ECONOMICS ELSEVIER Regional Science and Urban Economics 26 (1996) 613-643

The new urban landscape: Developers and edge cities

Vernon Henderson*, Arindam Mitra Department of Economics, Brown University, Providence, RI 02912, USA Received 1 December 1994; final version received 10 October 1995

Abstract

In this paper we model the decisions of an edge city developer who chooses business district capacity and location strategically to maximize profits. The de- veloper competes against a core city with historically given downtown capacity for employment and, implicitly, residential population. Moving nearer the core city enhances production efficiency by increasing the efficiency of the exchange of information between businesses in the core and edge city. On the other hand, it increases typical residential rents and commuting costs (and hence wages demanded by employees) and weakens the developer's local monopsony power. The develop- er's choice of location and capacity play out in a complex but fascinating fashion, depending on the historical capacity of the downtown.

Keywords: Edge city; Developers; Location theory; Information externalities; Chaos

JEL classification: L12; RI0; R14; R30; R52

1. Introduction

In 1991 Joel Garreau published Edge City, an important book that helped to popularize the notion of edge cities and helped to prompt changes in the way we think about American cities. Edge cities are new cities created since 1965, outside of major central, or core cities. They are centered around enormous tracts of mixed use office space. They are complete cities, offering

* Corresponding author. Tel.: 401-863-2886/1417; fax: 401-863-1970.

0166-0462/96/$15.00 (~) 1996 Elsevier Science B.V. All rights reserved PII S0166-0462(96)02136-9 614 V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 jobs, residences, shopping, and services for their inhabitants. Their export products are not traditional manufacturing. Rather, 'in Edge City, the offices are the factories of the Information Age... the finished product shipped back out, essentially, is cleverness'. Garreau identifies 123 existing and 77 emerging edge cities in the 35 largest U.S. metro areas and asserts that more people inhabit these cities, than traditional cities. An edge city is a planned controlled entity with 'not only patterns and rules but limits to its growth' (p. 81). Controlled development of office space also often involves control of the surrounding residential areas with shadow governments which levy taxes, provide local services, and control land use. Edge cities present a very different perspective on suburbanization and the role of land developers in the urban landscape than in the traditional urban literature. Similarly, they provide a new perspective on the spatial organization of production in metropolitan regions and how business district sizes are chosen. Finally, they present a fresh perspective on how segrega- tion by income or socio-economic class occurs. Traditional suburbs were based on the decentralization of economic activity from the central city as transport and production technologies progressed after 1920 with the introduction of the automobile and truck, intra-metro highways, and assembly line and continuous process production (e.g. Mills, 1972, and Muth, 1969). Economic activity spilled out of core cities, into nearby smaller towns, as a simple decentralization phenomenon, where the central city still retained the focal economic role. Economists also modelled bedroom suburbs, as involving the flight of higher income residents from central cities into more homogeneous suburban 'clubs' for the consumption of relatively high levels of local public services (Ellickson, 1971; Mills and Oates, 1975). Edge cities differ from these traditional suburbs. They are not simply bedroom communities or a product of urban decentralization and sprawl. They are the creation of strategically controlled office development, by large-scale land developers. For edge cities, developers make strategic choices of office space capacity, location vis-~-vis the central city or other edge cities, industry/job mix in allocating office space and perhaps popula- tion. Edge cities are imperfect competitors in metropolitan labor markets, given there are a limited number of major communities in any metro area. Edge cities are typically based around one of the great American ports such as Boston, Chicago, Houston, New York, and San Francisco. These traditional ports are passive entities with largely historically given capacities and little control over total population. While major portions of these cities may have been the construct of large land developers decades ago, all land has been sold off to atomistic holders over time, and time has eroded the constraints of the original planning and zoning process. Edge cities' location and capacity choices are current (not historical) and strategic because they v. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 615 engage the passive core city and potential competitors in a struggle for the metro area resources. In this paper we start by modeling the edge city phenonemon. We examine the strategic choices by one edge city for capacity, employment, and location vis-h-vis the core city. We explore questions like why edge city capacities may be large relative to historical downtowns with their advan- tages such as existing, paid-up infrastructure and better access to federal funding, and why edge city strategic locational choices give the appearance of randomness suggested in Garreau. This work tells us the key elements of the impact of edge city developers on metropolitan form. For comparison, we analyze how edge city configurations in metro areas might differ if they occurred only through the atomistic movements of workers, without the tight control exhibited by edge city developers. The final section of the paper suggests a broader research agenda on edge cities. Before turning to edge cities per se, we comment that, obviously, we are choosing to model the modern urban landscape as being determined by a sequence of strategic choices adopted by 'large agents' or land developers who engineer en masse reagglomerations of people. This is the general approach in one branch of the existing literature (e.g. Hamilton, 1975; Mills and Oates; 1975; Henderson, 1974, 1985; Pines, 1991; Rauch, 1993; Helsley and Strange, 1990; Mitra, 1994). This is in contrast to a different branch of the literature which views spatial configurations as determined by decisions made by atomistic agents, each with naive expectations (e.g. Beckmann, 1976; Fujita and Ogawa, 1980, 1982; and especially Krugman, 1993, 1996). We believe that the national land market is replete with 'large' private agents, particularly at the development stage. They initiate massive planned private developments on the scale of medium-size cities, manipulating the decisions of the atomistic agents, driving us to equilibria typically not even considered in a purely atomistic world. Developments may be purely private with the developer building all public infrastructure as well as private structures (e.g. office space). Structures may be sold off over time, so they return to atomistic holders and the public infrastructure may be gradually turned over to the collective/public domain. But the private developer may retain majority ownership for decades, and her initial plans will contractual- ly or institutionally define city development for decades. The role of (state or county level) public government is to be a repository and perhaps the enforcer of the developer's plans. Developments can also involve public and private participation, with various arrangements involving the financing and provision of public infrastructure and services. Sometimes the developer is even a government entity which buys up, develops, and sells off space for profit. Regardless, there is effectively one large agent in each of these scenarios who develops agglomerations on a massive scale. To back up these assertions and Garreau's assessment, we also gathered 616 V. Henderson, A, Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 information on edge cities. We were unable to uncover any central source of information on individual land developers in edge cities. Our information is based on telephone interviews and a review of documents. Our interviews suggested that all edge cities were originally the product of decisions by a single large agent; certainly we have uncovered no counter-examples of edge cities created purely through the decisions of atomistic agents. In Table 1 we present information on 10 edge cities-the name of the development, the planned office space, the 1993 office space and employment, and the name of the developer. Only three of these are among Garreau's 10 largest edge cities. 1 For the others, we were unable to untangle the development story. Most of the problem lies in disentangling space developed under the aegis of the original developer, compared with space developed by others who flocked to the scene and developed contiguous space over time. In Table 1, all developments, in terms of planned space, meet Garreau's 5 million square feet of office space, as a minimum to qualify as an edge city.

