,, where 1’ = (k--$)/(/c-- 1) > 0. Recall that p/t, was the fsonticr of the gcngraphic margin of land UC under integrated production. We now consider our principal rc
J.M. Hartwick, Intsmediate gook in land-use models
134
Rs -=:
h2u; Py-3.
2
=
=
h2u,3p ht, T_T{_F+T
ph"u;
hut2
I-- h3 [-p-j_!$!!(~)[!g-J
h2ujp ht, hat,, I -h2 -I-_.__{.F+__[7;j--]_~!]y}
ph’u;
R’ FM, 2n
Thu: when ii, = p/t,, then Is’ = R’, Small increases in t2, from the value oft, obtaining when U2 = p/t,, result in u2 < y/r, and R” < R’. Small increases in a lead to the same rest&. Decreases in t, and/or a resuit in R” > R’. Hence the theorem. One might ask if the above planning solutions will be replicated by decentralized market processes. Although we cannot precisely define the process of transition from one solution to the other, we can co&u& ihnt decentralized market decisions will lead to Ibe cflicient result. For example, consider the silua:ion when R’ -=cR’ and the segregated land use solution is in effect. Tn !hir: case, afi entrepreneur can move to thn i‘mn+:-_UBlrlr.~,lllarginally beyo,nd 1C2and start production of good 1 in an integrated fashion, pay’zero land rents, and make positic;e profits by seliing his output of good 1 at the town (center). ‘Integration will continue until the system of’ land use is completel> integrated. Thus market. processes will lead to the planning solution obtaining. Observe that at the point in lxhich both integrated and segregated solutions yIe!d she :;ame total rent, the slope of the r-e ..IAt ~~i~i;uuiCiii’~~~.~~lfor the inte.gratlsd soIu?;on wili be in value between the slopes of‘ the two segments of the rent schedule for the segregated solution. The followin,o numerical exnmple illustrates the transition from cme efficient solution to the ather as f2 is varied; tote that JI/C~ = 18.75: 1’ -1 f5, N = 0.2, 1, =4: Let us review how our re~‘t relates the observatioris 26‘Warner and Weher as presented in the introducGon. First we observe from our illustrative numerical example that for all values of ,t, greaier than 4.8, the integrated land LEE
J %f. Hart wick. Itz~erntediafe goods &Iland-use models
33s
Table 1 ‘r ranspdrt cost per unit distance
Distances
for intemxdiate good
Aggregate rents
-I--
u2
k
IE9
R’
t2
__-.15.81
15.03 .
7:73 7.54 . .
5:30
38.73 36.8CI
-.-
3 124.72 2822.08
732.42 732.42
. .
.
. .
Ii.94 18.47
747-A 7!0.34
73i42 732.42
.
12:97
. .
350:53
--
0.6 0.8
416 4.8 ,
.
732142
t;:2
prevailed and for ail values 1e:;s than 4.8 the spatially segregated land use pattern prevailed. Thus any technolo:icA change i F transportation which tuok the cost of commuting from above 4.8 to below 4.2 would release eco:,U-zrr,l
136
./MS Har:wick, intermediate goodf in land-use models
It should be noted that thle bs ‘:c condition indicating whether an integrated or segregated land use solution oh: lined carries over to Mills’ formulation of the problem. hlihs let Q, be fixed. FT,+ ask oneself which solution results in the least costs being incurred. It is easy to ,!ee that the %,ar?eamount of land is used under both the integrated and segregaied strategies XC: Q, fixed. Hence we can set a priori E2 = pl/tl (p’ will equal i:re betivered ::nit cost of good 1, the final good, say cc’).Given ti2 fixed and the t&.: .;~r!ditions: l(i)that Q1 = Q2, and (ii) that the rental price on land must be e~uc at the margin between competing us& in equilibrium, then the delivered C-I:. cost of the final good cy can be determined for the segregated solution. F&Y-V :‘s$ ci according as tl p [a + o]tz , u and o as defined above. Clearly thF; iand bse pattern which prevails will be the one for which unit costs are lea.sr, and this outcome depends on the same parameters as vie had for the previous analysis. (Mills condition should have been the same as the above.) 3. Some numerical exarnL&s with inputs substitutable In Hartwick and Hartwick (1974) a rmodel of land use was set out which incorporated technology in its activity analysis characterization and considered land in discrete units. Transportation of goods required inputs of land and capitaLs Parcels of land were annuluses of equal width centered on a particular point. Let us consider a two-good version of this model with one good intermediate to the other and each good produced with land and capital available iri unlimited quantities at price r. We shall select techniques so that the resulting iscqzlants ore convex to the origm. We allow transportation services produced directly from land and capital and in so doing make the model slightly more realistic compared with the one above. To this point we have considered the maximization of the value of output given a fixed delivered price for the final good at the center. In this model we examine a dual problem, namely, the maximization of total factor returns for achieving a prescribed level of output of the final good is in the center. 