CUER Working Paper Robert W. Helsley and William C. Strange Agglomeration, Opportunism, and the Organization of Production May 2003 CUER Working Paper 03-02

Agglomeration, Opportunism, and the Organization of Production

Robert W. HelsleyFaculty of Commerce and Business Administration

2053 Main MallUniversity of British Columbia

Vancouver, BC, V6T 1Z2Canada

and

William C. Strange*Rotman School of Management

105 St. GeorgeSt.University of TorontoToronto, ON M5S 3E6

Canada

May 19, 2003

COMMENTS ARE WELCOME

*Helsley is Watkinson Professor for Environmental and Land Management. Strange isRioCan Real Estate Investment Trust Professor of Real Estate and Urban Economics.We gratefully acknowledge the financial support of the Social Sciences and HumanitiesResearch Council of Canada, the UBC Center for Urban Economics and Real Estate, andthe Connaught Fund at the University of Toronto.

Abstract

Recent economic analysis of outsourcing has focused on its international dimension. Incontrast, this paper considers local outsourcing. The paper specifies and solves a modelwhere the organization of production (vertically integrated or not) and the location ofproduction (agglomerated or not) are jointly determined. The paper shows thatagglomeration reduces opportunism (a thick market effect), and so serves as a substitutefor integration. The paper also shows that the normative properties of equilibrium withlocal outsourcing are not as clear cut as for international outsourcing, since localagglomeration is achieved at the cost of congestion. Thus, there are competinginefficiencies: vertical disintegration has a positive externality but the agglomeration thatallows it has a negative externality. In some situations, markets may be too thick, whichwould not be the case for an increase in the thickness of a world market. Finally themany changes in economic and social circumstance that have been labeled"globalization" not only impact vertical integration. They also impact agglomeration tothe extent that it is a substitute. The choice is not between vertical integration andoutsourcing. It is instead the richer choice between vertical integration, internationaloutsourcing, national outsourcing, and local outsourcing. The most important implicationof this is that international outsourcing can have a profound impact on cities, withinternational disintegration substituting for local disintegration and reducing the benefitsof agglomeration.

1

I. Introduction

Some of the best papers on outsourcing begin with discussions of Barbie dolls

(Feenstra (1988), Grossman-Helpman (2002b)). Barbie's contribution to the

understanding of outsourcing is as follows: the doll is producing using plastic and hair

from Taiwan and Japan and assembled in Indonesia and Malaysia. The molds are made

in the U.S. Barbie is clothed by Chinese, and decorated in the U.S. In sum, Barbie is a

small plastic icon of international outsourcing. There are many other treatments

examples of outsourcing that are not much different from Barbie, including among others

cars. The bottom line of all this is that outsourcing has an important international

dimension. In fact, the term "outsourcing" has been used as synonymous with

international vertical disintegration.1

While it is undeniable that the international outsourcing of inputs is a

phenomenon of crucial importance, it is not true that outsourcing is an inherently global

phenomenon. This paper considers local outsourcing. There are many instances where

local sourcing of inputs is extensive. A particularly striking instance is business services.

Schwartz (1993) makes a compelling case that firms disproportionately purchase business

services from within their own city. Looking at central city companies and across all

services, 48.3% of service outsourcing was within a city. This proportion is much larger

for some services: legal counsel (72.5%), major banking relationship (54.2%), business

insurance brokerage (60.9%), and auditing (79.1%). Looking at demanders in New

York, Los Angeles, Chicago, and San Francisco, the figures are larger: all services

(65.2%), legal counsel (87.0%), major banking relationship (75.0%), insurance brokerage

(74.4%), and auditing (86.9%). (Schwartz, 1993, p. 293). The figures are slightly lower

but still high for companies located in the suburbs. For all cities, for instance: all

1 Feenstra and Hanson (1996, p. 240): "...outsourcing, by which we mean the import of intermediate inputsby domestic firms."

2

services (49.5%, the total of outsourcing from the central city or one of its satellite cities),

legal counsel (69.3%), major banking relationship (52.2%), business insurance brokerage

(54.3%), and auditing (70.7%). (Schwartz, 1995, p. 293) In Section II, we discuss the

facts on local outsourcing in detail, showing that there are many other instances of local

outsourcing, including the computer, television, film, and jewelry industries.

Outsourcing has been a topic of interest in several different areas of economics.

The urban economic analysis of outsourcing begins, of course, with Marshall (1890,

1920). His treatment of outsourcing is technological: when there are scale economies in

the provision of intermediate inputs and transportation costs in delivery to input

demanders, there is an increasing return associated with spatial agglomeration. In this

literature, Goldstein and Gronberg (1984) focus on cost complementarities and input

sharing, while Helsley and Strange (2002) show that input sharing in a probabilistic sense

leads to innovation. Duranton and Puga (2001) present a model where a new product

requires a different kind of input sharing than does a mature product.

Outsourcing is also considered in the literature on the theory of the firm. Coase

(1937) is seminal. Three excellent surveys of this literature are Williamson (1989),

Holmstrom and Tirole (1989), and Hart (1995). This is a large literature -- hence, the

many surveys -- and we will not attempt to completely review it here. They heart of the

literature is the determination of the boundary of a firm. The transactions cost theory of

the firm holds that opportunism and relationship specific investment are central to this

determination. When it is not possible to execute complete contracts, then the possibility

of hold-up leads to an inefficiently low level of relationship specific investment. This, in

turn leads to vertical integration, since hold-up is less likely to occur when both parties to

a relationship are part of a single firm. One kind of investment that is in peril of hold up

is site-specific investment. For example, when the costs of moving inputs are large, an

investment in input production is useful only to nearby demanders. It is thus site-

specific. Joskow (1985) shows that site-specificity leads to the vertical integration of

3

colocated coal-burning electricity generating plants and coal mines. Although this

suggests that transactions cost considerations may be an important force leading to

agglomeration, nearly all of the analysis is in the spirit of Marshall's technological

microfoundations and does not consider transactions costs. The paper that provides the

most complete treatment of the link between transactions costs and agglomeration is

Rotemberg and Saloner (2000). It considers worker incentives to acquire industry

specific training when firms cannot commit to reward their investments. The paper does

not, however, consider the choice between integrated and disintegrated production.2

The international economics literature has considered outsourcing as well. Three

recent papers (Grossman-Helpman (2002a,b) and McLaren (2000) have added to the

theory of the firm literature by providing models of transactions costs and the boundaries

of the firm in an industry equilibrium. These papers have established that market

thickness effects can lead to multiple equilibria in organizational form, vertically

integrated or not. McLaren has also shown that decreases in the costs of trade, by

increasing market thickness, can lead to greater outsourcing. Since market thickness is

an industry level public good, equilibrium is inefficient, with the amount of outsourcing

being below the efficient level. This analysis is conducted at the level of the nation or the

world, and it thus misses some important aspects of local outsourcing.