Table 1 Some edge city developments Edge city Developer Planned office Developed space Employ. space as of 1993 as of 1993 (million sq. ft.) (million sq. ft.) (thousands) Irvine, CA Irvine Co. 45.3 33 165

Tyson's Til Hazel 50 28 80 Corner, VA (Hazel & Peterson)

180/284, NJ Local govt. 15 5 75

Schaumburg, IL Local govt. 20 11 71

Reston, VA Til Hazel 30 18 45

Research U.N.C. 22 14 40 Triangle Pk., NC

Dearborn- Ford Land 8 4 25 Fairland, MI Devel.

Rariton Ctr., Visceglia 30 11 20 NJ Brothers

Park 10, Wolf 14 4 15 Houston, TX Companies

I Missing are O'Hare, IL, King of Prussia, PA, Meadowlands, NJ, Anaheim, CA, Edens Expressway, IL, West Los Angeles, CA, and LAX, CA. V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 617

As a reference point, cities such as Richmond, Spokane, Memphis, Wichita, Birmingham, Albany, and Little Rock have less than 5 million square feet of office space. Moreover, most office space in the listed edge cities is class A (highest quality) office space (e.g. 100% of Irvine's office space is class A). The CBDs of Atlanta, Baltimore, Cleveland, Buffalo, Detroit, Hartford, Indianapolis and Milwaukee have 8.5, 4.5, 9.5, 2.2, 4.4, 8.0, 4.0, and 3.0 million square feet of class A space, respectively. The very largest metro areas such as Washington, Los Angeles, Boston and Chicago have 17, 25, 34, and 40 million square feet of class A space, respectively.2 Note that our figures in Table 1 do not include space developed in nearby areas, because other economic activity has clustered around these edge cities. Current office space developed in the core part of these 10 edge cities varies from 4 to 33 million square feet, with total development planned to reach 8 to 50 million square feet over time. Most of these developers are private companies, reflecting the vision of one person or family. However, the developer listed as 'local government' for I80/284 is a 'for profit' development by the Townships of Morris, Parsippany and Whippany. The towns originally owned the land and have sold it off, tightly controlling the development through land use planning and zoning levels. For Schaumberg, again development is tightly controlled through the planning and zoning process.

2. An edge city model

In this section we adapt the Fujita and Ogawa (1980, 1982) model of urban spatial structure along a line to include large agents and a history (see also Kim, 1988, and Henderson and Slade, 1993). The model consists of the following components: • Spatial structure • Agents, behavior, and history • Production technology • Commuting costs and land rents • Worker supply • The maximization problem We look at each in turn.

2.1. Spatial Structure

Consider a region that consists of an historically given port city and one potential edge city. Fig. 1 illustrates one possible configuration. The region

z These numbers are from the National Association of Realtors. 618 V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643

Port City Population Edge City Population KI Ko -~ Commuting Direction Office Space Capacity tJ g,

I t • k\\\\\\\\ N\\\NN\\\\\\N\\"x"x'x.~l

0 A B Distance Y Fig. l. Spatial layout.

is linear, with a closed left-hand side at O, where the port is located. The edge city locates to the right of the port along the line at a distance chosen strategically. Each city consists of a business district, which is a point whose capacity is given by a level of infrastructure capital, K 0 for the port and K 1 for the edge city. The port city's level of infrastructure, K0, is given by history, while K 1 is chosen by the edge city developer. The port city's residential area occupies a line segment to the immediate right of O. The edge city's business district is at a distance y to the right of the port. Its residential area will occupy space to the right of y, and in some regions of parameter space to the left of y as well, as in Fig. 1. The distance y will be a critical choice variable. Residents in either city live on lots of fixed size and commute to work. The residential land market is competitive and land rents are remitted to absentee landowners. The endogenous number of resident-workers in the port city is A and they occupy a segment of the line of length A. Similarly, the endogenous number of resident-workers in the edge city is B and they occupy a length of line equal to B.

2.2. Agents, behavior, and history

The port city is passive. The location of its business district at the left-hand boundary of the region is fixed. The capacity of its business district, K0, is historically fixed. This implies that investments are irrevers- ible downwards; and that either there is no desire by atomistic investors to V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 619 increase K 0 (or, the port city's productivity of investment is less than the opportunity cost) or K 0 is irreversible at the margin upwards (investments are 'lumpy'--see below). We can think of capacity as vertical space--total high-rise office floor space--which is divided up among (identical) firms. Competitive firms in the business district hire workers on the basis of their required compensation and value of marginal product. The residual return to each port firm's share of infrastructure is paid to absentee atomistic capital owners, and the competitive firms earn zero profits. The edge city business district is owned by a single entrepreneur, the developer. By assumption, developers may own only one business district-- this permits competition among developers when there are multiple edge cities. This can reflect financial, institutional and/or technological con- straints, just as in the industrial organization literature when oligopolists do not buy each other out. Capacity for the edge city, K1, is chosen as a lump, all at once at one location, reflecting implicit economies of scale from concentrating investment at a point in time and space, rather than spreading investment out over time or space (Weitzman, 1970).3 The developer chooses a location for her business district at a distance y from the coast, a capacity for her district, K1, and a work-force B. The work-force may be set directly by the developer, through contracts with firms who rent 'vertical space' (a share of K1) and who each agree to employ a limited number of workers. Equivalently, the developer can implement the same B by carving up K1 into so-many business firms and charging a fixed fee per worker employed. This may be collected from the worker-residents, or the firms. Given the appropriate number of firms and fixed per-worker fees, the zero profit constraint will fix the firms' choices of employees at the desired B. By exhibiting control over aggregate edge city employment, the developer exercises monopsony power in labor markets, an important aspect of our model. The edge city developer chooses y, K 1 and B strategically to maximize profits. The developer correctly anticipates the reactions of the passive players in the model--port city competitive employers and renters and rentiers in land markets. Port city employers adjust their labor force in response to changing marginal productivities and opportunity costs of labor. Later in the paper we consider the impact of edge city developers also having to strategize against potential future edge cities, to either preclude their entry or minimize the damage they will do to profits and employment.

3 Below we specify investment costs as being prK1 for a lumpy investment K~. If K~ was achieved instead by a succession of k's over time or space, then the cost of each portion would be pKk ~', with 3' sufficientlygreater than 1 to rule out this option. 620 v. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643

2.3. Production technology

Firms in each business district are identical, with internal constant returns to scale technology, so we aggregate the firms in each business district and talk about district production. Firms interact with each other by com- municating messages, which affects their productivity. The communications could relate to knowledge or information transfers, or the firms could be producing information services which are intermediate inputs to other firms. These communications either decay with distance or, equivalently, the cost of conveyance increases with distance. Information decay draws firms to agglomerate together in space and potentially draws the port and edge cities closer together. Because we are focusing on 'Information Age' products such as R&D and modern services, information decay rather than shipping costs (for manufacturing) seems to be the relevant phenomenon. The spatial decay of information itself is documented in Jaffe et al. (1993) for R&D and is implicit in the industrial organization analysis of R&D firms. It is also implicit in the sharp slope to land price 4 and wage gradients (McMillen and Singell, 1992) in service- oriented business districts, where players on the fringe as opposed to the center are only willing to pay much lower prices because they are removed from the 'center of action'. In this paper we ignore the internal spatial structure of business districts but, at least, model information decay between business districts. It is simplest to treat information spillovers as external economies of scale to the firm. Firm efficiency depends on the given volume of undecayed messages received from other firms. Kim (1988) proves that external economy of scale formulations can be derived as reduced-form outcomes, where the volume of messages is a firm choice variable. Firms have a derived demand for messages from other firms, where messages are costly to purchase and move through space. To conserve on space, we adopt the reduced-form outcome. For the port and the edge city, respectively, outputs are

output (port) = Q(A + Bm(y))'Ko AI-'~ , (la)

output (edge) = Q(Am(y) + B)'K~B 1 '~ , (lb)

