Regional Science and Urban Economics 12 (1982) 405-923. North-Holland

LOCATION IN A CONGESTED CITY

David SEGAL and Thomas L, STEINMEIER* St. Antony’s College, Oxford, OX2 dJF, UK

Dartmouth College, Hanover, NN 03755, USA

Received July 1980, final version received January 1982

This paper analyzes optimal location decisions in EL city where both residential and cmploymcnt locations are endogenously determined. A model is constructed which includes both traffic congestion among commuters and colocational externalities among firms. The model establishes that if a location tax of the kind discussed by Koopmans and Beckman is imposed on both residential and employee locations, the private decisions of households and firms will produce an optimal pattern of location.

1, Introduction

This paper investigates market failure in the distribution of residential and employment locations within a city when traffic congestion and colocational externalities are present. While the notion of optimal residential density in the presence of congestion has received extensive coverage [see, for example. Riley (1974) and Oron, Pines and Sheshinski (‘1973)], the findings have been sensitive to the assumption of a single employment site, at the center. Mills (1972b), building on the earlier work of Koopmans and Beckman (1957), constructed a linear programming model with endogenous residential and employment decisions in a congested city. However, only illustrative numerical examples of the solution were given for this model; general characteristics of the solution were not derived.

The present paper is a reformulation of the Mills model in an optimal control framework which permits a much more extensive analysis of the nature of the solution. The model encompasses both the traffic congestion of commuting workers and the colocational externalities of the location decisions of employers. The paper demonstrates the existence and nature of a set of location taxes on firms and households, similar in form to those sought by Koopmans and Beckman, which enable an otherwise: competitive market city to obtain Pareto efficient results in its location decisions.

*The authors wish to express their appreciation to Leon Moses for suggesting thi?; paper anti to John M. Hartwick, William C. Whenton, and an anonymous referee for helpful comments on an earlier draft

0166--0462/82/000&0000/$02.75 @) 1982 North-Holland

406 D. Segal and TL. Steinmtder, Location in a congested ciry

We begin our analysis with a model for determining equilibrium locations of households and firms in a city. In the next section, an optimization model is formulated to obtain the residential and employment patterns that maximize the total welfare of the city, and the conditions of optimal&y are interpreted. In the third section, it is shown that a competitive equilibrium will not yield the optimal patterns in the presence of Pareto-relevant congesti -II. The competitive equiilbrium can, however, be modified so as to obtsin optimal patterns through a lbzation tax to be placed upon both hcjuseholds and employers. Other implicat ;ZL; of the mode1 are explored in section 4.

2. The model

The model of this section describes the simultaneous determination of residential and employment location within a city. The city is assumed to have hi households and an equal number of jobs. Each household is assumed to have one worker who must commute to a job; the price of elasticity of the demand for work trips is taken to be zero. Firms employ the jvorkers and use some of the city’s land in order to produce output. The net productivity of any particular firm is assumed to depend in part on its accessibility to other firms.. as well as to customers and suppliers in other cities.

The model considers three sources of city welfare. The first is the productivit:y of the workers in their jobs. A second is !he surplus that the households derive from the land on which they live. And finally, there are the transportation costs associated with the work trips between households and places of employment. The first tvro represent positive contributions to \vclfarc. while the third is a cost and mlist be subtracted.

We begin with a consideration of the surplus arising from residential land. The household’s demand for land at any location is a derived demand, dependent upon the price of land at that location,

where r/ is the quantity of land demanded by a household at a location s miles from the center of the city and p(s) is the price of land at that location. The demand funzlion 4 is considered to be uniform across households. impiying that the households are homogeneous with regard to incomes and tastes for land. The effects of income differentiation are explored in ;t Gmulation framework elsewhere.’

Since $5 Ic an ordinary clemrrnd function, it is monot~~nically declining in ~1 :md hence may be inverted.