This paper contributes to all of these literatures. First, using examples from

media industries, services, and traditional manufacturing the paper establishes that local

outsourcing matters too. Second, the paper shows that agglomeration reduces

opportunism (a thick market effect), and so serves as a substitute for integration. Third,

the paper shows that the normative properties of equilibrium with local outsourcing are

not as clearcut as for international outsourcing. Agglomeration is achieved at the cost of

congestion. Thus, there are competing inefficiencies: vertical disintegration has a

2 Dudey (1990, 1993) models retail concentration as a response to a similar kind of opportunism, but forobvious reasons it does not consider the possibility of vertical integration.

4

positive externality but the agglomeration that allows it has a negative externality. In

some situations, markets may be too thick, which would not be the case for an increase in

the thickness of a world market. Fourth, the many changes in economic and social

circumstance that have been labeled "globalization" not only impact vertical integration.

They also impact agglomeration to the extent that it is a substitute. The choice is not

between vertical integration and outsourcing. It is instead the richer choice between

vertical integration, international outsourcing, national outsourcing, and local

outsourcing. The most important implication of this is that international outsourcing will

can have a profound impact on locally disintegrated producers.

The remainder of the paper is organized as follows. Section II describes the

phenomenon of local outsourcing. Section III presents a stylized model that illustrates

the connection between the hold-up problem and agglomeration. Section IV considers

these topics in a more general setting, solving jointly for city size and organizational form

in the equilibrium and optimum. Section V extends this analysis to consider the choice

between vertical integration, local disintegration, and international disintegration. This

allows us to analyze the effects of increases in trade on cities.

5

II. Local outsourcing

We have already presented some evidence that local outsourcing is important for

business services. The purpose of this section is to show that local outsourcing is broadly

important.

We will begin with some historical cases showing that large cities and industrial

concentration are marked by vertical disintegration. In the review volume of the New

York Metropolitan Region Study, Vernon (1960, pp. 74-5) notes that "[e]xternal

economies industries" have smaller plant sizes in the New York Region than in the rest of

the country. In addition, New York has disproportionate employment in industries

dominated by single plant firms (p. 71). Vernon argues that this paints a picture of a

vertically disintegrated industrial system in what was at the time America's largest city.

Chapman and Ashton (1914) analyze the British cotton textile industry,

concluding that in the industry's most important city Lancashire, 82% of textile mills

were disintegrated with dedicated spinning an weaving operations. This contrasts to a

proportion of 49% in the rest of the country. Similarly, Allen (1929) finds Birmingham's

gun industry, which was concentrated there, to be very disintegrated. Eventually, this

industrial system was replaced by a mass-production industry characterized by vertical

disintegration.3

The relationship between vertical integration and city size or industrial

concentration is not merely a historical one. Scott (1988) notes that disintegration is the

common form of organization in the Los Angeles garment and printed circuit industries.

He observes that there is a positive correlation between distance to each industry's point

of focus and plant size in garments (p. 117). Similarly, Scott notes that there are close

geographic linkages between printed circuit manufacturers and laminate suppliers,

3 Both Chapman and Ashton (1914) and Allen (1929) are discussed in Scott (1988).

6

chemical suppliers, and subcontractors (p.110). This is a close parallel to Vernon's

analysis, where city size and industry concentration are associated with vertical

disintegration.

Scott (2000) points out that the jewelry industry in Los Angeles is essentially

confined to a three square block section of the city. This industry is characterized by

small companies that make extensive use of local input sources. Scott (2000, p.47) writes

that "on average respondents purchased 66.4% of all their inputs from other firms in the

Los Angeles area...as much as 23.6% of all their business consisted of work

subcontracted to them by other [local] firms." The situation is similar in Bangkok's

jewelry industry, with 61.9% of inputs local, 11.6% of business from subcontracts (Scott,

2000, p.53). The furniture industry in Los Angeles also has substantial subcontracting,

with 8-10% of business on average coming from this channel (Scott, 2000,p.68). The

multimedia industry in the Bay Area and Southern California also exhibits substantial

local subcontracting, with 79.3% of Bay Area subcontracting by Bay Area firms and

69.6% of Southern California by Southern California (Scott, 2000, p.147).

In her study of high technology, Saxenian (1994) notes that Boston and the

Silicon Valley have different industrial systems. The latter involves a lot more

outsourcing. These claims are supported through discussions of cases and through

interviews.

These case analyses of individual industries all suggest that local outsourcing is

sometimes important, and that its importance is greater in large cities or where an

industry is concentrated. These case results continue to hold studies that look across all

industries. For instance, Holmes (1999) finds that purchased input intensity -- the value

of purchased inputs divided by the value of total output -- is greater the greater is the

same-industry employment within 50 miles. Since these inputs were not obtained

internally, it suggests that there is less vertical integration where an industry is

7

concentrated.4 In an analysis of agglomeration economies, Rosenthal and Strange (2001)

show that a bigger external effect is exerted by small firms. Henderson (2001) shows that

a bigger external effect is received by small firms. Together, these two findings are

consistent with the idea that one aspect of agglomeration economies is the sharing of

inputs by small firms. In sum, all of this evidence suggests that in order to come to a full

understanding of outsourcing, one must consider the outsourcing that occurs locally.

4 Interestingly, in a later paper, Holmes and Stevens (2000) find that establishment size is larger where anindustry is concentrated. This result is identified using a "size coeffficient," equal to the ratio of the meanestablishment size (measured in value of output) at a location to the mean size in the sector across the US.This result contrasts with Vernon on the firm size - city size relationship. It does not, of course, directlyaddress the degree of vertical integration.

8

III. A simple model of outsourcing

A. Primitives

This section presents a simple model of the outsourcing decision. In the next

section we will use this model to examine how agglomeration influences outsourcing and

the choice between vertically integrated and disintegrated organizational forms.

There are two types of firms in the model. The first are final goods producers.

These firms employ specialized inputs in variable quantities to produce a homogeneous

output, which we take to be the numeraire. We refer to final goods producers as (input)

demanders.

