WORKING PAPER Department of Economics Tufts ...

WORKING PAPER

Product Competition and Upstream Flexible Specialization

Lynne Pepall Department of Economics Tufts University and George Norman Department of Economics Tufts University Discussion Paper 98-18 Department of Economics Tufts University

Department of Economics Tufts University Medford, MA 02155 (617) 627-3560

PRODUCT COMPETITION AND UPSTREAM FLEXIBLE SPECIALIZATION1

Lynne Pepall and George Norman Tufts University Medford MA 02155

ABSTRACT

This paper examines the relationship between a differentiated downstream market and a specialized upstream market. We analyze four different ways price competition takes place between the upstream and downstream sectors when the upstream market supplies specialized and complementary inputs to a downstream product differentiated market. The first is the benchmark case of decentralized markets, the second is a network of alliances among upstream suppliers, the third is partial vertical integration and the fourth is an upstream consortium of suppliers. We identify the perfect equilibrium for a symmetric model in each case.

The key factor affecting prices and the relative profitability of the different market

organizations is the degree of product differentiation among the downstream firms.

This factor is

important not only because it affects profitability downstream, but also because it affects indirectly the intensity of competition among upstream suppliers.

We also show that vertical foreclosure by the

partially integrated firms is not an equilibrium strategy.

JEL CLASSIFICATION: D40, L22.

KEYWORDS: Complementarities, Flexible Specialization, Networks, Integration.

1

Financial support from the German Marshall Fund is gratefully acknowledged.


1.

INTRODUCTION The Wall Street Journal recently reported that “More and more of the work in America is project

oriented, with a beginning, a middle, and an end. Projects lend themselves to a blend of traditional employees, contract workers and consultants, who combine into teams, do a job and then usually break up, with most of the players looking for their next gig.”2 The project-based approach to doing business has in fact long been practiced in the building trade and in Hollywood where independent specialized production companies create an alliance to work together jointly on a project. What is drawing current attention is that this approach is spreading, most notably in information services where high-tech contractors are the norm. In parallel with the spread of the project-based approach has been the emergence in recent years of increasingly varied relationships between upstream suppliers of specialized services and their parent downstream company. Most notable is that many large companies such as Shell, Xerox, AMR Corporation, and Scandinavian Airlines, are spinning off information technology, training and consulting services into separate divisions in order to market these specialized services to companies other than their parent company. Again other examples are found in Hollywood. The first television program developed by the partnership between the network ABC and its new studio Dreamworks was a show made for CBS. Similarly, another production company, Brillstein/Grey, has a deal with ABC to develop a comedy to air on NBC.

Other suppliers of prime-time programs, such as Paramount and Warner

Brothers, want to start their own networks. Some of the programs appearing on these new upstart networks are shows that NBC paid to develop. Fox Television develops programs for every network but also has a competitive network of its own. These changes in the upstream provision of specialized services to downstream markets bring to mind the notion of flexible specialization, a term coined by Piore and Sabel (1984) for the labor market. Flexible specialization refers to how a network of specialized upstream suppliers can facilitate the creation of alliances among these suppliers and can lead to a more decentralized system of production than full vertical integration of upstream and downstream production.

2

An interesting feature that

Cited in “Flying Solo: High Tech Nomads Write New Program for Future of Work”, The Wall Street Journal, page A1, August 19 1996.

2


characterizes the empirical studies by Piore and Sabel and others3 on flexible specialization is that the upstream suppliers supply their specialized services to downstream firms that produce differentiated products. This observation suggests to us that the degree of product differentiation in the downstream market may be an important factor affecting the spread of flexible specialization or, in current parlance, the project-based approach to organizing upstream and downstream production. More generally, product differentiation in the downstream sector will also affect how the upstream suppliers compete, and therefore, will affect the way these services are supplied. There is an extensive literature in industrial organization on product differentiation but this literature has focused primarily on price and product competition in the downstream market.4 There is an equally extensive literature on vertical relationships between upstream and downstream firms.5 However, this literature is typically characterized by the assumption that upstream firms produce perfect substitutes that they supply to downstream firms who also produce perfect substitutes.6 The relationship between a differentiated downstream market and a differentiated upstream market has received much less attention. This strikes us as surprising, since one of the most important means by which downstream firms differentiate their products is through the use of inputs that are themselves differentiated or specialized. The goal of this paper is to investigate more closely how the degree of product differentiation in the downstream affects the profitability of alternative methods of organizing transactions between upstream suppliers and their downstream customers. To do this, we develop a stylized model of an upstream sector that supplies specialized inputs to a product differentiated downstream sector. Our model captures two important features of upstream and downstream sectors that have been affected by the project-based approach. First, a project draws upon more than one upstream supplier, and the