LX Segal and TL. Steinmeier, Location in a congmed ritJ 407

Then S[&)], the sum of consumer and pr ,ducer surplus for the land occupied by a single household at s, will be given by

sCa(s)l = 1 PWx. 0

(3

The surplus per unit area of residential land is simply the surplus per household divided by the amoun: of land occupied per household and is

equal to ~cdsll/dd* Not all land is residential land, however. Let L(s) be the amount of

business land per employer at s, and let g(s) be the number of employees per unit area (i.e., the employment density) at s, and let K(s) be the capacity of the transportation system at s. In this case, the fraction of land occupied by businesses will be given by g(s)&), and if the amount of land required for the transportation system is proportional to K(s), the fraction of land occupied by this system will be given by k,K(s)/s, where li, is a constant of proportionality. The fraction of the land used for residential purposes is then determined as the residual 1 -g(s)L(s)- k,K(s)/s.

The surplus per unit area in the city is found by multiplying the surplus per unit area of residential land times the fraction of land used for residences. This quantity is [l -g(s)&) - k,K(s)/s]S[q(s)]/q(s). The total surplus throughout the city may be found by integrating this expression over the area of the city,

4 2fl(s[1 -g(s)L(s)] -4@(s)) ~$&, 0 L

(4)

where R is the radius of the city. The cost per mile for work trips is dependent upon the degree of

congestion in the transportation network at any point. The degree of congestion, in turn, depends upon the number of commuters in relation to the capacity of the transportation network. In fig. 1, let A be the area of the

408 D. &gal and T.L. Steinmeicr, Location in u congested city

city lying beyond the cordon at s, and let B be the interior area. The residential density at any point in the city is simply the fraction of land used for residential purposes divided by the amr. tint of land demanded by each household. Hence the number of households residing in A is given by

2n R x,? -&)L(x)l - ‘&‘) dx 1

-- 46)

t

and the number in R is

Similarly, the number of jobs in A is given by 2I7 1: xg(x)dx. If jobs tend to be more centrally located than residences, then of the

households that live in A, only 211 j: xg(x)dx can work there. The remaining number, &ven by

V@) = 2n j “’ -““);;;y - ‘oKtx) dx _ 2n i xg(x)ds, S S

(3

must commute inward from residences in .A to jobs in B, as indicated by the arrows in fig. 1. Since there are N residences and N jobs, V(s) is also the number of jobs in B which cannot be filled by people residing in B,

V(s)=2&cg(x)dx-2UJ R ‘cl -g(x)L(x)l -k,Kt2du . .

0 3 G)

These jobs are of course filled by people commuting from A Hence V(s) is the number of people who commute through the cordon

depicted in filg. 1. Congestion at this cordon depends upon the number of commuters relative to the capacity of the transportation network through the cordon, and transportation costs per mile reflect this congestion,2

‘Transportatioi; costs are assumed to be the same for all commuters at any given point in the LX!?, tmplying a constant value of time among them. This assumption is realxed in the wnulatlon model of a paper cited in footnote 1.

D. Segal and T.L. Steinmeier, Location in a congested city 409

where t($ is the variable cost per mile per commuter for transportation and K(s) is the capacity of the radial transportation system at s. The nature of the transportation system is left unspecified; there may be a single mode of transportation or some combination of several different modes. If more than one mode is involved, however, it is assumed that commuters alloczte themselves among the different modes so as to equalize the variable transportation costs among them.

The variable costs of commuting through any cordon are given by t(s) l V(s).

The total amount of such costs throughout the city is found by integrating this quantity over all values of s,

C = I” t(s) V($ds. 0

(6)

To this capacity may be then given by

must be added the cost of the transportation system itself. If the at s is K(s), then the annualized expense of maintaining this capacity deftned as r[K($j]. The total expense of maintaining the system is

E = 7 r[K(s)]ds. 0

(7)

The total transportrtion costs are simply the sum of the variable costs and the costs of maintaining the capacity.

‘I-=! {#)V(sj+rCK(s)]}ds.