Inputs are described by addresses on a characteristic space. We suppose that the

characteristic space is the unit interval, [0,1]. Each demander has an address yj Π[0,1]

that describes the address of the input that the demander is best suited to employ. In this

section we assume that there are just two demanders, with addresses y1 < y2, where Y ≡ y2

- y1 represents the distance between the firms. We consider a model with more than two

demanders in the next section.5

The second type of agent in the model are the input suppliers. Each supplier

creates one unit of a specialized input that may be sold to a demander. The type of input

that a supplier produces is described by an address x Π[0,1]. If the address of an input

does not exactly match the address of its employer, then the productivity of the input is

reduced. The loss in productivity is determined by the distance d = |x – y | between the

input supplier and the demander. The precise nature of this loss is described in detail

below. The cost to an input supplier of creating one unit of input is c > 0. We suppose

5 The assumption of a linear characteristic space is not essential. Our qualitative results would also hold fora circular characteristic space. We chose the unit line because it allows us to derive a number of results forthe case when there are two demanders and then extend these results readily to the case of an arbitrarynumber of demanders.

9

that input suppliers are small and that their mass, denoted by Q, is determined by free

entry.

Demanders can also create inputs internally. Obviously, internal inputs will be

designed to have characteristics that exactly match the demander's input requirements.

For these inputs, d = 0. The cost to a demander of creating one unit of an input internally

is assumed to be k > c. The cost difference could reflect a managerial or organizational

diseconomy associated with integrated production, or a competency that allowed

dedicated input suppliers to produce at lower cost. When a firm acquires some of its

inputs internally, we say that the firm is vertically integrated. When a firm acquires all of

its inputs externally, that is from input suppliers, we say that the firm is vertically

disintegrated. In this model, as in the related literature, outsourcing is synonymous with

vertical disintegration.

The timing of decisions and events is as follows:

Stage I: Input suppliers enter so long as their profits are non-negative. Each

active input suppliers chooses a location x to maximize profits and produces one

unit.

Stage II: Input prices and the allocation of inputs to demanders are determined

through bargaining. The bargaining process consists of take-it-or-leave-it offers

from demanders to input suppliers.

Stage III: Demanders choose levels of internal and external inputs to maximize

profits.

We use backward induction to characterize a subgame perfect Nash equilibrium of the

game, beginning with the outsourcing decision in Stage III.

B. Outsourcing

10

Let qj represent the quantity of inputs that demander j acquires from input

suppliers, and let qj0 represent the amount that j creates internally. Suppose all input

suppliers choose a common location x, and let dj = |x - yj| represent the distance between

the input suppliers and demander j.6 Finally, let pj be the price that demander j pays for a

unit of input. Then the profit of demander j is

pj = f(qj + qj0) -[a(dj) + pj]qj - kqj0, (1)

where f(⋅) is the production function, and a(dj) is adjustment cost for the inputs supplied

from x. We assume that f(⋅) is increasing and strictly concave, with f ¢¢¢(⋅) ≥ 0, and that

a(⋅) is increasing and strictly convex, with a(0) = 0. For future reference, let

vj ≡ f ¢(qj + qj0) - a(dj) (2)

represent the marginal value of qj. Since a(⋅) is increasing, an increase in dj reduces the

marginal value of qj.

In Stage III, each demander provides inputs internally if it is profitable to do so.

In this stage, demander j's problem is:

†

Maxqj0

pj subject to qj0 ≥ 0. (3)

There are two cases. If qj0 > 0, then the first-order condition for profit-maximiztion is

f ¢(qj + qj0) = k. (4)

6 Later we will show that input suppliers choose a common location in equilibrium.

11

Conversely, qj0 = 0 is a maximizing choice when

f ¢(qj) < k. (5)

When internal provision is profitable, the maximizing level of internal provision equates

the marginal benefit and the marginal cost. Internal provision is not profitable when the

marginal benefit is less than marginal cost. Then, all inputs are supplied externally, that

is, outsourced. In the former case, demander j is vertically integrated; in the latter case

demander j is vertically disintegrated.

C. The pricing and allocation of inputs

1. Bargaining and maximizing bids

In Stage II, demanders make take-it-or-leave-it offers to each input supplier,

offering to purchase the supplier's input at a specific price. Since production costs are

sunk at this point, each input supplier accepts the highest bid for its input, provided the

bid is non-negative. This leads to the familiar problem of ex post opportunism:

demanders have no incentive to offer prices that would support efficient external input

production. Indeed, in the absence of competition between demanders, the equilibrium

bid for any input would be zero. However, in this model, there can be more than one

bidder for each input, and competition between bidders serves to limit opportunism. To

obtain a particular input, a demander must make a non-negative bid that is larger than the

bid of any rival.

12

A demander's optimal bid is determined by the marginal value of the input. If

demander j' bids pj', it is profit maximizing for demander j to bid pj = pj' + e and obtain the

input, provided

∂pj/∂qj = f '(qj + qj0) - a(dj) - pj ≥ 0, (6)

that is, provided pj' + e £ vj. In the limit, this implies that a demander bids an amount for

each input equal to the marginal value of the input to the rival, provided such a bid is

both positive and not larger than the bidder's marginal value. This implies

pj = f ¢(qj' + qj'0) - a(dj') = vj', (7)

for vj ≥ vj' > 0. However, if vj < vj', then pj = 0. As in a standard English auction, the

winning bid is determined by the value of the good to the second-highest bidder.

A few remarks on the specification of the model are useful at this point. We have

supposed that input demand is variable and that it exhibits a diminishing marginal

product. This means that in many cases, both demanders would derive positive marginal

value from additional supplies of the input. Because of this, both demanders are

available as outside options to input suppliers, reducing the ability of other demanders to

opportunistically set low input prices. This can be contrasted with a fixed input demand

model, where each demander requires, for instance, one unit of input. In this case, once a

particular demander has obtained its required input, it no longer provides an outside

option in the bargain between another demander an an input supplier. This allows other

demanders to behave opportunistically. This problem arises in Grossman and Helpman's

(2002a) matching model. They deal with it by supposing that a fraction of matches

dissolves before production takes place. In the limit as this fraction becomes small, the

presence of other demanders curbs opportunism. The problem also arises in McLaren's

13

(2000) matching model. He deals with it by sequencing the decisions and events so that

outside options still have expected value when decisions are made. Our paper shows that

another way to deal with the problem of the vanishing outside option -- and a reasonably

natural one -- is to suppose variable input demands.

Variable input demand seems to be an appropriate assumption for many of the

cases discussed in Section II. A jewelry manufacturer, for instance, chooses among

various inputs -- both goods and services -- in order to create final products. The

opportunity to substitute is great. Similarly, a multimedia producer can substitute among

input goods and services. Even an automobile manufacturer has variable input demand in

the sense that the total number of cars produced can vary. In fact, it has been argued that

this connection (Klein (1988)) is one of the forces that led to the famous merger of G.M.

and Fisher Body.7

2. Input allocation

Since each active supplier produces one unit, the total quantity of external inputs

supplied equals the mass of active suppliers, Q. In what follows, we focus on the case

where the demanders are not "saturated" with external inputs. By this we mean that the

external input market clears, so q1 + q2 = Q, and the marginal value of an external input is

positive for both demanders.8 Clearly, this is the only case that is relevant in equilibrium,

since input suppliers would not operate knowing that they would receive a price of zero

for their inputs.