4 In Henderson and Poole (1989) a land price gradient for the modern service-oriented Providence CBD is given for the mid-1980s. Price per square foot for vacant land varies threefold between the CBD center and 0.4 miles away. V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 621 where

m(y) = max[O, 1 - cy], (2)

Q is a constant term, ~ is the degree of economies of scale associated with information received, A and B are employment levels in the port and edge city, respectively, K 0 and K 1 are capital stocks in the port and edge cities, a and 1 - a are factor shares, y is the distance between business districts, and c is the decay rate per unit distance of messages. The use of m(y) indicates that if y is too large, then the business districts move out of communication, where 1 - cy < 0 and m(- ) = 0. So for the port, each firm gets a volume of communications proportional to A from other port firms; but the volume from the edge city is discounted by cy, so in total B(1- cy) are received, unless 1 - cy ~ O, in which case no messages from the edge city are received. Total net communications improve firm efficiency according to a degree of scale economies, ~. We utilize a linear rather than exponential decay function because exponential decay functions favor bipolar outcomes (see Henderson and Slade, 1993), and linear decay allows the edge city to move totally out of communication, which will turn out to be an interesting strategic choice. For Eqs. (la) and (lb), production efficiency in both the port and edge cities is enhanced by lowering y. However, the developer's choice of y occurs in a monopsonistic situation (see later), where from the developer's point of view, lowering y has the undesirable side-effect of enhancing the labor market power of the competitor port city. That consideration plays an important role in the analysis developed in the paper.

2.4. Commuting costs and land rents

Commuting and land rent levels experienced by residents depend on the exact spatial configurations involved. The opportunity cost of land to the region is zero. Land is owned by passive absentee rentiers. The cost of commuting a unit distance (there and back) is t. In each city the rent gradient is determined competitively so that workers have no incentive to switch residential locations. These assumptions, along with the fixed lot size assumption, yield standard linear rent gradients. Fig. 2 illustrates the three sets of possible regimes that arise in our context. Further division of these regime sets, into corner vs. interior solutions, yields a total of six regimes. Once the developer has chosen y, K 1, and B, residents of the core and edge cities locate themselves in competitive land markets. Thus, generally a unique residential spatial equilibrium and 622 V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643

I. No Cross-Commuting: Symmetrical edge city a.) Exterior location : out of communication I-cy=O - 1/2B

Rent Gradient

O-----v---- A I/~e Y i/Ee y Distance

b.)ln communication: touching borders I-cy_>O ~; Regime Ib y= A+ I/2B Rent Gradient

O -----~ A I/2B Y I/ZB 2. No Cross- Commuting Y Distance b. Interior solution: asymmetric edge city $ Rent Gradient I-cy>O y < A + I/2 B

0 ~- -e, Y B

Y Distance

Fig. 2. Urban configurations.

regime result, foreseen by the developer (see later). The three sets of spatial regimes are related to how close the developer locates to the port, or how small y is. With distant locations, the port and edge cities will have independent residential land markets. At intermediate locations, the two residential land markets will be interdependent. At close-in locations, the two cities will share a common residential market. For given edge and port city populations, we will show that as y moves inward, the required per-resident expenditures on land rents and commuting costs will rise, which is a disadvantage for the developer in reducing y. The three sets of spatial regimes are depicted in Fig. 1 and defined next. The rent gradient (rent per unit land) for each configuration is illustrated. V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 623

3. Cross- Commuting a.)Boundarysolution: one sided edge city $ Rent ~ Regime3a Gradient

I-cy>O

commute to port

O' ~ "y ~ Distance b.) Interior solution: edge city as a neighborhood $ Rent ~ Regime 3b Gradient

I - cy > O ~ y

0 A B" Distance c.) Monocentri city Rent $ Regime 3c Gradient ~ I-cy>O

oy A+B Distance Fig. 2. (Continued).

The derivation of rent gradients and total per-resident expenditure on commuting plus rent costs are footnoted. (1) Regime set 1: Distant location, with symmetrical edge city. In Fig. 2, part 1, symmetry requires y ~>A + (1/2)B, so land markets in the two cities are independent. The A residents of the port commute inward to the port. The B residents of the edge city are located symmetrically about y and commute to y. The maximum commuting distance from the edge city edge is (1/2)B. Rent at the edge city edge is zero. The sum of the person 624 v. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 rents plus commuting costs, which equals the height of the rent gradient at y (where that resident has no commuting costs since he lives next to work) is 5 R 1 = (1/2)Bt. We distinguish two cases, with symmetrical edge cities: (a) The edge city is out of communication, as in part (la) of Fig. 2. y is sufficiently large that 1 - cy <~ O, in which case the expression is bounded, so m(y) = O. (b) The edge city is in communication range, but remains symmetric with borders touching, as in part (lb), of Fig. 2. In theory, the edge city could be in communication without borders touching, but we have been unable to isolate an equilibrium example of this. (2) Regime set 2: Intermediate location and asymmetric edge city. In part 2 of Fig. 2 we have a 'cross-commuting: interior solution', where the residential land markets impinge on each other because the edge city moves towards the port to enhance received communications. This inward move gives the edge city an asymmetric shape, where y < A + (1/2)B. In the line segment B to the right of e, all residents commute to y. To the left of e, where the land markets intersect, all A workers commute to the port. For the two cities, per-person rents plus commuting costs are given in Table 2.6 By inspection of Table 2, the move from regime 1 to 2 increases rents plus commuting costs, ceteris paribus, given R 0 = t(3A + B - 2y) > tA for y < A + (1/2)B, and R 1 = t(A + B-y) > (1/2)Bt for y

5 The person living at A from the port must have the same rent (zero at the boundary) plus commuting costs, tA, as a person at a distance s from the port with rents R(s) and commuting cost is, so that both have equal utility levels (equal expenditures on all other goods after commuting costs and rents on the lots of fixed size). Thus R(s) = t(A - s), and R(0) -= R 0 = At. The port has a rent gradient with slope -t, which rises from zero to At at the city center. Maximum rents, maximum commuting costs, and the sum of rents plus commuting costs for any resident, denoted by R0, are each At. Edge cities will be symmetrical. Given that, by the same reasoning, maximum rents (at y), maximum commuting costs (at either edge), and the sum of rents plus commuting costs per person, denoted by R 1, are (1/2)Bt. 6 In the edge city rents start at zero at the right edge and rise to (A + B - y) • t at location y. A + B - y is the maximum distance commuted from the right edge (at distance A + B) into y. Total rents plus commuting costs, R 1, are (A + B -y)t everywhere in the edge city. Rents then fall from y to e as workers commute increasingly longer distances out to y. At e the maximum commuting distance out to y is y - e, so rents decline by t(y-A) from (A + B - y)t (at y). Rents at e are thus t(2A + B - 2y). At e the maximum commuting distance to the port is A, so rents plus commuting costs in the port are t(2A + B - 2y) + tA, and thus R 0 = t(3A + B - 2y). V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 625