The productivity per employee in a firm depends upon L(s), the amount of business land per employee at s. Productivity, as noted earlier, is also affected by the accessibility to other firms in the same city and to customers and suppliers in other cities. The greater tile average distance bctwecn the firm in question and other firms in the ctty, the higher will be its costs. When a firm finds itself at a greater distant.: from other firms, either (i) it must be incurring higher transportation charb’l=s, or (ii) it must be giving greater preference to nearby customers and suppliers (even though the sdppfies of the firm might be of slightly lower quality or the prices that nearby customers are willing to pay might be slightly lower). Whichever may be the case, the output per employee, net of transportation charges, will be lower. Similar reasoning can be applied to firms that are relatively further from the shipping points to other cities. These ideas can be expressed in the following

410 D. Segul and 'lIL. Steinmeier, Location in a congested tit)

relationship for productivity:

X(s)= f(L)- kLD,-k,D,, . 1%

where x is the output per worker at s, D, is the average distance to other firms in the same city, and D, is the distance to the shipping points to other cities. It should be noted that this formulation abstracts from congestion costs for business shipping. This is consistent with the general notion that the really significant congestion costs are borne by commuters during the commuter rush hours.

The distance D, to the shipping points of course depends upon where these poin!s are. If the point is a railroad yard near the center of the city, then the relevant distance would be s, the distance to the center of the city. If the shipping points are scattered around a peripheral interstate, then the relevant distance would become R-s, the distance to the boundary of the city. Both kinds of shipping points can be available in tire same city with firms in the center of the city using the central rail facility and those elsewhere using shipping points on the peripheral interstate. In any of these cases, however, the distance to the nearest shipping point is related to the distance from the city center, so that D, can be expressed as D,(s).

The average distance D,, to other firms may be easily calculated if the transportation network is assumed to be comprised of radial and circum6xntial routes. In fig. 2, let 2 be a firm located at a distance s from the center of the city, and let any other point in the city be denoted by Its distance from the center of the city and by the angle it forms with Z. Thus, if point 1’ is a distance _v from the center of the city and forms an angle 0 with %. it would be denoted by L~,O].

D. Segai and ll. Steinmeiar, Location in ~1 congested city 411

The shortest distance between Z and Y depends upon the location of Y Call this distance d(y, 0). If y <s and fb 2 radians, so that Y lies in the region labeled A, then the shortest route from 2 to Y is to follow the radtai route from 2 until it intersects the circumferential through U, and then to take the circumferential to I! This path is indicated by the dashed line in fig. 2, and its distance is given by

44(y, w = (s - y) + yu, (10)

where the first term on the right-hand side is the distance along the radial route and the second is the distance along the circumferential. If JQ s and 6~2 radians, then Y is in region B, and the shortest route fr WI Z now is to follow the circumferential route through Z until it intersects the radial route through Y, and then to take the radial route to Y. The distance of this route IS

&(y, 0) = so + (y - s). (W

Finally, if 0>2, the shortest route is to take a radial route from % to the city center and another radial route from there to YI Points with fb2 arc in region C, and for these routes the distance is given by

d,(y, 0) = s + I’. (12)

The employment-weighted average distance from % to other firms in the city is found by multiplying the employment density at every point by the respective distance from Z and integrating over the area of the city:

where N is the total crnploymcnt in the city and it equals fo

Let us define the functions:

Then.

D. Segal and ZL. Steinmeier, Locution in a congested city

The total output in the city may then br. found by substituting for D,, in eq. (9). multiplying by the employment density, and integrating,

D,_ and

- 41 (F{s),iA’) + 47~) - X,D,(s)]sds.

The problem of city-wide allocative eficiency is to choose the gradients q(s), L(s). g(s). and K(s) so as to maximize the sum of production (XT) and the total surplus attributable to residential land (s7.) less the transportation costs associated with work trips (77, subject to the constraints in eqs. (3), (5). and (14). and subject to the endpoint condition,

which says that the price of land at the city’s boundary must equal P,, the

price of the surrounding agricultural land. This is a well-formulated problem in optimal control theory, and the details of the solution are presented in the appendix.

The solution to the optimization problem generates four marginal conditions. each of which has a straightforward and plausible interpretation. -T‘hcy are as follows:

The left-hand side of this condition is the additional surplus that accrues if a hrjuschoid moics slightly further from the center of the city, and tht right- h;ind +lo is the additional transportation costs resulting from such a move,

D. Segal artd T.L. Steinmrier, Location in a congested city 413

including both the private cost and the cost imposed on other commuters through congestion. Ey. (18) requires that these be equal for an optimal distribution of households and employment.3

f’= P. (19)

This condition requires that the marginal productivity of land in production be equal to its price. By differentiating eq. (3) with respect to 4, we see that the price of land is also equal to the additional surplus resulting from an increment in the size of residential plots.