3

Piore and Sabel studied flexible specialization in the ceramics, textiles and knitwear industries in Emilia-Romagna regions of Italy. Subsequent studies were done on other industries were done by Goodman and Bamford (1989), Pyke, Beccattini and Sengenberger (1990), Storper and Christopherson (1987) and Saxenian (1992). 4 See Beath and Katsoulacos (1991) for a survey of the economic theory of product differentiation, and Thisse and Norman (1995) for a collection of many of the most important papers. 5 Some of the notable contributions include Bolton and Whinston (1993), Bonanno and Vickers (1988), Gaudet and Van Long (1996), Hart and Tirole (1990), Ordover, Saloner and Salop (1990) and Salinger (1988). 6 Exceptions are Bolton and Whinston (1993) who investigate the vertical relationship between an upstream firm who supplies two downstream firms selling in completely differentiated markets and Bonanno and Vickers (1988) who consider two firms which produce differentiated products that compete with each other. They investigate the profitability of selling directly to consumers compared with that achieved by selling through independent retailers.

3


upstream suppliers who come together in a project typically supply complementary specialized inputs that are combined to produce the good sold in the downstream market.7 Second, different combinations of these complementary specialized inputs are used to produce different downstream products. We begin our analysis with the benchmark case of a fully decentralized organization of the upstream and downstream markets. For this case the downstream producers buy their differentiated and complementary inputs at (linear) prices set by independently owned and managed upstream suppliers.8 The downstream producers produce differentiated goods and compete in prices for consumers of their products. The equilibrium set of upstream and downstream prices for the decentralized case is affected by two coordination failures: (i) a pecuniary externality between the complementary input suppliers, and (ii) double marginalization between the upstream and downstream producers.

These coordination

failures give rise to profit incentives for the downstream producers and the upstream suppliers to create different horizontal and/or vertical relationships. We then consider three alternative ways of transacting business between the upstream and the downstream that at least partially internalize the pecuniary externalities and/or remove the effect of double marginalization.9 The first is horizontal networking among upstream suppliers, an organizational form that is perhaps closest in spirit to flexible specialization as described in Piore and Sabel (1982). Networking allows the suppliers of complementary inputs to create alliances and to coordinate the prices set to a downstream customer. Such alliances facilitate coordination horizontally but not vertically, and thus we assume that an alliance is constrained to charge linear prices to the downstream customer. As a

Complementarity is associated with the recent literature on product networks, which builds on the Cournot (1838) model of complementary duopoly. Cournot showed that joint ownership of two complementary products reduced the sum of the two prices relative to those prices set by two independent producers because joint ownership internalizes the pecuniary externality associated with the decentralized production and trading of complementary goods. Such pecuniary externalities in the context of product networks are called indirect network externalities. Economides and Salop (1992) generalize the Cournot model to the case of multiple producers of differentiated downstream products and analyze how indirect network externalities affect competition and integration among the producers of complementary goods in a product network. In this paper, we also consider multiple producers of differentiated complementary goods. However, in contrast to Economides and Salop (1992), the differentiated complementary goods are intermediate input goods demanded by downstream firms to produce a set of differentiated final products. 7

8

Each upstream supplier has potentially many customers operating in different downstream markets. Moreover a downstream firm relies on more than one upstream supplier. Nonlinear pricing is, therefore, less appropriate than in the case studied, for example, by Hart and Tirole (1990), particularly if upstream and downstream firms are fully decentralized. 9 Our comparison of the different organizational structures is comparative static, in that we identify sufficient but not necessary conditions under which particular organizational structures will emerge. We leave to subsequent analysis a full consideration of the endogenous determination of organizational structure.