Table 2 Per-person rents plus commuting costs (height of rent gradient at y or 0) Port city Edge city (So) (s,) Regime set 1, no cross-commuting: tA ½tB independent land markets

Regime set 2, no cross-commuting: t(3A + B - 2y) t(A + B - y) land markets intersect

Regime set 3, cross-commuting: t(A + B) t(A + B - y)

2 for each city. From Table 2, for a given A and B, the move from regime 2 to 3 raises rents plus commuting costs per person, as y moves from a value greater than A to one less than A. For regime set 3, we distinguish three cases. (a) The edge city is one-sided at a boundary solution, where all workers to the right of the edge city commute to it and all workers to the left of it commute to the port. At such a boundary, while having moved closer to the port, the edge city retains some 'independence'; all activity at y and beyond is its own activity. (b) The edge city is now internal, so, in some sense, it is just a neighborhood and there is actual cross-commuting. Of those people to the right of y, B go to y and A - y go to the port. This is cross-commuting since some residents (living beyond y and going to the port) commute past others' work places at y.7 Residents to the left of y all commute to the port. Thus the edge city is a 'neighborhood' within the realm of the port city, as defined in Muth (1969). (c) The edge city moves all the way into the city center, simply adding office space to the existing port. The whole region is a monocentric city, where the edge city has little 'independence'. In summary, for regime switches, for the same A and B, ceteris paribus, moves inward increase rents plus commuting costs. We state 'ceteris paribus' because this is not a comparison of different equilibria (for any A and B, generally there will be only one equilibrium regime), but a comparison of calculations of R 0 and R~.

7 A cross-commuting pattern where there is also a region to the left of y, where workers may commute either in or out, cannot be sustained by an equilibrium in land markets. 626 V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996)~613-643

2.5. Worker supply

The urban region in which the edge city and port are located is effectively a small congested open economy, subject to migratory flows. As a general case, we assume there is an increasing opportunity cost to the region of hiring worker-residents, where required per-resident compensation, or wages net of rents and commuting costs is given by

opportunity cost = (A + B) ~ , z > 0. (3)

Because of potential communications among adjacent cities, when multi- ple cities form there can be on-going net economies to increasing regional population. With a constant opportunity cost of labor, that can create analytical problems (Henderson and Slade, 1993). An increasing oppor- tunity cost can represent a regional congestion function related to quality of life, or just simply the increasing costs of hiring workers from other regions. This increasing opportunity cost of labor function also gives the edge city developer her degree of monopsony power, since she chooses her own total employment. For completeness, we note that the shape of the labor supply function, (A + B) ~, and the form of technology (e.g. Q, E, and a) in Eq. (1) describe the strength of this urban region in competing for workers with other regions. Regions may be specialized, as in system of cities and have different technologies (Henderson, 1974) and/or have different amenity endowments affecting the shape of (A + B) z. These factors will determine the general size of this region compared with other regions. In that context, we examine how large the edge city will be relative to the port city. But if we were to look across regions, edge city sizes relative to each other would also be affected by overall regional characteristics (the values of Q, e, a, z, etc.).

2.6. The maximization problem

The edge city developer chooses location, y, capacity K 1 and employment B so as to maximize profits. The developer's profits are

H = Q(Am(y) + B)~K"B '-" --prK -- B(R, + (A + B)~). (4)

The first term is total business district output from Eq. (1), the second is the cost of edge city capacity, where the per-unit cost of capital is Pr, and the third term is total labor costs. Note that the price of business district output V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 627 is normalized at 1. Total labor costs are the total edge city labor force, B, multiplied by the per-person required labor compensation. The latter consists of the per-unit opportunity cost of labor, (A + B) ~, which is the consumption of all other goods in Eq. (3) that labor must net, plus the per-person residential cost of living in the edge city, R 1. The developer maximizes (4) subject to constraints. The first constraint involves the determination of the port city work-force, A, where the port city wage equals the per-person cost of labor. In the port city, wage equals the value of the marginal product from Eq. (lb), Q(1-a)(A+ B(m(y))) KoA . Labor cost is the per-person net opportunity cost, (B + A) z, plus R 0 in Table 3. The constraint is

(1 - a)Q(A + Bm(y))'Ko A-'~ - (R 0 + (A + B) ~) = 0. (5) The developer is subject to non-negativity constraints and the constraint that the contribution of the port to edge city productivity never becomes negative. A, B, y, K 1/> 0, (6) re(y) = max[0, 1 - cy]. (2) Finally, there are inequality constraints according to the spatial regime, which also ensure the validity of the R 0 and R~ specification in Table 3 for that regime. In regime 1, parts (a) and (b), where port and edge cities have independent residential areas, the constraint is

y>~A+(1)B. (7, i)

This is binding in part (b) of regime 1. In regime 2, where there is still no cross-commuting but an interior solution where residential areas intersect, the constraint is

y<~A+(1)B. (7, ii)

Again this binds at the limit where regime 2 becomes regime l(b). Finally, for the cross-commuting equilibria in regime 3, the constraint is y ~< A. (7, iii) This constraint binds in part (a) of regime 3. Given (7,i)-(7,iii), we have a maximization problem, with switching regimes of constraints. To solve the problem, we solve three separate maximization problems, and evaluate outcomes within each regime when constraints (5)-(7) are binding. That yields a total of six solutions. We then calculate total profits for each of the solutions and pick the one with the maximum profits. 628 V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643