S’[q(s)] = P.

Together, this equation and eq. (19) imply that the incremental value of must be the same whether it is used for business or residential purposes.

r’ + 27&P = - V(&/X).

land

(20)

In this equation, J is the annualized cost of maintaining an additional unit of transportation capacity, and 2&P is the rental value of the land that the additional unit would require. Hence, the left-hand side of this equation is the total marginal cost of enlarging the transportation capacity, and the condition simply requires that this cost at any point in the city should just be balanced by the reduced transportation costs that it makes possible. If the price of land increases toward the center of the city, this condition implies that the optimal amount of congestion should also increase toward the center of the city.

This equation may be interpreted as follows. If an employment location is moved a unit distance further from the center of the city, then t + V(i%/W) is the resulting savings in commuter transportation costs, both private and external. Such a move would also mean that L/q households would be displaced from the new location and moved inward to the old location. This would result in further savings in transportation costs of (t + 1/(&/C V))( Liy). Hence, the quantity on the left-hand side represents the total savings in commuter transportation costs if one employment location were exchanged for L/q residential locations a unit distance more distant.

“If the optimal solution involves no commuting through the cordon at S, then ey. (IX) wrll hold with inequality in(;tead of equality, and the price gradient of land will he determined entirely by eq. (21).

()n the right-hand side of eq. (9). the term within the brackets is the increase in the employment-weighted average distance to other firms which occurs when an employment location is moved a unit distance further from the center of the city. This is most easily seen with reference to fig. 2. If Z is moved out one unit, the radial part of the route to points in A will be increased by one unit, while the length of the circumferential part of the route will remain unchanged. The Iength of the route to points in B will similarly be increased by one unit, but for points in C, the shorter radial distance will on the average be exactly offset by the increased circumferential distance. Since the pcrcentagc of employment in A is 4G/21rN and the percentage in B is 2(n- 2)/2x. the employment-weighted average increase in route lengths will be 4G,& Y + 2(7t - 2)/27r.

T’hc final term on the right-hand side is thus the reduction in net productivity arising from the decreased accessibility to otker firms. The factor arises because the effect is symmetric -in the other firms. The second terms is the change in net productivity due to the changed accessibi1it.y to the shipping points to other cities. The sum of the two terms is thus the total cta;tngc in productivity that would occur if one employment location were cx:hangcd with L/y household locations at a unit distance further out, and the cl.mdition in eq. (9) requires thal this just be balanced by the savings in transh>ortation costs resulting from such an exchange.J

P.~ills and Hartwick have shown in simple models with constant trrinsportatirzn costs that the optima! city is either completely segregated with ;I business district surrounded by a residential district, or completely integrated. with al! workers living adjacent to their worksitcs. Mills furthe! found that in a linear programmini: model with congestion, numcricai simulations continued to generate only these polar results. Such results are certainly possible in the prcscnt mode!. F’or instance, if the business transport costs are low enough relative to commuting costs, the model will generate the integrated solution in which no commuting takes plncc. Likewise, il

husincss transport costs are high relative to commuting costs. the mode! will gcneratc a segregated solution in which business transport costs arc rcduccd to ;1 minimum.

The present mode! also appears to allow for solutions bctwccn thcsc two poltir C;FCS. In this respect. the critical diffcrencc tft’twcen it ;tnd t!ltb mode\

(I!’ Mulls is that in tne lLl~!!s mode!. both business tr;tnsport and cc,nlmuters

Imposed congestion costs on each other; in the present mode! commuters

impose external costs only on other commuters. With appropriate parameter va!ues in the present model, the high land costs in the n&r-downtown rcsldcntia! area may result in an optimal transportation system which :lllow~

D. &gal and 7X. Steinmeier, Location in a cwgcsted city 415

a substantial amount of congestion durin? the commuter rush hours. This will in turn produce a steep land price graaient in the area, Businesses, which are not subject to the commuter congestio .A costs, will thus have an incentive to move outward. Near the perimeter, however, land prices, congestion, and the land price gradient are all less, so that businesses will not have an incentive to move all the way out to the perimeter. The result would bc a partially integrated city, with a central business district surrounded by an area of both firms and households surrounded further by a residential area.