6 There are three possible solutions one might expect for different parameters of demand, the technology of production and the technology of transportation : (i) an integrated land use allocation iu which houses (workers) and workplaces are both in annulus i in amounts such that no workers commute across any boundary of an annulus, (ii) a segregated solution in which no annulus has both houses and production activity lying with it except for a single annulus between two land areas specialized in different uses, (iii) a mixed land use allocation in which at least two boundaries have flows of workers moving across in cquiliVhis model was similar to that in Hartwick and Hartwisl, (!c17-92). !;i ilik euriier lnCk.el Iand a priori into a grid of square parcris, each parcel equal in area to another. Vhe model is set out in brief in the appendi/r to the paper. A detailed description of the general model is in Hartwick and Hartwick (1974b). was divided
J.&f. Hnr twick, Intermediate
goods iIf land-use models
137
briwm and also in the same equilibrium, at least one annulus, with positive levels
of output for the export good, has no workers entering or exiting from it in equilibrium. In fact, we can rule out case (iii) ba!cause of the linear homogeneity of the technology and demands for intermediate goods. in the examples below, we observe only two possible er.juilibria: (i) the one we called the integrated land use pitttern, and (ii) the segregated land use pattern. Tb: experiment reported aelow was one in which the physical transportation services demanded by each unit flow of housing or workers was decreased relative to the analogous magnitude per unit flow of the final good being exported. Thus in the 6 runs performed (only 5
PBl PB2 PB3 PB4 TB5 PB6
0.1 0.175 0.250 0.250 0.250 0.30
0.5 0.5 0.5 0.25 0.1 0.1
For the intermediate good, good 2, the demands for transportation fall through run 1 to 6 relative to the demands made b!r the final good, good 1. Our general finding is that iha integrated land use becomes segregated as the transportation costs on intermediate goods fall below a certain value relative to those transported across the border between :my two annuluses. IY PB5, laborhousing is located only in annuluses 10 and 1i, and flows to the ;o:alions of the production of good 1, annuluses 2 to 9. Annulus 10 transships lab.Jr to annulus 2-9. Residential activity has become suburbanized. PB6 is similar to PBS in the sense that all residential activity is suburbanized. However, in PB6, all residences are concentrated in a single annulus rather than being divided between two as was t’re case in PBS. Tile outputs are summarized in fig. 2. Each square represents an annulus. Along the horizontal axis we indicate the proportions of the area of an annulus occllpied by an activity. There are four activities possible: transpol tation, good 1, good 2, and agriculture. The vertical axis indicates the capital intensity of the technique used for the relevant activity. Transportation, good 1 and good 2 can be produced with three different techniques, each technique corresponding to a different ratio of capital to land in production. The highest ratio of capil:.d to land is indicated by the activity ‘occupying’ the complete vertical dimension for an annulus. Thus in P 3, in a.lnulus 1, transportation 15 produced with tlvn techniques; the maximum capital to land ti:chnique and the technique with m intermediate value for the ratio of capital to land. Again, for example, we see that in annulus P and in P 1, t~a~sportat~oi~ occu ies alil the Iani; availa
31725
P63
24138
i
1914
0.817
1267
0.506
0.771
0.236
11224
12.560
Il.200
12 240
i3600
12560
Fig. L
3 555
2;27
? ve
1.652
1.362
1154
0646
0594
fgjgp#q~Q~~
2 676
1173
i--
0103
0.!52
0348
0.0
0.0
0.0
0.0
J.M. Hartwick, Intermediate goods in land-use models