3. Solutions

The edge city developer directly chooses a location, y, a capacity, K1, and an employment level B. The choices depend on the strength of the core city, as represented by its historically given capacity, K 0. To understand the forces at work in the model, we analyze how edge city choices are affected by changes in the port city history, K 0. Specifically, we analyze experimen- tally edge city choices of K1, y, and B, as K 0 is increased. To understand the economic forces at work, we first consider the choice of y. For a given historical capacity, in choosing y the developer is choosing to what extent she wants to interact with the core city. There are three considerations. (1) From Table 2, as the developer lowers y and moves her center towards the port that raises the commuting and land rent costs which her workers incur and must be compensated for. So it raises her costs. ~(2) On the other hand, lowering y means that communications with the port are less decayed and edge city productivity rises. (3) But, as a third factor, lowering y also increases port city productivity and port city competitiveness, thus reducing the edge city's monopsony power. These three forces define the tradeoffs the developer makes in choosing her location, y, along with her capacity and employment. As we will see, the strategic choice of y, as port city historical capacity, K0, is increased, varies in a very complex manner given the three forces at work. In particular, an infinitesimal change in K 0 can illicit a large discrete change in y. In terms of the developer's choice of her capacity and employment, the situation appears a little more straightforward. In general, as K 0 is made larger, the edge city developer will choose a smaller development with lower investment, KI, and employment B. Why? The higher K 0 reduces the potential monopsony power of the developer in competing for labor, for a given regional supply function of labor. However, as we increase K 0, the decline in K 1 and B is non-monotonic. K1 and B may increase discretely if y varies discretely in response to an infinitesimal change in K 0. Our model is too complex to solve analytically, given any solution for a particular set of parameter values, and K 0 involves comparing profits across six potential regimes, some of which for those parameter values may not have an interior solution. We explore the properties of the model through simulation. The simulation results suggest fairly clear general patterns of the type appropriate to simulation: the existence of unusual features to out- comes. Edge city choices fit that description. In the simulations, for any set of parameter values we start with K 0 at a miniscule level and raise it incrementally until it reaches a value where edge V.. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 629 city profits become non-positive. 8 As noted earlier, from the six potential solutions for any K 0' we pick the profit-maximizing spatial configuration. We then vary K 0 to see how the outcomes are affected. In essence we are presenting a set of comparative static results for each region of parameter space to see how A, B, K1, and y vary, as K 0 values, or core city history, varies. To keep the number of comparisons within reason, we fix the non-critical parameters at Q=8, at=0.3, E=0.125 and PK=0.10. Q=8 ensures sufficient profits for the edge city to exist in this region of parameter space. a and • are chosen as 'reasonable' empirical magnitudes for labor share and the degree of scale economies. PK = 0.10 is a normalization relative to the price of output (= 1), affecting the scale of measure of K units. The opportunity cost of labor parameter, z, is also fixed at one, although we report some results later in the paper for higher values of z. That leaves the spatial decay and commuting cost parameters, t and c. These are varied to illustrate the potential for all spatial regimes in Fig. 2. In the text, our results are summarized graphically. The appendix illustrates the numerical simulation results for one set of parameter values. A more complete presentation of the numerical simulation results are in a longer version of this paper (Henderson and Mitra, 1993). We start with the results concerning the edge city's choice of location, not because it is the 'most important' decision, but because it helps to define the nature of capacity and employment choices.

3.1. Location choices

The most unexpected result in our work concerns how the edge city's optimizing choice of location changes as K 0 is increased. The key features are contained in Fig. 3, parts (a)-(d). In Fig. 3, for visual effect, we put location on the horizontal axis and port city capacity on the vertical axis. The graph illustrates what happens to y as K 0 is increased, y is bounded above as shown, at its value where 1 - cy = 0. Beyond that point, re(y) = O, so y is indeterminant and irrelevant; the edge city is out of communication. Fig. 3 gives an overview result: there is a zig-zag or 'chaotic' pattern to location choice. In Fig. 3, regimes are labelled. The values of y within a regime as K 0 varies are depicted by solid lines. The switch points for regimes occur along the dashed lines, with the two values of y at the switch value of K 0 marked by dots. As K 0 is increased, y may increase or decrease

8 Within the context of our region (given regional Q, a, e and z), as the port size increases generally the edge city size declines. Across regions, edge city sizes may rise with port city sizes according to the historical regional configurations of/(o'S, Q's, z's, etc. 630 V.. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643

(a) (c=.03,t=.15) 4600 33 out of communication threshold regime 3c 4000 % ~_~_.?_b____

~ 3000

o u me 30 u~ 2000 ~ I000

regime 1o

0 IO 20 30 40 50 Distonce (y)

35 out of (b) regime 3c (C=.081,t=.45) communication threshold

A 4000 0

>, regime Io "5 g 3000

¢.) 2000 n

I000 regime 1o

regime Ibm-"

I 2 3 4 5 6 7 8 9 I0 II 12 13

Distonce (y)

Fig. 3. (a) Location choices (c = 0.03, t = 1.5). (b) Location choices (c = 0.081, t = 0.45). (c) Location choices (c = 0.06, t = 0.65). (d) Location choices (c = 0.06, t = 0.25). continuously within a regime; but then, with a regime switch, y may change 'catastrophically' in a spatial sense, not unlike the results in Krugman (1993) or the older non-economic urban models of Forrester (1969). By spatial V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 631

(C) (c=.06, t= .65)

4600 16.6 out of communication regime la threshold ------4 4000 A 0

~ 3000

U 2O0O t,. no 1000 °Z

regime I0

0 ~ 6 9 12 15 18 Distance ( y )

( c:.06, t =.25)

(d) 16,6 out of 4600 communication threshold

4000 % regime 3c

o 3000

o U

u 2000

o

regime Io 1000 regime 2 q;~..~'~___~--_ -:

regime la

0 0 3 6 9 12 15 18 Distance ( y )

Fig. 3. (Continued).

chaos we mean not just that y may jump incrementally in or out with an infinitesimal change in K0, which induces a regime switch, but that the edge city may jump from some interior location outward, to the extreme where it 632 V.. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 is out of communication, and then at a modestly higher K 0 jump back in again. It is particularly instructive to look at parts (b) and (d) of Fig. 3. In part (b), while the edge city is in communication at low values of K 0, twice as K 0 is increased it moves strategically out of communication. In the second episode this occurs when the succeeding regime moves all the way into the port. So the edge city goes from complete flight to complete surrender with an infinitesimal increase in K 0. The zig-zag pattern is a fascinating general feature of location choice, which is achieved by having large development agents in land markets. It is not surprising that Garreau is vague about the determinants of edge city locations, suggesting that they are accidents of where particular highways intersect. Yet underlying the zig-zag patterns are strategic choices by the edge city developer, involving very consistent patterns of behavior. The specific patterns of locational choice as K 0 is increased in all simulations are as follows. The first two points deal with regime switch points and the last deals with how y varies within each regime as K 0 is increased. (i) With one critical set of exceptions, the pattern of regime switches starts at regime l(a) when K 0 is small and then move's successively inwards at regime switch points from l(a) to l(b) to 2 to 3(a) to 3(b) to 3(c),9 in Fig. 2. That is to say, the edge city starts off strategically out of communication (regime l(a)) when the port is weakest and moves inward with successive regime switches as K 0 is increased. Ultimately, in regime 3(c), when K 0 is very large, the weak edge city is sucked into the port city business district, becoming just another downtown office building. However, in any region of parameter space, some regimes may not occur. That is, as K 0 is increased, in the sequence of regimes some regimes may be skipped (see the Appendix for an illustration). (ii) There is a critical exception to the neat pattern laid out in (i). When moving between any pair of regime switches in (i), the first regime, l(a), can intervene, with the edge city moving strategically out of communication. Why? As K 0 is increased within a regime, edge city rents plus commuting costs build up. Before switching further inward to enhance communication, the edge city may flee outwards to revert to minimization of rents plus commuting costs. 1° This phenomenon is partially responsible for the zig-zag