Another model which avoids the polar results of the Mills and Hartwick models, even though it assumes constant transportation costs, is described in a paper by Goldstein and Moses. The Goldstein-Moses model assumes two activities, and this is the means which enables it to generate a solution in location. However, there is a crucial diRerence. In the Mills and Hartwick models, and in the model developed in this paper, there must be equal numbers of residential and employment locations, and it takes the output of one residential location to provide the input to one employment location. The Goldstein--Moses model does not enforce these restrictions on its two activities, and this is the means which enable: it to generate a solution in which the activities are segregated in one part of the city and integrated in another. If these restrictions are included, the Goldstein-Moses model would exhibit the same kind of polarity as do the Mills and Hartwick models.

3. lncentivcs for efficient private decisions

Eqs. (1X) to (21) give the requirements for ;rn effjcicnt pattern of residential and business location within a city. However, thesc location decisions are usually private decisions, The conditions for efficiency do not tell us anything about whcthcr priv;ltc inccntivcs will fulfil them. It is to this question that WC now turn. In the following discussion, WC will assume that the public offGals act so as to insure that the optimal lcvcl of highway capacity is maintained.

To kc in cquihbrium at a distance s from the ccntcr of the city, ii household must not be able to grin by reducing transportation costs with a !ocatirm inside s; nor can it be ahlc to gain by reducing rent with :I location beyond s. This yields the well-known equilibrium condition fix the runt and

density gradients 01’ a city;

t -- q(df’ds). (22)

The rcquirument for an optimal distribution of residential location is given by cq. (18). The two equations differ by ;ln additive fr:ctor of V(?r/W), which is simply the tlxternal costs imposed upon other commuters by the ;iddition 01’ on2 more c’Wntnutcr A ;tny particular. loc;rtiorr in the city. It is cvidcnt

416 D. Segal and T.L. Steinmeier. Location in a congested city

that the equilibrium residential location pattern established by private incentives wiil not be optimal unless the private incentives are altered to

reflect external congestion costs. Firms will be in equilibrium only when any savings in wages and land

costs resulting from an outward movement would just be offset by a reduction in net productivity. The savings in wages would be -d~/ds, where w is the wage level at a distance s from the center of the city, and the savings in land costs would be - L(dP/ds). The change in productivity arising from an altered accessibility to the shipping points to other cities would be k,B,. Finally, the total loss in productivity due to the reduced accessibility to other firms in the city is given by 2kL[4Gj2nN + 2(n- 2)/27r]; unless the elasticities of suppliers and customers are infinite, however, the firm in question will not bear the fuI1 burden of the loss in productivity.

To amplify this point, let us consider shipments between the firm in question and some other firm which is now further away because of the move. The effects of the additional transportation costs due to the greater distance can be analyzed in much the same manner as the imposition of a tax. The quantity of goods shipped between the two firms will fall, and the incidence of the economic burden will be divided between the firms in a way v:hich depends upon their relative elasticities of supply and demand. The part of the burden which falls on the other firm may be referred to as a ‘colocational externality’?

if, on the average. the supply and demand elasticities of the firm in question arc comparable to those of its suppliers and customers, then the cost to the firm of an outward movement will be approximately equal to the coiocational externalities. Both costs would be about

kL[4G/2nN +2(x - 2)&r],

and the resukng equilibrium condition for the firm would be

d\r* dP - --& - L -& = k,

4G 2(n - 2) T rti + --r-- 1 + IQ,,. (23)

For ~~ploycc~, equilibrium requires that any wage ,.dvantage which may kc: found by commuting further into the city must just be balanced by the private costs of the additional comrmltc,

- d\t’: d.q = t. (24)

ll~osc equilibrium conditions for employers and employees, plus the equilibrium condition for households given in eq. (22), yield tho following

D. in a congested city

condition for an equilibrium pattern of employment locasion:

417

The equilibrium condition differs from the optimal condition given in eq. (21) both by an additive factor of kt[4G/2nN + 2@-- 2)/2n] and by INI additional factor of V(ik/W). The first term represents the colocational exrernality and the second the congestion externality.