7 This interpretation of the Fisher body case is controversial. See Freeland (2000), Klein (2000), andCasadesus-Masanell and Spulber (2000) for some of the issues of contention.8 There are a number of possible corner solutions. For example, it could be the case that vj = 0 for one orboth firms, given x and Q, with q1 + q2 £ Q. Then the equilibrium price of an externally supplied inputwould be zero, and no input suppliers would enter.

14

Input allocation and pricing are jointly determined in equilibrium, as the

following result shows:

Proposition 1: If the market for external inputs is not saturated, then inputs are allocated

between demanders so that their marginal values are equal. Also, the price of an input

equals both the highest and second highest marginal value of the input.

Proof: Consider some arbitrary allocation of Q between the two demanders. Suppose,

without loss of generality, that at this allocation, v1 > v2. Then demander 1 can

successfully bid p1 = v2 + e for the marginal input and increase his profit. A parallel

argument applies to the case where v2 > v1. Thus, profit maximization implies that the

allocation of Q between the demanders must satisfy

f ¢(q1 + q10) - a(d1) = f ¢(Q - q1 + q20) - a(d2). (8)

(7) then implies that the equilibrium price of an input equals the input's common

marginal value. QED.

Input allocation and pricing are illustrated in Figure 1. The horizontal axis depicts

the distance between the demanders in the characteristic space. The curve descending

from the y1- axis is v1 = f ¢(q1 + q10) - a(d1). The curve ascending toward the y2- axis is v2

= f ¢(Q - q1 + q20) - a(d2). Profit maximization requires that the bids and input allocations

be such that v1 and v2 intersect at the given value of x. The common value of v1 and v2 at

this point gives the equilibrium input price,

p = f¢(q1 + q10) – a(d1) = f¢(Q – q1 + q20) – a(d2). (9)

15

Most of the comparative statics of q1 and p are clear from the figure. For

example, an increase in Q, holding all other factors constant, decreases v2, by the

concavity of f(⋅). This implies that q1 must increase. The new intersection occurs at a

lower value of vj, indicating that p must fall. The comparative statics are presented

formally in the Appendix.

The most interesting comparative static result concerns changes in x, the location

of the input suppliers. An increase in x, holding all other factors constant, increases d1

and decreases d2, which decreases v1 and increases v2. Thus, q1 must decrease. However,

the comparative static for price is ambiguous. From (9),

)qqQ(f)qq(f

)d(a)qqQ(f)d(a)qq(f)d(a

x

q)qq(f

x

p0

210

11

10

2120

111

1011

+-¢¢++¢¢

¢+-¢¢-¢+¢¢=¢-

∂

∂+¢¢=

∂

∂ . (10)

In the case where the demanders have equal levels of internal input provision, q10 = q20,

(8) implies q1 = Q/2 at d1 = d2, the midpoint of the interval. Then, (10) implies

†

∂p∂x at x = (y 2 +y 2 )/ 2

= 0 . (11)

A sufficient, but not necessary, condition for p(⋅) to be strictly concave in x is that the

third derivative of f'''(⋅) ≥ 0 , as we have assumed. Then, in the identical demander case,

the midpoint of the interval is the location at which the input price reaches a global

maximum. This relationship between location and input price will obviously influence

the suppliers' location choices, one of the questions to which we now turn.

D. The entry and location of input suppliers

16

In Stage I, input suppliers enter the market individually, and active suppliers

choose a location x to maximize profit. With free entry, the equilibrium mass of input

suppliers is determined by a zero profit condition. The equilibrium location must have

the additional property that no supplier would earn greater than zero profit at any other

location.

There are three forms that the equilibrium may take. First, it is possible that

enough input suppliers enter that there is no further incentive for internal provision of

inputs, a fully disintegrated equilibrium. Second, it is possible that no input suppliers

would enter, in which case the equilibrium involves fully integrated production. Third, it

is possible that input suppliers enter, but not in sufficient mass to completely discourage

internal provision. Since there is both internal provision and outsourcing, this is a mixed

equilibrium.

Thus far, we have supposed that input suppliers all choose the same location in

the characteristic space, x. We are now able to prove this claim.

Proposition 2: All input suppliers between any two demanders choose the same location.

Proof: Consider any arbitrary allocation of input mass Q across locations. Bidding and

profit-maximization imply that each input is allocated to the highest, that is, the closest

demander. The price that each input receives is its value to the second-highest bidder.

Let ( 1q̂ ,Q - 1q̂ ) by the resulting allocation of inputs to demanders. Given this allocation,

define x̂ to be the location where the two demanders' marginal values are equal. Since p

= Min{v1, v2}, the location x̂ gives the input supplier the highest price. This means that

any input supplier at any other location would be better off moving to x̂ , taking the

allocation of inputs to suppliers as fixed. Thus, no allocation with any input suppliers at

any location other than x̂ can be an equilibrium. By the same logic, if all input suppliers

were located at x̂ , none would have an incentive to deviate unilaterally. QED.

17

Proposition 2 shows that all input suppliers will pick the same location. However,

since the location at which p is maximized varies with the allocation of inputs to the

demanders, there are many locations that are consistent with equilibrium even when

demanders are symmetric in their internal input choices. If q1 > q2, then x̂ < (y1 + y2)/2,

while if q1 < q2, then x̂ > (y1 + y2)/2. If q1 = q2, then x̂ = (y1 + y2)/2, the midpoint of the

characteristic space. In this case, q1 = q2 = Q/2, and each input supplier maximizes profit

by choosing x = (y1 + y2)/2, giving a distance to either entrepreneur of Y/2. Since the

problem we are considering is completely symmetric, the symmetric solution seems most

natural. Also, when internal put choices are equal for the demanders, all of the

asymmetric solutions for input supplier location involve both lower input prices (recall

(13)) and lower levels of input supply. Thus, in a sense, the asymmetric solutions are

dominated. For both of these reasons, we will focus on the symmetric outcome in what

follows.

Proposition 1 describes the allocation and pricing of inputs. Proposition 2

characterizes input supplier locations. The task that remains is to determine the mass of

input suppliers, Q. As discussed above, the equilibrium may involve complete

disintegration, complete integration, or partial disintegration. In each of these cases,

input suppliers enter until the marginal supplier earns zero profit.