9 We did not find a dominant solution where y is chosen so the edge city is in communication (1 - cy >0) but borders are not touching, y > A + (1/2)B. This solution was, in one instance, within the limits of accuracy of the simulations, equivalent to a corner (either 1 - cy = 0 and y > A + (1/2)B or 1 - cy >0 and y = A + (1/2)B), but never dominant. 10 In addition, when we are within a 'corner' configuration as in regime l(b) or 3(a), the growth of the port pushing edge city out may push the edge city out of communication. V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 633 pattern in Fig. 3. For example, in Fig. 3(b), in moving between regime 3(b) and 3(c), before the port city neighborhood is sucked into the port business district, the edge city flees outwards, moving out of communication into a small independent city with very low rents plus commuting costs. 11 But when K 0 is increased further, the edge city becomes too weak (too small in terms of its own isolated external economies) to sustain itself, and at some critical K 0 is sucked all the way into the port city business district. (iii) Within regimes, three different patterns obtain. First, if the edge city is at one of the two corner regimes---either a symmetric shape with touching borders in regime l(b) (y = A + (1/2)B) or a one-sided shape in regime 3(a) (y = A)--then as K 0 is increased within that regime, the port city population and area grow, pushing the edge city outwards. Thus, within a corner regime, as K 0 increases, y grows. Secondly, and in contrast, within an interior regime, where the edge city is either asymmetric in shape (regime 2) or a neighborhood (regime 3(b)), as K 0 is increased, the edge city moves inwards and y declines. That is to say, within regimes, as K 0 is increased, for the interior regime vs. corner regime, changes in y are opposite in sign. This also helps to contribute to the zig-zag patterns in Fig. 3. Finally, and obviously, if the edge city is out of communication, as in regime l(a), as K 0 is increased within the regime, y remains at its boundary (y >1 1/c). Also, if y = 0 in regime 3(c), then further changes in K 0 have no impact on y. Finally, we comment on the comparative static effects for any K 0 of changing the commuting cost parameter, t, or the spatial decay parameter, c. Since outcomes follow such a chaotic pattern with regime switches, it is difficult to do general comparative statics. However, as explained in more detail in Henderson and Mitra (1993), we can state that for the spatial decay parameter, c, it appears that the impact of increasing c is simply to draw locational choices in, because the loss of productivity sharply reduces regional population. Also the boundary where re(y) = 1 - cy = 0 moves in. 12 In terms of comparative statics for changes in t, we could not make general statements.

11 It is important to note that as K o is varied over the regimes 3(b), 3(c) and l(a) in Fig. 3(b), the model solves for all three regimes throughout and we are picking the regime for the various K 0 that yields the highest profits. That is to say, the discontinuities and catastrophic changes in y do not occur because the model fails to solve for some of these regimes as K 0 varies. They are part of the optimizing behavior. 12 This inward shift can be seen in Fig. 3, where the scale of the bottom axis, y, changes as c changes. For low c (part (a)), maximal distances are much greater, as are the distances traversed. For high c (part (b)), all locational choices take place over a much smaller distance interval. 634 V. Henderson, A. Mitra / Reg. S¢i. Urban Econ. 26 (1996) 613-643

3.2. Edge city profits, capacity, and labor force

The switches in locational regimes set the stage for analyzing the patterns of edge city capacity choices because they are influenced by port history. For small K0, the edge city invests in a large capacity and chooses a large employment relative to the core city. As K 0 is increased, the edge city is disadvantaged and, in all simulations, profits decline monotonically. How- ever, the edge city uses its capacity and employment choices to 'resist' (in a comparative statics sense) the increased potential of the core. With respect to capacity, in general K 1 and employment, B, decline as K o is increased, but the decline is non-monotonic. First, in all simulations, K~ and B move together, as one might expect. Secondly, at certain locational regime switches K 1 and B may jump up a modest amount, rather than down; and, within regimes, on occasion K 1 and B may actually increase rather than decrease over some interval. Finally, there appears to be no strict pattern between location and capacity choices when K 0 is increased. While K 1 may increase as K 0 is increased, in a switch to the out-of- communication regime for example, it may also decrease. In summary, the key result is that while K 1 and B generally decline as K 0 is increased, the decline is neither monotonic nor continuous. For one set of parameters (corresponding to those in the appendix), the values of A, B, K~, and H are plotted against K 0 in Fig. 4. Further plots of K~ against K 0 are given in Fig. 5. One notable aspect of Figs. 4 and 5 is that the edge city may maintain approximately the same capacity for large variations in port city capacity. While developer profits always decline monotonically, the developer's strategy can be basically to maintain size and only move location as K 0 is increased. For example in Fig. 4, during regime 3(a), K 0 increases by 200%, while K 1 declines by only 11%. In fact, at the end of regime 3(a), where K 0 ~-3800, K 1 is larger than at K 0 ~-1200 (just before the switch into regime 3(a)). During regime 3(a) the edge city moves significantly, increasing its distance from the port by almost 70%. In Fig. 5, the same results apply. This would suggest that core cities with very different capacities and populations could have edge cities with very similar capacities, or square feet of office space. However, as K 0 gets relatively very large, at some point in the regime switch there is usually a precipitous drop in edge city capacity. Finally, we note how regional population, the sum of port and edge city populations, changes as K 0 is increased. First, as K 0 is increased, in general, regional population increases, because the monopsonistic power of the edge city declines. Since the impact of monopsony is to restrict hiring, a weakening of monopsony power should result in employment expansion. However, the impact of regime changes is also to quite dramatically alter regional population. Fig. 6 illustrates two typical parametric cases. Regional V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 635

.-- 3000 U Edge City Copocity o o ¢.) 2000 1 I I I I U I000 e-----___.._ KI

"1o I.,rJ 0 I I I I I 30 Populo t ions

t- 2C A

°mO

o n I0 ?

0 , I i I | Profits 200 ~,,,~e g i me Io 150 Oq- ~,~eg i me Ib I00 ,.~,.~ m e 2

50 regime la 0 i i t | 0 ~ooo zooo 3000 4000 4600 Port City Copocity (K o) Fig. 4. Edge city profits, capacity and population (c = 0.06, t = 0.65, z = 1). 636 V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643

4000 C=.06,t =.65 C-.06, t=.25 h 3000 ...... , ...... ~ ~ ...... c..o8,.,=.,5 ----,.o ...... t : I _>'2000 '~'L ' ...... 1 : I |

I000

0 I000 2000 3000 4000 4600 Port City Population (K o)

Fig. 5. Edge city capacity choices.

*~'S~ I j. A+B 'r" I A+B 01 /./I ~. ~" "J' o

Q. 0 0. 20 (c=.06, 1= .65 )

---- (c=.08l, t=.45)

15

I I ! I I Jooo 2000 3000 4000 4600 Port City Copocily (K o)

Fig. 6. Regional population. V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 637 population can experience big shifts up and down with regime switches and can decrease during a regime.