There are two potential methods to modify private incenGes so as to induce an optional distribution of residential and employment locations. The first method is the familiar congestion toll z(s) equal to the external congestion costs V(i%/W). Such a toll would alter the private commuting costs in eq. (22) so that

(26)

Thus, private incentives for residential location would yield the same condition as is necessary for social optimality. Also, employees would have to consider the tolls in the balance between transport costs and wages,

dw

-ds- =t+*=t+VC

W’ (27)

Substituting eqs. (25) and (26) into eq. (32), the equilibrium condition for firm location, yields

But this condition differs from eq. (21) by an additive factor of kJ4C’;/2nN + 2(71- 2)/27r]. Thus, some additional measures beyond the t ,-qcsticln toll are required to compensate for the colocational externality.

An alternative to the congestion toll is a location tax on businesses and households! Let L,,(S) be the location tax to be levied on a household at

6The idea of a location tax is mentioned by Oron, Pines and Sheshinski (1973, p 627). Rut. ;1> is true for others who have written on the subject, they do not pursue the concept.

418 D. Segal and T.L. Stuinmeier, Locutiort in a congvstld city

radius s, regardless of the amount of land it owns there. The equilibrium of the household, balancing off location costs against transportation costs, will be determined by the condition

t = - q(dP/ds) - i,,.

If the gradient of the residential at every point in the city, so thai

i, = v (apt)?

(28)

tax is equal to the external congestion costs

(2%

then eq. (28) is identical to eq. (N), and the conditions for household equilibrium and social eRicienr:y are the same.

Similarly, let Lb(s) be a location tax per employee on emp’royers. Equilibrium for employers who balance off wages, shipping costs, and the location tax will now require that

dw _- -- - ds

4G 2(X-2) 27LN+-2F 1 + k,d,. i- i,.

If the gradient of the business location tax is equal to the colocational externality less the external congestion costs,

+2(x-2) _~ ?t --

2rr 1 (71/ then eqs. (24) (28) and (29) imply eq. (21) the eficicncy condition for employment location. Thus, unlike congestion tolls, ~1 strt of’ Ioctrrion WXS, .hilar in nature to those sought by Koopmans and Beckmctn, is dde to corrwt simulf dnt3wdy _fiv congest ion externalities and colocat ionaf vs tcrndit ies.

Suppose that the business location tax, L,,(S), were levied on employees rather than employers. The equilibrium condition for employees balancing wages. commuting expenses, and the location tax would then be

t = i, = - dw/ds. (32)

If this tax satisfies eq. (31), as before, then eqs. (28) and (29) imply eq. (21). The condition for social efficiency is then achieved by the tax regardless of whether it is imposed on employers or employees. In either ease the effective net cost to employers and the effective net wage to employees will he the

same. ‘The only difference is that the wage actually paid will be different in the two cases by the amount of the tax.

D. Segal and ‘;I_?,. Steinmeier, Location in a congested city 419

Since V(&/dV)> 0, it follows that &, > 0, and this encourages households to move inward toward the center of the city. If u(at/W)k,[4G/2nN +2(n -2)/2x], then it would follow that t, ~0, and this would encourage businesses to move out. In this case, the excessive congestion resulting from the unpriced externality is believed both by moving workers in and by moving jobs out. If, on the other hand, V(at/N’) +[4G/27~‘V + 2(71- 2)/271] and &, >O, then firms are encouraged to move inward to relieve the more important colocation externality.

Finally, it bears mentioning that the corrective effects of the location taxes are achieved through the distance derivatives i,, and i,; zqs. (30) and (32) determine the levels t,, and k$, only up to an additive constant. This leaves open the possibility that either location tax may be negative at some points in the city. At such points, the tax can be reinterpreted as a subsidy, and the rest of the analysis remains unaffected.’

4. Dynamic aspects of residential and employment location

Several concluding observations can be made concerning the model WC have developed and applications to which it might be put. Although the analysis is comparative statics, the model can be used to $,hed light on some dynamic urban processes, such as suburbanization.