139
uses the techniques with a minimum possible capital to land ratio and the technique with an intermediate value for the ratio of capital to land. The numbers beneath each square representing an annulus are shadow prices on land or land rents net of the agricultural price for land. These rents generally decline with the distance the annulus lies frieci the center except for the case of annuluses, not at the certcer. which specialize in transportation. For example, annulus 3 in PB3 exhibis the anomalous rent value. The reason for these anomolies is that the resources used in transporting a unit of good across a boundary are divided into equal parts and are iequi .d?dsupplied by the two annuluses on either side of the boundary in the same factor proportions. Thus in annulus 3 of PB3, the resources used are entirely determined by the flows from and to the adjoining annuluses. This subarea in ‘t sense is not free to select its own activity in a way which minimizes t!-iecosts of production within the particular annulus. It should be emphasized that the se rinomolies in the rent values only occur in annuluses completely specialized m the transportation activity. What might we expect with more than one final and more than one intermediate good? There will now be the possibility of different intermediate goods ‘integrating’ with different final goods and depending on the technology of production and on transport costs, we can expect different patterns of integration and segregation of land t,ses in difIerent areas. Exogenous changes in parameters can integrate or segrer;ate two activities at the same site. These results are apparent in light of the result; for one final and one intermediate good and of the strticture of the activity analysis land use models with many commodities. 4. Conclusions Th: fuildamental characteristic of intermediate goods in location analysis is ihat they are 4 necessary element of a model which is to exhibit a spatial mixing of activities at various sites given constant returns to scale. When all commodities are final goods we observe the Von Thiinen pattern of segregated land use. Nowever, given that intermediate goods are in a model, not every allocation will display the mixing or spatial integration of land uses. In the fixed coefficient mode! we determined the precise conditions under which one allocation versus another would occur. The naidre of technology and relative transportation costs for tht: final and the intermediate good were crucial. In particular, we noted in all mcfdels that, given an efficient integrated land use patkrn, some decline; in the rela?ite cost of transporting the intermediate good would eventually lead to the segregated land use pattern being efficient. Such land use changes have been discussed by urban historians. They ide~t& 4 the intermediate good a’; labor required for rh-: production of the final good. We have analyzed formaliv what appear to be a fundamental transformation of cities in history - the changt from a spatially integrated land !~se pattern to a spatially segregated one. One might ask what are the reasons why contemporary North American citrss
J.&f. Hartwick, Intermediate good9 in land-use models Table 2
Total rents
Distances
0.158107E-k02 0.1502558402 0.143146E+02 0.136680Ef02 0.130773Et02 0.125355E-t02 0.1203688 + 02 0.115763E-k02 O.l11497E+O2 0.1C7534E+ 02 0.183843E+r)2 G.l00398E+02 0.971734E+01 0.941497EfOl 0.913084E+Ol G.886337E+ oi 0.861112E+Ol 0.837,828 +Ol 0.814737E+Oo1 0.793373E+Of 0.773102E+Ol O.75384OE+ 01 0.355515E+ 01 0.71806OE+Ol 0.701414E-t01 0.685522E + 01 0.67033.1E+01 0.6558058+ 01 0.64189tE f 01 0.628558E +01 0.615766E-701 0.6032828 + 31 0.5916838+ 01 0.580335E + 01 O.S69413E+Ol 0.558895E+Ol 0.548758E+Ol 0.588983E -1-01 0.52955OE-t 01,
Transport cost for intermediate goods
a2
R”
R’
t2
0.387281E+G2 0.368048E+O2 0.350636E-k 102 0.334797E-k02 0.320326E+02 0.307055E-k102 0,29484OE+ 02 0.2835598 + 02 0.273llGE+02 0.263404E + 02 0.25436448-I-02 0.245923E + 02 0.238025E-k 02 0.320619E+O2 0.223459E+ 02 0.2i7107EfO2 0.210928E+02 0.206091E+02 0.199569E f 02 O.l94336E+02 0.18937OE+ 02 O.l84652E+02 0.180164E+02 O.l75888E+02 O.l71811E+02 O.l67918E+02 0.16419bE+O2 0.160639Ef02 O.l572?1E+02 O.l53965E+O2 0.150831Et02 O.l47823E+02 O.l44932E+G2 O.l42152E+02 0.139477E + 02 h13690lEf02 0.134418E+02 0.1320238+@2 0.129713E+K!