4. Uncontrolled development

How do our edge city configurations compare with those where edge city development is uncontrolled? That is to say, what happens if there is no developer who controls investment and job levels in the edge city business district? To make the comparison for any port city historical capacity K 0, we use the same location as in the developer model. There is no competitive determination of y in a static setting. So we pick the same y values as the developer picked for each K 0 and observe what happens to edge city sizes and capacities if they are competitively determined. An alternative would be to perform a Krugman (1996) like experiment. We could start with small random edge city populations and capacities at, say, two dozen potentially fixed y locations, specify naive dynamic adjustment mechanisms for both population and investment flows among the y's and the port, let the model run, and observe which of the potential y's is the winner--the edge city where population and investment ultimately agglomerates. Here we content ourselves with the simpler experiment with a single predetermined y (which would be the winner in a Krugman experiment if enough of the initial arbitrary allocation of employment and investment across potential y's was massed there). The model is solved by using marginal productivity equal to required compensation conditions to infer A, B and K 1. These conditions are

Pr = aQ(A(1 - cy) + B)'KI-IB 1-~ , (8a)

(A + B) ~ + g I = (1 -- a)Q(A(1 - cy) + B)'K~B-~' , (8b)

(A + B) z + R 0 = (1 - a)Q(A + B(1 - cy))'Ko A-'~ . (8c) Eq. (8a) equates the marginal product of edge city capital to the price of capital. Eqs. (8b) and (8c) equate the marginal product of labor in the edge and port cities to labor's required compensation--its opportunity cost in terms of all other goods plus its expenditures on rents plus commuting. In solving the model, as before, the determination of R 0 and R~ from Table 2 is critical. Table 3 gives three sample results, comparing the edge city developer outcome with the outcome under uncontrolled development. The top part of the table lists the common parameters and the K 0 and y values, as well as profits to the developer under the developer solutions. The rest of the table compares developer values (top part) for A, B, K1, the regime, and the 638 V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643

Table 3 Competition: Urban Sprawl

Developer outcome

Parameters K 0 y II Return to A B K l Regime capital in port (1) c = 0.06 t = 0.45 400 14.6 166.5 61.7 5.5 18.2 2887.5 lb (2) c = 0.03 t = 0.15 2500 16.6 57.2 268.2 16.6 16.1 2677.0 3a (3) c = 0.03 t = 0.15 3500 7.2 22.3 392.4 23.8 9.7 1650.6 3b

Competitive outcome Parameters A B Return to K~ Regime capital in port (1) 1.8 30.5 27.1 5284.0 2 (2) 11.9 25.4 212.1 4406.3 2 (3) 18.4 17.9 329.4 3115.3 3b

5000

4000

v 3000 KI 8 Ko 2000

I000

0 • 0 5 I0 15y 20 25 30 35 40 45 50 Distance

(a) Controlled Development

5000 Kj 4OOO

3000

U Ko 2000

1000

O • I3 "\NI N\ \ \\\\\\\\\\\\\\\\\~ I Q 5 IO 15y 20 25 30 35 40 45 50 Distance A B (b) Uncontrolled Development

Fig. 7. Controlled development versis sprawl (c = 0.03, t = 0.15, K 0 = 2500, y = 16.6). v. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 639

return to port city capital owners with those for the no-developer, or competitive outcome (bottom part of Table 3). These examples and a variety of others we calculated illustrate a consistent theme, which is illustrated in Fig. 7, for example (2) of Table 3. Control by the edge city developer constrains edge city and regional populations, as well as edge city investment in infrastructure. Loss of that monopsonistic control causes regional population to expand, with edge city population and capacity increasing dramatically and port city population declining, although less dramatically. The result is 'urban sprawl'--enormous edge communities that encroach upon the traditional city. A second consistent result concerns profits. By definition of the equilibrium in the competitive outcome, edge city profits fall to zero. But the core city's 'short-run profits', or residual return to capital, also decline. While this may not universally be the case, since the port could gain if it does not have to compete against a monopsonist in labor markets, in fact we could not construct an example where the port was better off when the edge city behaved effectively as a competitor. Often, the move from controlled to uncontrolled development causes a regime switch owing to the expansion in edge city and regional population. In example (2) in Fig. 7 we switch from regime 3(a) (one-sided edge city) under developer control to regime 2 with an asymmetric edge city under competition. In example (1) in Table 4 with a small K 0 we move from a symmetric edge city in regime l(b) under developer control to an asymmet- ric city in an interior solution in regime 2. Edge city population increases by 75%; port city population declines by 65%.

5. Extensions

5.1. Segregation by income

If workers are heterogeneous in terms of skills and hence are paid different incomes, we can expect the edge city developer to strategize to obtain the best mix of workers for herself. Whether these are relatively high-skill or low-skill people will depend on two considerations: (a) differences in lot size consumption (and implied per-person rent plus commuting costs within the edge city) and (b) differences in information spiUovers associated with high- vs. low-skill workers in the production process. The developer potentially can influence skill mix in the edge city in one of two ways. She can either influence skill composition in the work-place by designing/designating office space appropriate for high- vs. low-skill service workers, or, if the residential government for the edge city is private and thus part of her company, she can design a package of public service 640 V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 qualities and charges that will appeal only to higher skill workers, such as in a product vertical differentiation model. While these are non-trivial phenomena to model, especially if the spatial dimensionality of the problem is to be maintained, it is clear that edge cities act strategically to separate off segments of the population. Our understand- ing of segregation would be improved by modelling these edge city activities.

5.2. Multiple edge cities

What happens if there are multiple edge cities? One might reason that, with multiple edge cities, the type of zig-zag location patterns illustrated in Fig. 3 for a single edge city developer would be greatly constrained. This turns out not to be the case. To show this, we started to model location choices when there are two edge city developers, in a multi-stage game. At stage one, developer 1, chooses location Yt and capacity K 1. At stage two, developer 2 chooses location Y2 and capacity K 2. At stage 3 the two developers choose their employment levels, B and C. At stage 4 competitive markets clear, thus determining the anticipated port city employment, A, and residential rent gradients. The first developer strategizes against both the passive port and the second developer. Imposing subgame perfection on a simple extensive form game, from Henderson and Slade (1993) we know that generally the first developer, by strategic choices of yl and Kt, can preclude entry of the second developer through a strategic choice of 'excess' capacity, K~. This strategic choice is costly, but leaves the first developer as the sole edge city. The alternative to preclusion is acquiescence, where y~ and K 1 are chosen to permit entry. While these acquiescence choices are less costly per se (i.e. K~ is smaller), the first developer must share the market with the second. The first developer compares the outcomes (her profits) under preclusion with acquiescence and chooses that branch of the game tree that yields the highest profits. With two edge cities and one port, the number of spatial configurations escalates to 23, which have to be evaluated under both preclusion and acquiescence. Given this, the number of stages in the game, and the complexity of the reaction functions, programming the entire problem was beyond the scope of this work. However, we did program a couple of acquiescence situations and thought about the problem in general. What did we learn? To preclude entry, the first developer might opt for extreme choices. She might locate far from the port, so a potential entrant could not communicate with both her and the port. Alternatively, she might locate within communi- cation in a symmetric shape, but not with borders touching. The space V. Henderson, A. Mitra / Reg. Sci. Urban Econ. 26 (1996) 613-643 641 between the port and the first edge city would be too small for a potential entrant, so this would force a potential entrant to locate beyond the first edge city, out of communication with the port. Under acquiescence, to enhance its first mover advantage, the first developer could also locate out of communication or could locate close in, to take greater advantage of the port's communications. But that would change as K 0 changes. Experimental results under acquiescence indicate an even richer pattern of regime switches than with a single edge city, where small changes in port city capacity, K 0, can prompt various 'catastrophic' changes. As before, the first edge city can also move from a location very close to the port, beyond communication, and then back again within a small interval of K 0 values. Indeed, having a second developer 'constrains' the first developer's choices, but that enhances the potential for location and capacity choices to zig-zag.