4


result, networking upstream eases the pecuniary externality but does not solve the problem of double marginalization between the upstream and downstream. The second arrangement that we consider is partial vertical integration in which some, but not all, downstream producers own their specialized upstream suppliers. For this case we must also consider how an integrated firm internally organizes itself since, with competition in the downstream market, it is not at all clear that marginal cost pricing of the upstream division’s services will be desirable. In other words some form of vertical separation10 in the internal organization of the integrated firm can help soften price competition in the downstream market. Therefore we consider three different forms of partial vertical integration.11 The first is analogous to the traditional U-form of internal organization, whereas the remaining two are variations on the M-form of internal organization. i)

Centralized integration in which the upstream division is given no autonomy over the pricing of input services it supplies to the downstream unit, in which case they will be supplied at marginal cost.

ii)

Divisional integration in which the upstream unit is established as a separate division and profit center and sets the prices of its services to maximize its divisional profit.

iii)

Divisional integration in which the upstream unit is established as a separate division and profit center but in this case it sets prices to maximize the integrated firm’s profit. An important result that emerges from our analysis of each organizational form of partial vertical

integration is that vertical foreclosure of the non-integrated downstream rivals is never an equilibrium strategy. In all three cases it is profitable for the upstream divisions of the integrated firm to supply their services to downstream customers outside of the parent company. This appears to be consistent with the stylized facts when upstream divisions of companies, such as Shell and Xerox seek out downstream customers outside the parent company. However, the result does contrast with the analysis of Ordover,

The incentive for vertical separation between manufacturers and retailers is explored in Bonanno and Vickers (1988). The logic of their argument is relevant in this context as well. 11 We note here that there is an alternative way to interpret the different forms of partial vertical integration that we consider. The alternative reflects the view noted, for example, in Tirole (1987), that the benefits of vertical integration can often be realized through some type of non-linear pricing scheme. Similarly here, the benefits of partial vertical integration could be achieved if we allow an alliance of upstream firms to apply non-linear prices to those downstream customers for whom that alliance is the sole supplier. 10

5


Saloner and Salop (1990). The difference is that in our model the exit of the foreclosed downstream firms would increase competition in the downstream market so as to make foreclosure unprofitable. Even though the specialized upstream suppliers are in some sense monopoly suppliers to their downstream customers, neither networking upstream or partial vertical integration confer on these suppliers the ability to set monopoly prices for their specialized services. There is indirect competition between the specialized upstream suppliers, which is mediated through the competition in the downstream markets. The third arrangement that we consider allows the upstream suppliers to exploit more fully their monopoly power.

It is a full consortium of the upstream suppliers. The upstream

suppliers in the consortium are able to set a two-part tariff to all of the downstream firms and are able to coordinate the prices charged to each downstream buyer with the result that the consortium can use nonlinear pricing to extend its monopoly power into the downstream market. We show that the different methods of organizing transactions between the upstream and downstream affect prices and profits in a manner that is, in some cases, Pareto-improving.

For

example, a move away from the decentralized benchmark case always benefits consumers through lower prices for the downstream products. What is, perhaps, surprising is that the upstream and downstream producers do not necessarily gain from closer ties between them. Our analysis makes clear that the relative profitability of the different relationships between the upstream suppliers and the downstream producers depends upon the degree of product differentiation in the downstream sector. The reason that this factor turns out to be so important is that it affects the intensity of the indirect competition among the upstream suppliers of specialized inputs, and hence their incentive to develop closer ties with each other and with the downstream market. In the next section we describe our stylized model of upstream and downstream production. The following four sections analyze each of the different ways in which competition is structured between the upstream and downstream markets. The relative effects of these market structures on prices and profits are detailed in Section 7 and Section 8 presents our summary and conclusions. 2.