In this regard, the model suggests that when incomes grow within a city the following concatenation of events is likely to emerge, ce~~is p~‘bus. First, the demand for land will shift outward, raising [in terms of eq. (22)] the function &)==$[p(s)] at the original pattern of prices. rhis provides impetus for a declining residential gradient over time, which has been widely reported.

At the original employment gradient, this growth in rc:;idential suburbanization will raise the number of commuters who must pass through any given cordon [see eq. (S)]. And unless transportation capacity is increased proportionately, there will be increased roadway congestion, which in turn will cause an increase in private transportation costs.

An increase in transportation costs will tend to make the wage jiraciicnt steeper [eq. (24)]. It may also raise the rent gradient [eq. (22)], although such atl outcome could be partially or totally offset by increases in y. The stccpcr wage gradient and, to the extent it occurs, the steeper rent gradient, will

7A possible difliculty may arise at the periphery of the city. If, for example, l,,,(R) >O, and if there is no tax on the surrounding farmland, then people will seek to avoid the tax by moving to the countryside, even though they would incur higher transportation COSIS in doing SO. 7711s kind of reaction would cleatly be inefficient. and it may be avoided with either of two

procedures: (1) Adjust thti level of the location tax structure 30 th;lt I,,,(K) 0. This of COII~SC mphes a subsidy for all interior locatic-ns in the city. (2) E’xtcnd the location tax into the countryside. If the latter were done, it would be necessary that )i,,l< I in the countryside so that the people there would not be induced into suboptimal rclocatim decisions.

420 D. Segal and ‘EL. Steinmeier, Location in a congested city

upset the locational equilibria of firms [as described in eq. (23)3, caus@ them 10 move outward. This will produce declining employment gradients

over time. What is particularly interesting about this sequence is that it suggests the

possibility that employment may have followed residential housing to the suburbs rather than vice versa. On a priori grounds we offer credence for Mills’ empirical suggestion that ‘the movement of people to the suburbs has attracted manufacturing employment rather than vice versa’. [Mills (1970, p. 121.1 It should be noted that our finding is based on equilibrium conditions with neither congestion tolls nor location taxes [i.e., on eqs. (22) through C4)].

There is a second result deriving from the model here that accords with empirical observation. Recall that in eq. (23) we used the term kt to denote the amount by which productivity falls off when the average distance among firms is increased. It is reasonable to expect that kL is higher for non- manufacturing activities within a city (wholesaling, retailing and services) than for manufacturing, owing to the fact that intra-urban shipping cosfs per unit of value added tend to be higher for such activities. (This is especially true of wholesaling in which firms transact mainly with other firms.) Accordingly we would expect to find non-manufacturing activity more centrally located than manufacttzing, which is precisely what is reported in the literature [e.g., Mills (1972b, p. 40 ff.)].

A final comment relates to our finding (at the end of section 3) that location taxes may be more efficacious than congestion tolls in enhancing the productivity of cities with a trai%c congestion problem. The reason for this. ;is noted. was that the location tax corrects simultaneously for colocational and congestion externalities while the congestion toll adjusts only for the latter.

An insight into this important result may be drawn from the theory of economic policy, which tells us that a necessary condition for all policy goals !o hc achieved is that there must he at least as many policy instruments as thcrr are target variables. The location tax has two policy tools, taxes (or subsidies) on the two endpoints of the journey to work. It thus offers the possibility to reach two targets (which it does): reduction (1) of the congestion externality, and (2) of the colocational axternality. The congestion !oll. on the other hand, is a tax on the trip itself, rather than a separate tax on the origin and destination. Accordingly, as a policy instrument it can deal u ith only one of the externalities.

Appendix

The constraints relevant to the maximization problem are form:Jized as fi~llov~s. t-‘ir5.L define

D. Segnl and T.L. Steinmeier, Location in a congested city 421

where _< is the distance derivative, d.u/ds, for any variable x. Next differentiate eq. (3),

s(s) = p cm1 46)

= P IIds)

1 -g(s)L(s)] - k&s) V(s) = 2n sg(s) - ---

1 l_ll

I .

Y(S)

In the optimization, it will be convenient to regard Q(s) as the clontrol variable rather than q(s). For any given endpoint condition on q(s), Q(S) and q(s) are uniquely related through eq. (17). Such an endpoint condition is given by eq. (I 7),

(A.3

P, = P[a(R)].