Acknowledgements

We acknowledge the helpful comments of William Strange on various drafts of this paper.

Appendix: Simulation results

Each table is divided into two parts. In part (a) edge city profits are reported for each value of K 0, for each spatial configuration for which there is a solution. NS marks cases where there is no solution. Neg stands for solutions with negative profit. For each K 0' the dominant (highest profit) solution is boxed. In part (b) sizes (A and B), edge city capacity (KI) and distance (y) are recorded for these dominant boxed solutions. IN marks the cases for regimes 1, 3, and 7 where y is indeterminant--a value such that (1 -cy)<~ O, with (1 -cy) being bounded at zero in all equations. In Table A the different regimes mark the following specified spatial configurations: • Regime 1: Strategic no communications (regime l(a)). • Regime 2: Symmetric edge city, communications (regime l(b)). • Regimes 3 and 7: Symmetric edge city, no communications (regime l(a)). • Regime 4: Interior asymmetric city (regime 2). • Regime 5: One-sided edge city (regime 3(a)). • Regime 6: Interior cross-commuting (regime 3(b)). • Regime 8: Monocentric city (regime 3(c)). 642 V. Henderson, A, Mitra Reg. Sci, Urban Econ. 26 (1996) 613-643

Table A(a) K 0 Profits No cross-commuting Cross-commuting Regime Regime Regimes Regime Regime Regime Regime 1 2 3 and 7 4 5 6 8 100 202.30 193.25 NS NS 153.76 NS 137.30 250 169.13 164.64 NS NS 137.46 NS 109.39 400 146.98 145.29 NS NS 125.45 NS 90.41 600 125.47 125.55 NS NS 112.72 NS 71.87 800 109.11 109.57 NS NS 102.18 NS 57.74 880 103.52 103.87 NS 104.19 98.39 NS 53.00 960 98.35 98.49 NS 99.69 94.75 NS 48.66 1120 NS NS 89.08 92.13 88.11 NS 41.10 1200 NS NS 84.90 88.88 85.26 NS 37.61 1300 NS NS 80.01 NS 81.22 NS 33.72 2000 NS NS 53.55 NS 58.86 NS 14.75 3800 NS NS 16.07 NS 17.35 NS 0.00 4400 NS NS 8.47 NS 6.08 NS Neg 4600 NS NS 6.29 NS 2.48 NS Neg

Base parameters: Q=8;a =0.3;e=O.125;px=O.1;z= l

Varied parameters: c = 0.06; t = 0.65 Regime 1: Strategic no communications. Regime 4: Interior asymmetric city. Regime 2: Symmetric edge city, communications. Regime 5: One-sided edge city. Regimes 3 and 7: Symmetric edge city, Regime 6: Interior cross-commuting. no communications. Regime 8: Monocentric city.

Table A(b) K 0 Converged values: A, B, y, K~ A B y K~ 100 1.758 17.296 IN 2696.111 250 3.758 17.079 IN 2656.289 400 5.265 16.581 IN 2565.244 600 7.410 15.273 15.047 2347.700 800 8.457 14.948 15.931 2280.202 880 8.575 13.638 13.373 2080.646 960 8.696 12.979 12.030 1981.618 1120 8.960 12.095 10.273 1849.570 1200 9.088 11.478 9.592 1797.962 1300 8.283 13.423 8.283 2098.680 2000 10.160 13.119 10.160 2040.530 3800 13.646 12.310 13.646 1865.845 4400 19.977 8.119 IN 1105.717 4600 20.413 7.793 IN 1053.580

Q=8; t~ =0.3; E =0.125; pK =0.1; z=l c = 0.06; t = 0.65 V. Henderson, A. Mitra I Reg. Sci. Urban Econ. 26 (1996) 613-643 643

References

Beckmann, M.J., 1976, Spatial equilibrium in dispersed city, in: G.J. Papageorgiou, ed., Mathematical land use theory (Lexington Books, Lexington, KY) 99-116. Ellickson, B., 1971, Jurisdictional fragmentation and residential choice, American Economic Review 61, 334-339. Forrester, J.W., 1969, Urban dynamics (MIT Press, Cambridge, MA). Fujita, M. and H. Ogawa, 1980, Equilibrium land use patterns in a non-monocentric city, Journal of Regional Science 20, no. 4, 455-475. Fujita, M. and H. Ogawa, 1982, Multiple equilibria and structural transition of non-mono- centric urban configurations, Regional Science and Urban Economics 12, 161-196. Garreau, J., 1991, Edge city (Doubleday, New York). Hamilton, B., 1975, Zoning and property taxation in a system of local governments, Urban Studies 205-211. Helsley, R.W. and W.C. Strange, 1990, Matching and agglomeration economies in a system of cities, Regional Science and Urban Economics 189-212. Henderson, J.V., 1974, The sizes and types of cities, American Economic Review 64, 640-656. Henderson, J.V., 1985, The Tiebout model: Bring back the entrepreneurs, Journal of Political Economy 93, 248-264. Henderson, J.V. and E. Slade, 1993, Development games in non-monocentric cities, Journal of Urban Economics 34, 207-230. Henderson, J.V. and A. Mitra, 1993, Edge city economics, Brown University Working Paper No. 93-26. Henderson, J.V. and N. Poole, 1989, Principles of economics (D.C. Heath, Lexington MA). Jaffe, A.B., M. Trajtenberg and R. Henderson, 1993, Geographic localization of knowledge spiilovers as evidenced by patent citations, Quarterly Journal of Economics 108, 577-598. Kim, H.S., 1988, Optimal and equilibrium land use pattern in a city--a non-monocentric approach, Brown University Ph.D. Dissertation. Krugman, P., 1993, First nature, second nature, and metropolitan area location, Journal of Regional Science 33, 129-144. Krugman, P., 1996, Spillovers, agglomeration and growth, in: Japan and the World Economy, forthcoming. McMillen, D.P. and L.D. Singell, 1992, Work location, residential location and the intraurban wage gradient, Journal of Urban Economics 32, 195-213. Mills, E.S., 1972, Studies in the structure of the urban economy (Johns Hopkins Press, Baltimore, MD). Mills, E.S. and W.E. Oates, eds., 1975, Fiscal zoning and land use (D.C. Heath, Lexington MA). Mitra, A., 1994, Dynamic externalities and industrial location, Brown University, mimeo. Muth, R., 1969, Cities and housing (University of Chicago Press, Chicago, IL). Pines, D., 1991, Tiebout without politics, Regional Science and Urban Economics 21,469-489. Rauch, J., 1993, Does history matter when it only matters a little, Quarterly Journal of Economics 108, 843-867. Weitzman, M.L., 1970, Optimal growth with scale economies in the creation of overhead capital, Review of Economic Studies 37, 555-570.