In the disintegrated case, the zero profit condition is

f ¢(Q/2) - a(Y/2) - c = 0. (12)

The equilibrium mass of input suppliers Q, implicitly defined by (12), is a decreasing

function of the cost of external input production and of the distance between the

demanders. From (5), qj0 = 0 requires f ¢(Q/2) < k. Combining this with (12) implies that

a fully disintegrated equilibrium will obtain when

18

†

k - c > a Y2Ê Ë

ˆ ¯ . (13)

Intuitively, demanders will be disintegated in equilibrium when the cost disadvantage of

internal production is larger than the loss associated with imperfect input specialization.

The disintegrated equilibrium is illustrated in Figure 2. Given Q and x = (y1+y2)/2, the

input is allocated so that marginal valuations are equal. Q then adjusts so that marginal

value equals marginal cost. Marginal cost plus the adjustment costs in equilibrium must

be less than the cost of integrated production, which holds as Figure 2 is drawn. The

bottom panel of the figure gives the price that an individual input supplier would earn

taking as given the locations of all other input suppliers. As it must be, it is maximized at

x = (y1+y2)/2.

Turning now to the integrated case, if no input supplier could enter at the profit

maximizing location and earn non-negative profit,

0c2

Ya)q(f 0j

19

In this case, the demanders' cost disadvantage is small relative to the adjustment costs

that would be incurred in the disintegrated equilibrium. The integrated equilibrium is

illustrated in Figure 3.9

The final possibility is a mixed equilibrium.10 This is a knife edge in this model.

In this case, zero profit for input suppliers requires

0c2

Ya

2

Qqf 0j =-˜

¯

ˆÁË

Ê-˜¯

ˆÁË

Ê +¢ , (16)

where the level of internal provision satisfies f ¢(qj0 + Q/2) = k, from (4). Combining this

with (16) implies that a mixed equilibrium obtains when

†

k - c = a Y2Ê Ë

ˆ ¯ . (17)

An equilibrium with vertically integrated demanders and active input suppliers can only

exist when the cost disadvantage associated with integrated production exactly equals the

loss associated with imperfect input specialization.

This analysis may be summarized as follows:

Proposition 3: The symmetric equilibrium is disintegrated, integrated, or mixed as k - c

is greater than, less than, or equal to a(Y/2).

Proof: See above.

9 The key difference between this figure and Figure 2 is that the cost of integrated production is now belowwhat would have been the adjustment costs and marginal cost of disintegrated production at a candidatedisintegrated equilibrium.10 The situation where a demander both produces an input itself and purchases it at arm's length is referredto in the vertical integration literature as a "tapered integration."

20

E. Interpretation

Proposition 3 shows that the organizational nature of the equilibrium -- integrated

or not -- depends on the cost advantage of outsourced production, the cost of adjusting an

input, and the density of input demanders. Both the cost advantage and the adjustment

cost are technological. If the technology is such that adjustment costs are small at a given

distance, then outsourced production need have only a small cost advantage in order to

lead to the disintegrated equilibrium. If adjustment costs are large, however, then the

cost advantage would need to be greater. By itself, this suggests that goods where

production is a matter of routine and adjustment costs are therefore small can readily

disintegrate production. Goods where adjustment is costly are less likely to be able to

operate in a disintegrated fashion.

The density of input demanders, on the other hand, is not determined by the

technological nature of production but by the decisions of those involved in the

production. When input demanders are similar, Y = y2 - y1 is small, which may allow

disintegration even when production is not routine, and so adjustment costs are large. As

discussed in the next section, the distance between demanders is more likely to be large

in a big city. This is consistent with Vernon (1960), who makes much of the industrial

organization in the New York Metropolitan Region in industries where production is

"unpredicatable" such as high-fashion apparel. We will return to this issue in the next

section.

It is worth pointing out that the organizational nature of the equilibrium is

uniquely determined in our model. There is either disintegration or integration,

depending on the values of the model's key parameters. This is a consequence of the

structure of the game. The key assumptions are variable input demand and that

demanders are allowed to supplement outsourced inputs with internal production after the

input suppliers make their decisions. If we had instead supposed that there were fixed

21

input demands and that demanders must choose whether or not to produce the input

internally prior to the operation of the input market as in Grossman and Helpman (2002),

then there would have been multiple equilibria. If demander 2 chose in advance to obtain

all its inputs internally, then demander 1 would do the same since the absence of

competition for inputs would discourage the entry of independent input suppliers.

This brings home the point that the integration decision of a rival can affect the

choice of another demander in some situations. Another situation where this can occur is

when the two demanders are in some way different. For instance, suppose that demander

1 can produce internally at cost k1, while the cost for demander 2 is k2 > k1. If k1 is

sufficiently close to c, then demander 1 will acquire the input internally. This, in turn,

will discourage independent input supplier entry, forcing demander 2 to integrate as well.

IV. Agglomeration and opportunism

A. Extending the simple model

The previous section used a two-demander model to characterize the equilibrium

outsourcing decision. This section extends the two-demander model to the case of an

arbitrary number of demanders, J ≥ 2. We begin by taking J as exogenous, but we later

consider the optimum and equilibrium number of demanders. Extending the model to

more than two demanders allows us to examine the relationship between agglomeration,

opportunism and outsourcing.

One of the merits of the model described in Section III is that its extension to

more than two demanders is quite natural. For any input supplier, only the bids of its

nearest demanders are relevant, since any more distant demander would be outbid by one

of these. Consequently, a supplier in a market with many demanders faces exactly the

same choices as in the two-demander case: it enters if profits are non-negative and

22

chooses a location between two demanders to maximize profits. Also as before, input

demanders choose whether to procure inputs internally or externally, and in the case of

external provision, they bid against at most one other demander for the input of a

particular supplier.

It is natural to assume that the distance between two firms in the characteristic

space is smaller when activity is concentrated in a city or region – agglomeration

increases density, in this case the density of input demanders. As suggested by the

analysis of Section III, a higher density of input demanders leads to greater competition

for inputs, and this limits the exercise of opportunism in the external input market. By

limiting the degree of opportunism, agglomeration permits external input markets to

form, leading to a positive relationship between the density of economic activity and

outsourcing, or to a negative relationship between agglomeration and vertical integration.

To formalize these results, we suppose that the demanders are evenly spaced on

the unit line with addresses y1, y2,..., yJ, where yj = (j – 1)/(J – 1). To make use of prior

analysis, we continue to denote by Y the distance between any pair of demanders.