T HE MODEL We assume that the upstream market contains four monopoly suppliers of differentiated brands

of two basic complementary input goods or services. Specifically, we suppose that there are 2 firms

6


each supplying a differentiated or specialized input xj, and 2 firms each supplying a differentiated input zk, j, k = 1, 2. The two inputs are compatible with each other and the marginal cost of each specialized input is assumed to be constant and equal, implying that these costs can be normalized to zero. Each downstream firm produces at most one differentiated product using a constant returns to scale, fixed proportions production technology. Product differentiation arises because the downstream firms use the specialized upstream inputs in different combinations. There are four potential differentiated goods that can be produced for the downstream market. One unit of the differentiated product jk, where jk = 11, 12, 21 or 22, is produced by combining one unit of input brand xj with one unit of input brand zk (j, k = 1, 2). Specifically, firm 11 produces q11 using inputs (x1,z1), firm 12 produces q12 using inputs (x1,z2), firm 21 produces q21 using inputs (x2,z1) and firm 22 produces q22 using inputs (x2,z2). We can refer interchangeably to downstream product jk and downstream firm jk. The constant-marginal-cost and fixed-proportions-technology assumptions are characteristic of much of the literature which examines the relationship between the upstream and downstream sectors. In the interests of analytical tractability we make the further simplifying assumption that downstream demand is linear in prices. We show in the Appendix that the price effects of different upstream/downstream arrangements can be extended to a more general demand specification. The profit effects are more difficult to generalize. It seems reasonable to expect, however, that our qualitative conclusions will also extend to more general specifications, particularly in view of the importance to these effects of the degree of product differentiation in the downstream market. Demand for downstream products is assumed to be linear and symmetric with the demand for a differentiated good jk in the downstream market being described by: (1)

q jk = 1 − p jk −

(

)

1 p jk − p , γ

where p =

1 4

22

∑p

jk

(jk = 11, 12, 21, 22).

jk =11

The parameter γ measures the degree of product differentiation, or substitution, among the four products in the downstream market. As γ increases the products become more differentiated in consumption. When γ → ∞ the products are completely differentiated - each firm in the downstream is a single-product monopolist - whereas when γ → 0 the downstream products are perfect substitutes in consumption. A

7


major advantage of our simplifying assumptions is that the relative merits of the different vertical relationships between upstream and downstream firms are completely determined by γ. Price competition occurs in two stages. The upstream suppliers set input prices in the first stage. Let v = (v1, v2) denote the vector of prices of the specialized input goods x = (x1, x2) and w = (w1, w2) denote the vector of prices of the specialized input goods z = (z1, z2). In the second stage, downstream firms, facing these input prices, produce differentiated products and compete in prices p = (pjk), jk = 11, 12, 21, 22, for their consumers. We solve for the subgame perfect Nash equilibrium to this two-stage price game for each of the market structures. 3.

T HE BENCHMARK CASE OF DECENTRALIZED PRICING: The first case we consider is the benchmark case of decentralized price setting. This will, of

course, be characterized by two coordination problems: a pecuniary externality between upstream suppliers of complementary inputs and double marginalization between the upstream and downstream producers. The problem facing a downstream firm jk in the second stage subgame is to choose the profit-maximizing price p ∗jk for its product given the prices it faces for its upstream inputs. The solution to this problem can be written as: (2)

{ (

)(

p ∗jk = argmax q jk p jk , p p jk − v j − w k

)}

j, k = 1,2.

The solution to each firm's maximization problem leads to a best response function in prices from which the set of equilibrium downstream prices in the stage two subgame can be found. The equilibrium prices

{

}

p ∗ (v, w ) = p ∗jk (v, w ) are detailed in the Appendix. Given the set of equilibrium prices p*(v,w) from the second stage subgame, an upstream firm U xj derives the input demand function x ∗j (v, w ) that it faces in the first stage of the game. Note that an upstream firm U xj supplies two downstream firms with its specialized input xj. Similarly an upstream firm Ukz derives its input demand function z k∗ ( v, w ) .

In other words, the derived demands facing each

upstream supplier are:

8


( ) ( ) (p (v, w)) + q (p (v, w))

x ∗j ( v, w ) = q j 1 p ∗ ( v, w ) + q j 2 p∗ ( v, w ) (3)

z k∗ ( v, w ) = q 1k

∗

(j, k = 1,2).

∗

2k

In the first stage of the game the firms U xj and U kz , j, k = 1,..,n, choose independently their input prices vj and wk, respectively, to maximize profits defined as:

(4)

Π xj (v, w ) = v j x ∗j (v, w ) Π kz (v, w ) = w k z k∗ (v, w )

The input price reaction functions (see Appendix) indicate that the input prices of complementary inputs are strategic substitutes while the input prices of competing inputs are strategic complements. Solving the first-order conditions from (4) gives the Nash equilibrium input prices, which in our symmetric example are identical across all four inputs and are described by12: (5)

v ∗d =

2 γ ( 8 γ + 7) 48γ 2 + 46γ + 3

.