0.3124728+04 0.2822038 + 04 0.251%36E-I-04 0.2335188+04 0.2137698 + 04 0.196J23E-t04 0.181105E-kG4 0.1675128+04 0.155394EfO4 O.l44545E+O4 O.l34793E+O4 0.125996E A-04 O.l18033E+O4 O.l10802E+-04 0.104215Ei G4 0.98199iE+O4 0.92689lE-k04 0.8763028 + 04 0.829745E+ 04 0.786801E+O4 0.747108E+O4 0.710343E+o4 0.676228E + 04 O.tA4512E+ 04 0.614976E$04 0.537425E + 04 0.561685E+O4 0.5376OOE-k04 0.515032E+04 0.4938568-t-04 0.473959E + 04 0.4552415~04 0.4376108 + 04 0.420984E + 04 0.4052888 f 04 0.390454E -I-04 0.376419E c 04 0.363328E+04 0.3505288+04
0.732422B + 03 0.732422E-k 03 0.7324228-k 03 0.732422E + 03 0.732422E $03 0.7324228+03 0,732422E+ 03 0.7325228 -t 03 0.7324228 + 03 0.732422E+ 03 0.7324228 i- 03 0.7324228 + 03 0.732422E~t03 0.732422L:+O3 0.7324228+03 0.7324228+03 0.7324228 + 03 0.732422E f 03 0.7324228 -I-03 0.732422E + 03 0.732422E+03 0.7324228 + 03 0.732422E.t 03 0.732422E+ 03 0.732422E-t- 03 0.732422E + 03 0.732422E + 03 0.732422E + 03 3.732422E+03 0.732422E + 03 0.732422E + 03 0.732422E+O3 0.7324228 -i-03 0.7324228 -I-03 0.732422E + 03 0.7324226 -403 0.732422E -I-03 0.732422E f 03 0.73242X? + 03
0.6 0.6 1.0 1.2 1.4
-_1__
:*: 210 2.2 2.4 2.6 z*: 3:2 :.t 3:8 44.0 :: 4:6 4.8 5.0 0.2 0.4 0.6 0.8 6.0 0.2 ::: :: 0:2 0.4 0.6 0.8 ::; -
do not display a clearly defined segregated land use pattern. I believe the two primary reasons are that in modern cities there are a multiplicity of final goods and intermediate goods with some sectors spatially integrated an segregated, and second! yn that people of the sarrle ;ncomc classes h income elasticities of demand 1’~ reGlentia1 spxe and ccpm!a~ufinr. element causes a city to display considerable heterogeneity of resi to give an appearance of integrated residential land use. For an example of ahe
J. M. Hurtwick,
htermeciiote
goods irt land-use m-de/s
!41
first reason in operation, one could scrutinize the example reported in Hartwick and Martwick (1974a). A wide vaiizty of land use pttterns are possible for difl’er- 1 ent specifications of the technology and transport costs. To this point. we have restricted our attention to patterns of land use in long-run equilibrium. Some might argue that the neglect of the durability of structures and the related disequilibrium paths of adjustment in urban growth represents a major omission. In the opinion of the writer, our propositions require inessential changes when this omission is considered in detail. Considerable debate would no doubt arise on the precise nattile of the path of adjustment from one equilibrium to another. Appendix
Land in this model is simply Van Thtinen’s homogeneous plain with a center. We divide a priori land into n annuluses of Pqual radial distance or width. Commodities must be shipped at rrositive transportation costs to where they are used (intermediate goods) cr where they are to be exported (final goods more to the center). Land must be bid away from agricultural use at rent R, per unit, and capital must be bid away from alternate use.; at r dollars per unit. The driving force causing the city to exist is the exugenou ;ly given demand for exports of the fina! good. The non-transportation activities must be allocated to each parcel 0,’ land agriculture, the final good {good 1) and housing (good 2), plus transportation. Each unit of goods 1 and 2 crossing the boundary between two annuluses requires land and capital to be expended. These are the costs of transport. The allocation of activities must be such as to minimize the value of land and capital used in the city net of the value of transportation costs. The dual problem is the maximization of the value of exported goods net the value of land used for urban (nqn-agricultura!) activity. The objective function t3 be minimized with respect to activity levels and intersite flows is