Keeping the characteristic space of constant unit length means that Y = yj+1 – yj = 1/(J –

1). As noted above, an increase in the number of demanders is associated with a decrease

in the distance between any two. As in Section III, we restrict out attention to symmetric

outcomes with input supply locations that are exactly between the demanders. This

means that the input supply locations x1, x2,...,xJ-1 satisfy xj = (2j – 1)/(2(J – 1)), implying

that the distance between an input supplier and its nearest demanders is d = 1/(2(J –1)).

As in our earlier analysis, the distance between any pair of input suppliers may be

sufficiently large that opportunism prevents the operation of a disintegrated input market.

This will be the case when there are few demanders and Y is large. Stated more

formally:

Proposition 4. Vertical integration requires a sufficiently large population of firms.

23

Proof: Following the analysis in Section III, a disintegrated input market cannot operate

when an input supplied at the midpoint between two demanders involves adjustment and

production costs that exceed the cost of internal production. Substituting for Y = 1/(J –

1) in (13), a fully disintegrated equilibrium with many demanders will obtain when

˜̃¯

ˆÁÁË

Ê

->-

)1J(2

1ack . (18)

(18) defines a critical level of J, denoted

†

˜ J such that for J <

†

˜ J , equilibrium involves

integrated production. For J ≥

†

˜ J , equilibrium involves disintegrated production.

QED

Proposition 4 implies that agglomeration is a substitute for vertical integration.

When the population of firms is small, as it would be outside an agglomeration, there is at

best a weak outside option to spare an input supplier from the opportunism of

downstream firms. This result accords nicely with the cases and facts discussed in

Section II. For instance, the historical results on textile production in Britain and garment

production in New York City are exactly in this spirit: concentrations of industry are

accompanied by vertical disintegration. This is also consistent with Holmes' (1999)

finding that purchased input intensity is positively related to local industrial

concentration.

For, J >

†

˜ J , a symmetric Nash equilibrium with disintegrated production is as

discussed in Section III: the price of any input equals the highest and second highest

marginal value of the input and the marginal cost of input production; input supplies are

allocated among demanders so that the marginal values are equal and aggregate supply

24

equals aggregate demand; and inputs are supplied at the location exactly between the

input demanders. The location condition has already been given. The other conditions

simplify to:

†

¢ f (q) - a Y2Ê Ë

ˆ ¯ - c = 0 , (19)

and

†

(J -1)Q - Jq = 0 . (20)

(20) is a restatement of the zero profit condition, where

†

p = ¢ f (q) - a Y2Ê Ë

ˆ ¯ (21)

is the equilibrium price of an input, as in Section III. (21) ensures that the aggregate

input supply, (J – 1)Q, equals aggregate input demand, Jq.

For future reference, (21) defines a relationship between the input quantity, q, and

the number of demanders, J. Differentiating, and recalling that Y = 1/(J – 1), we have

†

dqdJ = -

a' Y2

Ê Ë

ˆ ¯

2(J -1)2 ¢ ¢ f (q) > 0 . (22)

As J grows, adjustment cost for an external input decreases, and this increases the

equilibrium input price. The reduction in opportunism allows greater supply of the input

to each demander. Together, (22) and (24) imply

25

†

dQdJ =

JJ -1

dqdJ -

q(J -1)2 . (23)

Each supply address provides a share 1/(J – 1) of the aggregate input demand. Although

aggregate demand rises with J, the share supplied from any address falls with J. As a

result, the impact of an increase in J on supply at a particular address is ambiguous.11

In this model, J represents the agglomeration of demanders into a particular city.

To see how J affect demander profits, note that the profit of any demander in the

disintegrated equilibrium is

†

p = f(q) - a Y2Ê Ë

ˆ ¯ - p

Ê Ë Á

ˆ ¯ ˜ q , (24)

from (1). Differentiating with respect to J, and using (21) and (22) implies

†

dpdJ =

¢ a Y2

Ê Ë

ˆ ¯ q

2(J -1)2 > 0 . (25)

An increase in the number of rival input demanders increases profit by reducing

opportunism. This effect occurs despite the competition between demanders for inputs.

The reduction in opportunism results in an increase in q for each demander by (22), and

this leads to an increase in profit as input demanders agglomerate.

This is an agglomeration economy that is strikingly non-Marshallian. As is well

known, Marshall (1890,1920) identifies knowledge spillovers, labor market pooling, and

input sharing as being sources of external increasing returns. The cases of knowledge

11 It is worth noting that there are many equivalent equilibria. Total production must allow each demanderto receive q. However, as long as there is no cross-purchasing, with two demanders buying inputs that arenearer the other than to themselves, there are many ways to allocate inputs to suppliers and meet theequilibrium conditions.

26

spillovers and labor market pooling are not especially relevant here, but the case of input

sharing is.12 The key to the Marshallian analysis of input sharing is scale economies. As

in Smith (1776,2001), the division of labor is limited by the extent of the market.

Agglomeration allows inputs to be shared in the sense that scale economies can be

realized with an extensive market, but not when production takes place in isolation. In

this model, however, there are no scale economies involved in input production. Instead,

the gains from agglomeration come from a reduction in opportunism. The agglomeration

economy is based, therefore, on transactions costs rather than on the purely technological

agglomeration economies considered by Marshall. In a sense, this agglomeration

economy can be seen as being Williamsonian, rather than Marshallian.

This is unique. Nearly the entire microfoundations literature to date has

considered Marshallian agglomeration economies. See Quigley (1998) for a survey.

Goldstein and Gronberg (1984) is typical. They consider input sharing in cities, arguing

that input sharing leads to geographically defined economies of scope. They note that

large cities allow vertical disintegration because of the availability of an "urban

warehouse" (p. 95) . However, although they cite Williamson's work, they do not make a

connection between opportunism and relationship specific investment, instead focusing

on the properties of cost curves. A recent exception in its focus on transactions costs and

agglomeration is Rotemberg and Saloner (2002), who consider the incentive of a worker

to acquire industry-specific human capital. In their model, the Bertrand competition for

workers by employers ensures that workers will be paid their marginal products, which in

turn encourages them to acquire human capital. Because of the Bertrand competition, it

does not matter how many employers are clustered as long as their are at least two. This

is not, of course, true in our model. In addition, Rotemberg and Saloner do not consider

12 See Glaeser (1999) or Helsley and Strange (1990) respectively for models of knowledge spillovers andlabor market pooling.

27

the degree to which agglomeration and vertical integrations are strategic alternatives in

dealing with opportunism.