Since P is monotonic, this condition fixes the endpoint condition q(R). The Hamiltonian for the optimization is formed as folIows:

where the I//~ arc the auxiliary vatkbles. The auxiliary equations are:

(A-7)

422 D. &gal and T,L. Steinmeier, Location in a congested city

4 3= - (‘H/?G = (4/N)kr,gs ‘, 04.9)

The first-order conditions for the optimal gradients Q(s), L(s), and K(s), and R(S) are found by setting llH/i?Q, i?H,Mt, i?H/iX, and i?H/ag equal to zero.

?H/‘SQ=e, ++,P=O, (A.12)

?H/?L= 2Tcgsf’ - 2nsgWq) + II/ 5 2ns(g/q) = 0, (A.13)

?H,‘aK = - 2nk,(§/q) - t@ V/iX) - r’ + &2n(ko/q), (A.14)

?Hfi)I:=s(2nf --k,[Js(G/N- l)-4(F/N)+4xs]-2k,D,

-2nsL(s/q)+~~2ns$~,271s2+~$2n:;(1 + L/q)=O. (AM)

The first-order conditions may be rewritten, eliminating the auxiliary variables, as follows. For eq. (A.12), first, differentiate the equation, substitute from eqs. (A. I), (A.7), and (AS), and simplify

s--&P,=O. (A&)

Differentiating this equation, substituting from eqs. (17), (US), and (24), and rearranging yields

- q(d P/ds) = t + I/ (A/(? v). (A. 17)

For eq. (A.1 3), divide by 27rgs, substitute from eq. (A. MI), and simplify

_f’ = p. (AM)

flq. tA.14) may be interprctcd simply by substituting in from cq. (AM) and rc;lrrangrng

r’ + hk, P = - k’( ?tW). (A.19)

E‘or cq. (A.1 51, first divide by s, differentiate, substitute from cqs. (A.2), (A.3). LLI), (A.10). (AM), ;rnd (A.181 and simplify

L3. Segal and 7IL. Steinmcier, Location in a congested city 423

To find ti4, we integrate eq. (A. lo),

where a is the constant of integration. From eq. (14) in the text, it is evident that F(O)==0 and that F(R) is not constrained by an endpoint condition, This implies *JR) = 0 by the transversal&y conditions, which implies a = 4k, since .Y = G(R). Hence,

t+b4 = 4kJ 1 - G/N).

Substituting this expression and eq. (A.1 1) back into eq. (A.20) and rearranging yieids

(,+ V-$-)( 1+;)=2k,[~+~!]+k,ri,. (A.21)

Goldstein, G.S. and L.N. Moses, 1975, Interdependence and the location of economic activities, Journal of Urban Economics 2,63-84.

Hartwick, John M., 1976, Intermediate goods and the spatial integration of land uses, Regional Science and Urban Economics 6, 127-145.

Koopmans, Tjalling C. and Martin Beckman, 1957, Assignment problems and the location of economic activities, Econometrica 25, 53-76.

Mills, Edwin S., 197Oa, The efIiciency of spatial competition, The Regional Science Association Papers 25.71-82.

Mills, Edwin S., 197Ob, Urban density functions, Urban Studies 7. Mills, Edwin S., 1972a, Studies in the structure of the urban economy (The Johns Hopkins

Press, far Resources for the Future, Baltimore, MD). Mills, Edwin S., 1972b, Markets and efficient resource allocation in brban areas, Swedish

Journal of Economics 74, 100-113. Oron, Yitzhak, Ovid Pines and Eytan SheshiL:.ki. 1973, Optimum vs. equilibrium land use

management arrd congestion toll, The Bell Journal of Economics and Management Science 4, 602-618.

Riley, John G., 1974, Optimal residential density and road transportation, Journal of Urban Economics 1, 230-249.

&gal, David, 1977, Urban economics (Richard D. Irwin, Homewood, 11 1. Segal, David and Thomas L. Steinmeier, 1980, The incidence of congestion and congestion tolls,

Journal of llrban Economics 7, Jan., 42 62.