where the variables in the pri nal linear progr’uu are defined as follows: S
=
index ranging from I to produce commodity
to S, q,
indicating the technique or process used
J.M. Hartwick, Intermediate goods ir! hd-use
142
d R RA W %q.=
z,(i)
%i)
x,‘!+‘(i)
=
= =
= =
models
index ranging from 1 to 2 indicating the boundary of the ith aunulus over which the indicated flows are moving, exogencausly given rental price of a un.Aof capital, exogenously given rental price of a unit of agricultural land, exogcnously given wage rate for a man-week of labor, input uf commodity k required to produce a unit of commodity 5.with the 8th technique, k ranges from 1 to !’+ 3 where cornmoLt ICS l to 4 are not primary, input r+ 1 is land, input r+2 is capital and input r-t3 is labor; commodity r is housing and ar,4j is set equal to u~+~,~,~in order to identify one worker with one hoi!Gn;; unit; q ranges from 1 to r plus one additional commodity, transy!Jrtr ion indexed to b, coefficient indicating the relative economic attractiveness of annulus (i) for exporting commodity 6, relative to other transportation nodes, coefficient taking on values of 1 if annulus (i) is a transportation node and 0 if it is not a node. Only for i = i will 6 = 1 in this paper, the activity level of the transportation sector used to transport aggregated commodity flows ficross the (i, if 13boundary of annulus (i) using the sth transportation technique, flow of the qth commodity 3eaving annulus (i) across the dth boundary, coefficient indicating the amount of the transportation commodity Tequired to move a unit of commodity q from one annulus to 6 coniIguous annulus, amount of commodity q exported from annulus ii), qs activity level in annulus (i) or the gross output of commodity q when the sth technique is annulus (i) is operated.
If no urban activities are assigned to an area in the system, then that arca remains in agricultural WC. Some peripheral annuluses may not bid away from agriculture for use in the city. Agricultural activity can be considered as all othx activities operated at zero activity levels. Now each annulus must be in flow equilibrium for each of the 2 non-transportation and non-agricultural activities. That is, we have the: following constraints:
where the first two terms indicate flows of commodity q to ammlus (1’)from each ofthe two continguous annuluxs, the third term is the gross output of comrrodity y lin annulus (i), the fourth term is total intermediate use of commodiry q by
J.M. Hartwick, Intermediatf
3
goods iE land-use models
143
other activities in an&us (i), the fifth term iii&“,_ -dates total flows of commodity q from annulus (i) to its neighboring annuluses, and the sixth term indicates the export or import of cohmodity q from annulus (ij for the case in which (i) is a transportation node. The exports of each commodity must at least satisfy the final demands whiayh have been directed to the urban are?. for the commodity in question. The resulting constraints are c &-)x,(i) 3 Yq, 4 = 1,2, i
j
z
(A.3)
1, - - l, n.