The agglomeration force contained in this model is present for all levels of

activity: here the benefits of agglomeration are boundless. This kind of result is not

without parallel. Modern growth theory has argued the possibility of boundless

aggregate growth, based on, for instance, the infinite possibilities of technological

improvement. While the possibility of growth without bound is certainly not universally

acknowledged, it is at the very least a reasonable possibility. The boundless growth of a

city, on the other hand, is unreasonable. Adding to a city results in congestion costs in a

way that increasing the size of the aggregate economy does not. The key difference is

that in aggregate, it may be possible to replicate, while within a city replication is

impossible. The next section addresses this fundamental difference between the

properties of macro and micro growth, which is one of the main differences between this

paper and Grossman-Helpman (2002) and McLaren (2000).

B. Congestion and the number of demanders

Obviously, cities do not grow without bound. Local outsourcing is limited by

fixed factors in a way that global outsourcing may not be. There are many fixed factors

that play this limiting role. Accessible land becomes scarce as cities grow. Traffic

systems become congested, increasing the costs of transportation. Air quality may

decline with increasing industrial agglomeration, increasing production costs for some

types of activities.

For simplicity, we incorporate congestion effects into the model by supposing that

the adjustment cost term is an increasing function of both the distance between supplier

and demander, d, and the aggregate number of final goods producers, J: a(d,J). Since

28

transportation costs are an important component of the costs of using many local external

inputs, this simple specification seems defensible.

(21) and (22) continue to characterize a symmetric Nash equilibrium with

disintegrated production. The only difference is that in (21), the adjustment cost function

now depends on both d and J, as discussed above. With a slight change in notation, the

relationship between input use and the number of demanders becomes

†

dqdJ = -

∂a∂d

∂d∂J

+∂a∂J

¢ ¢ f (q) , (26)

where ∂d/∂J = -1/(2(J – 1)2. Similarly, the relationship between the profit of a demander

and the number of demanders becomes

†

∂p

∂J= -q

∂a∂d

∂d∂J

+∂a∂J

Ê Ë Á

ˆ ¯ ˜ . (27)

The first term in either expression is the positive impact of agglomeration through the

Williamsonian agglomeration economy discussed earlier. The second term represents the

negative impact of congestion. If the congestion effect eventually dominates the

Williamsonian agglomeration economy, then both input use and the profit of an

individual demander will decline with further increases in J. With congestion arising

from the presence of fixed local factors, the benefits of agglomeration are bounded.

This naturally leads us to examine the determinants of the number of final goods

producers. It is easiest to begin with an optimum program. Suppose that there are many

potential city sites in the economy, and that a planner is charged with the task of

allocating a large number of final goods producers to city sites to maximize welfare.

Suppose further that the planner cannot directly control the locations or production

29

decisions of input demanders or suppliers. Thus, we are considering a constrained

optimum in which the planner must respect the incentives of individual agents.

Under these conditions, the planner should choose for each site the value of J that

maximizes the profit of an individual demander. There are two regimes. In the first,

production is integrate and the optimum number of firms is J = 1. In this model,

whenever there is integration, congestion costs encourage spatial decentralization since

there is a cost to agglomeration but no benefit. In the second regime, firms are vertically

disintegrated. In this case, the optimum number of firms J* satisfies ∂p/∂J = 0, or

†

-∂a∂d

∂d∂J

=∂a∂J

, (28)

from (27).13 At J*, the marginal benefit of another input demander, the increase in profit

associated with the reduction in opportunism, is just balanced by the marginal cost of

another input demander, the decrease in profit due to higher congestion. Supposing the

function p(J) to be strictly concave over the range J > J~ ensures that the second-order

conditions hold for (28). In this case, the choice between the two regimes depends on the

global level of demander profit.

It is most natural to assume that the equilibrium number of demanders is

determined by free entry. Let p(J) represent the profit of an individual demander in a

site with (J – 1) other demanders in the disintegrated equilibrium, and let Ĵ represent the

zero profit number of demanders. From (24), ignoring integer problems,

†

ˆ J solves

)Ĵ,d(a2)q(fq

)q(f=¢+ , (29)

13 There is, in fact, a third mixed regime. As noted in Section III, this requires J = J

~. This cannot be an

optimum, however, because with congestion profits would be higher at both J = 1 and J = J*.

30

where we have substituted for p from (21).

There are many cases to consider. We will focus on the case where congestion

costs are low enough that p(J*) > p(1), so the optimum involves vertical disintegration.

The relationship of equilibrium to optimum depends on the level of profit evaluated at J*.

It is possible that even though demander profit is maximized at J*, the maximized level

of profit is negative. In this uninteresting case, the equilibrium number of firms is zero.

It is also possible that p(J*) = 0 and J* ≥

†

˜ J . In this knife-edge case, also uninteresting,

the equilibrium and optimum allocations coincide.

The interesting case is where p(J*) > 0, which is depicted in Figure 4. In this

case, under the assumption of concave p(J), there may be two candidate values for

†

ˆ J ,

corresponding to the two roots of p(

†

ˆ J ) = 0. However, even if both roots are positive,

only the larger of these, the one that lies on the downward-sloping portion of p(J), is

relevant. Any root smaller than J* is an unstable equilibrium, since profit is increasing in

the number of demanders at that point. This implies that, if p(J*) > 0, then

†

ˆ J > J*.

Individual demanders ignore the impact of their decision on the level of profit, and this

leads to excessive entry. This means that with congestion arising from fixed factors,

local outsourcing can involve input markets that are too "thick," in marked contrast to

many of the results in the literature on national and international outsourcing.

V. Globalization and cities

The paper has thus far considered two related aspects of the organization of

production: whether production should involve vertical integration and whether firms

should agglomerate. There have been several recent papers that have considered the

parallel issue of the international vertical disintegraton of production. As noted earlier,

Feenstra (1998) documents the increasing international component of vertical

disintegration and reviews its sources and consequences, especially for wages. McLaren

31

(2000) and Grossman and Helpman (2002a,b) present models that illuminate the process.

The heart of these models is that increasing international openness leads input markets to

be thicker. Thicker markets allow a disintegrated equilibrium, although the complete

integration of production remains an equilibrium as well. Thus, reductions in trade

barriers, part of the broad phenomenon of globalization, are naturally accompanied by an

increase in vertical disintegration. Feenstra (1998, p. 31) writes that increases in

international outsourcing "represent a breakdown in the vertically integrated model of

production -- the so-called 'Fordist' production, exemplified by the automobile industry --

on which American manufacturing was built." The choice is thus between integrated

production and disintegrated production, with the latter sometimes involving offshore

input production. Agglomeration is not part of this analysis at all.