The 2nd commodity, hcusing is treated as purely intermediate to the urban economy since we equate labor with housing and assume that associated with each worker fs one hotlse. and that ail workers reside in the urban area. The I* i (3rd) sector i:$the transportation sector denoted by the subscript t, The demand for transportation services derives from the flows of commrditics (including housing/labor) moving between annuluses. The production of transportation services requin:s inputs of capital and land, transformable from processes in a finite array. More capital-intensive transportation producticn corresponds to, say, the tise of subways rather than automobiles. The process = tIra,+l,r,,,a,.i2,,,~ ) indicates the requirement of land (ar+l ,t,,) and ca&al ar+ ? 2,r,s) for producing one unit of transportation services with technique S. On the demand side, since, 1, is the amount of transportation good required in one annulus to move a unit of commodity q between two adjacent annuluses, then t,#(i)+I@i+ I)) is the amount of transportation good required produced in annulus (i) in order to transport the floes ofq across the (i, i+ 1) boundary of annulus (i). An identical amount of transportation good must be produced in annulus (i+ 1) in order to accomodate the incoming flow to annulus i+ 1 across the i, i+ 1 frontier. We thus get the following additional ctinstraints:
1 xj:i+‘ii)s
f
t,(Ti(i)+T,2(i+ 1))
2 0,
i = 1, . . ., M-
4=1
1.
(A.4)
Finally, all land in each annulus must be exhausted in use by the production ot commoditie:; I and 2, clnd/or transportation, and!or bsriculture. The land use the constraints
for 0 fixed and 0 < 0 5 I, where in (A.5), the firsi Lerm indicates land syortation, the second term production other than agricult
J.iW. Hartwick, Interm~Late goods in land-use models
144
land use in transportation, and the inequality indicates the land can be left in agr,-iculture (when the relevant slack variable is not zero).
ilatafor probienzs
in text
Max. number of rings : 12 (if 12 rings do not appear, then the remaining rings are filled with agricultural activity). Number of goods: 2 (1 export good and housing). Number ofcapital intensities: 3. Product ion technology ----~ Good I
Good 2
Land
Capital
0.25 &O
0.6 0.66
2.5 2.5
Height =: 1 output
1
0.0
2
0.0
.--~--iieight = 2 o!atput 1 2 --- _I_-__ Height = 3 Ourpui 1 2
--0.25 0.0
0.0
0.0
0.48 0.2
3.2 4.0 -
0.0 0.0
0.25 0.0
0.4 0.1
4.0 6.0
Final demand for export good: 100 units. Transportation
production technology
Congestion kvel _-I_-
Land
Capital
1 2 3
0.3 X::
0.2 2.4 4.0
Commodity transportation
3.2 3.3
II I’ 0.1 rl ,= 0.175 tt = 0.250
12 = 0.5 t2 = 0.5 tz = 0.5
PI3 3.4
?, = 0.250
II = 0.25
PB PB PI;
3.1
weights
This problem in spite of these differences in b:sic coeficients has an identical solution to th?t of PB 3.3. PB
3.5
Cl = 0.25
tr .:=0.1
PB
3.6
t1 = 0.30
bz = 0.1
References Green, J.R., 1974, Vertical integration and ~:ssurance of markets, Discussion Paper no. 383 (Harvard Institute of Economic Research, Cambridge, MA). Hartwick, P.G. and JX Hartwick, 197&r, EZicient resource allocation in a multinucleated city with intermediate goods, Quarterly Journ, ofEconomics, May, 340-352. Hartwick, J.M. and P.G. Hartwick, 1974b, The activity analysis approach to nrban model I building, Papers Regional Science Association, Nov., 75-85. Mills, E.S., 1370, Theefficiency of spatial competition, Papers Regional Science Association 75. 71-82. Mills, E.S., 1973, Studies in the structure of the urban economy !Re=ources for the Future, Washington, DC). Solow. R.M., 1973, On equitibrium models of urban locatio “, in: J.M. Parkin, ed., Essays in modem econtimics (Longs ans, London). Warner, S.B., 1962, Streetcar suburbs: The process of growth ix Boston 189t.LI%0 iHarv3rd University and MIT Press. Cambridge, MA). %‘e~,,-r, A., 1899, The growth of cities in the nineteent:] century (Cornell University Pre’ss, Ithaca, NY).