We will conclude the paper by adding an international dimension to our model,

thus offering a unified analysis of globalization and agglomeration. As will be seen, the

analysis reaches sharply different conclusions about the effects of globalization than do

models of home and offshore production that ignore agglomeration.

We will begin with the model of agglomeration and vertical integration from

Section IV. The key modification will be that a demander now has the option to obtain

inputs internationally. We suppose that the transportation cost per unit of input obtained

internationally to be t. These transportation costs should be broadly conceived, including

both actual transportation costs and various border costs. Gravity models of trade (i.e.,

Baier and Bergstrand (2001)) show these costs to be important. We suppose that the cost

of producing the input internationally remains c and that the cost of integrated production

remains k. It is possible, perhaps likely, that the costs would differ. Taking the home

market as a developed country, the direction of the difference is not completely clear,

however, with low labor costs in less developed countries weighed against high

productivity in developed countries. In contrast, the effects of globalization on market

thickness are unambiguous. We will denote by JW the number of suppliers a demander

32

has access to on the world market. Since the demander could always make use of local

suppliers, it is natural to suppose that the equilibrium number of demanders in a city

†

ˆ J <

JW.

Knife-edges aside, a demander now has three choices: produce internally,

disintegrate locally, and disintegrate internationally. The cost per unit of input under

internal production is k, as before. Recognizing that the equilibrium distance to a

supplier is 1/[2(

†

ˆ J -1)], the per unit cost under local disintegration is c + a(1/[2(

†

ˆ J -1)]), also

as before. Similarly, the per unit cost under international disintegration is c+ a(1/[2(JW -

1)]) + t. Thus internal production is chosen when

k-c < min { a(1/[2(

†

ˆ J -1)]) , a(1/[2(JW -1)]) + t}, (30)

while local disintegration occurs when

a(1/[2(

†

ˆ J -1)]) < min {k-c, a(1/[2(JW -1)]) + t}. (31)

International disintegration takes place when

a(1/[2(JW -1)]) + t < min {k-c, a(1/[2(

†

ˆ J -1)]) }. (32)

The choice between these alternatives is illustrated in Figure 5. The figure the

organization of production that would arise in equilibrium as a function of k-c and t, the

cost advantage of outsourcing and transportation costs. There are two ways to read the

figure. First, looking at an industry with a particular value of k-c, the organization of

production may depend on the cost of transportation. For high values of k-c, high

transportation cost industries are outsourced locally. For low transportation costs, the

outsourcing is international. For these industries, the cost advantage of outsourced

33

production is too great for vertical integration to occur. For intermediate values of k-c,

specifically for aW ≡ a(1/[2(JW -1)]) < k-c < a(1/[2(

†

ˆ J -1)]) ≡ aL, high transportation costs

imply vertical integration, while low transportation cost industries outsource

internationally. Finally, for low values of k-c, the cost advantage of outsourcing is so low

that vertical integration is always the best remedy for the problems of opportunism.

The second way to look at the figure is to consider the impact of the cost

advantage of outsourcing for given transporation costs. The case of high transportation

costs gives vertical integration for low k-c and local outsourcing for high values. This is

the case considered in Proposition 3. For t < aL - aW, the choice is between international

outsourcing and vertical integration.

It has been argued that international outsourcing has arisen in part as a response to

declines in transport costs (see Baier and Bergstrand (2001)). Looking at Figure 5, shows

that only a small fraction of the newly globalized industries were formerly vertically

integrated. Industries with large values of k-c instead make the transition from locally

outsourced production. then the transition is not from integrated production to

internationally outsourced production. This means that international outsourcing is not

simply a devolution of previously integrated firms, but also has fundamental impacts on

the spatial organization of production.

This result sheds new light on the effects of globalization on North American

industry. The integrate-or-disintegrate-internationally analysis that underlies Feenstra is

neutral with regard to the impact of globalization on cities. This section's analysis

suggests that by allowing international disintegration, globalization will have a profound

effect on cities. The input supplier-input demander nexus that led to agglomeration is

replaced, for low enough global transportation cost, with a supply chain characterized by

offshore suppliers and disagglomerated demanders.

34

Appendix

The comparative statics of q1 are:

0)qqQ(f)qq(f

)d(a)d(a

x

q0

210

11

211 <+-¢¢++¢¢

¢+¢=

∂

∂ (A.1)

0)qqQ(f)qq(f

)qQ(f

Q

q0

210

11

11 >+-¢¢++¢¢

-¢¢=

∂

∂ (A.2)

2,1j, 0)qqQ(f)qq(f

)d(a

y

q0

210

11

j

j

1 =<+-¢¢++¢¢

¢-=

∂

∂ . (A.3)

The comparative statics of price are:

)qqQ(f)qq(f

)d(a)qqQ(f)d(a)qq(f)d(a

x

q)qq(f

x

p0

210

11

10

2120

111

1011

+-¢¢++¢¢

¢+-¢¢-¢+¢¢=¢-

∂

∂+¢¢=

∂

∂ , (A.4)

0)qqQ(f)qq(f

)qqQ(f)qq(f

Q

q)qq(f

Q

p0

210

11

021

01110

11 <+-¢¢++¢¢

+-¢¢+¢¢=

∂

∂+¢¢=

∂

∂ , (A.5)

0)qqQ(f)qq(f

)d(a)qqQ(f)d(a

y

q)qq(f

y

p0

210

11

10

211

1

1011

1

>+-¢¢++¢¢

¢+-¢¢=¢+

∂

∂+¢¢=

∂

∂ , (A.6)

0)qqQ(f)qq(f

)d(a)qq(f

y

q)qq(f

y

p0

210

11

20

11

2

1011

2

<+-¢¢++¢¢

¢+¢¢-=

∂

∂+¢¢=

∂

∂ . (A.7)

35

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Figure 1. The price and allocation of external inputs

y1 y2

f'(q1)-a(x-y1)

f'(Q-q1)-a(y2-x)

x

v1

p

v2

41

Figure 2. A symmetric equilibrium with vertically disintegrated production

y1 y2(y2+y1)/2

f'(q1)-a(x-y1)f'(Q-q1)-a(y2-x)

k

x

y1 y2(y2+y1)/2

f'(q1)-a(x-y1)f'(Q-q1)-a(y2-x)

x

v

c

p

42

Figure 3. A fully integrated equilibrium

x

y1 y2

f'(q1)-a(x-y1)f'(Q-q1)-a(y2-x)

k

(y2+y1)/2

43

Figure 4. Optimum and equilibrium city size

J*J

v1

J~ Ĵ

44

Figure 5. Globalization and the organization of production.

aWk-c

t

LocalDisintegration

InternationalDisintegration

VerticalIntegration

aL

aL-aW