Substituting (5) in the reaction functions for output prices gives the Nash equilibrium prices: (6)

p ∗d =

(

8 γ 40 γ 2 + 49 γ + 12

(8γ + 3)(48γ

2

)

+ 46 γ + 3

)

As we would expect, ∂v ∗d ∂γ > 0 and ∂p ∗d ∂γ > 0 . The Nash equilibrium input and output prices are both increasing functions of the degree of product differentiation in the downstream market. As γ → 0, the products in the downstream market are less differentiated in consumption, downstream price competition becomes tougher and the equilibrium input price v ∗d → 0 , which is, of course in our example, marginal cost. However, even a small degree of product differentiation is sufficient to generate 13

a sharp increase in the Nash equilibrium downstream prices.

By contrast, as γ → ∞ the downstream

products are increasingly differentiated. Price competition in the downstream becomes softer, and as a result the equilibrium input prices v ∗d → 1 3 and the equilibrium output prices p ∗d → 5 6 .

The superscript ‘d’ refers to the decentralized case. It is easy to show that the elasticity of upstream and downstream prices with respect to γ is unity when γ = 0 and tends to zero as γ increases. 12 13

9


The impact of increased product differentiation on upstream and downstream profits is less clear-cut, as can be seen from Figure 1. On the one hand, softer price competition in the downstream increases upstream and downstream profits.

On the other hand, an equilibrium input price v ∗d >0

reflects two coordination failures that dissipate these profits. First, there is the pecuniary externality. The two upstream firms producing complementary differentiated inputs (xj ,zk ) could earn higher profits if they both were to reduce their prices but the decentralized method of transacting business does not allow for such coordination. Secondly, there is the familiar problem of double marginalization in the pricing of the inputs sold to the downstream firms. Both market failures are greater the larger is γ, which is a measure of the degree of monopoly power in the downstream market. For γ > 0.219 (0.121) the profit-reducing effects of the coordination failures offset the profitincreasing effects of product differentiation and Nash equilibrium profits of the upstream (downstream) firms fall as γ increases. The upstream firms are, however, always more profitable than the downstream firms because of the monopoly power they enjoy. The difference in profits between the upstream and downstream is greatest when γ is relatively small (equal to 0.112). In the limit as γ → ∞ we find that Π U∗d → 1 9 ; Π D∗d → 1 36. (Figure 1 near here) 4.

AN UPSTREAM SUPPLIER NETWORK: Coordination failures associated with our decentralized benchmark provide upstream and

downstream producers with an incentive to consider other methods of transacting business. We begin first by assuming that the upstream suppliers establish a supplier network, which facilitates the formation of alliances among suppliers of complementary inputs. An upstream supplier network allows for coordination of the pricing of complementary inputs and, as a result, at least partially internalizes the pecuniary externality in the upstream markets. We assume, however, that the members of the supplier network continue to employ linear pricing, so that the problem of double marginalization remains.

10


An important feature of a supplier network is that it facilitates multiple alliances for any one upstream firm.14

Two upstream suppliers come together to work on producing a specialized input

package for a downstream client. A crucial feature of our analysis is that these suppliers may be working at the same time with other partners on other specialized input packages for different downstream clients. As a result, upstream suppliers in a network alliance choose their input prices in a way that differs from the decentralized case. In particular, there is coordination among complementary suppliers of inputs. Cooperation is not, however, without competition. Each upstream supplier has many partners, and the different partnerships supply different downstream firms who are in competition with each other. Consider an alliance between two complementary upstream suppliers, say U jx and U kz . The stage two subgame is the same as in the decentralized case (equation (2) above). Downstream firms, facing the input prices {v,w} chosen by the upstream suppliers in the network, buy their inputs and compete in price for customers in the downstream market. However, the network of alliances affects stage one of the game.

Because of their alliance, the upstream suppliers are able to coordinate the

marketing of the specialized package of inputs (xj,zk) demanded by their downstream customer. The alliance's profit is defined by: (7)

(

)

Π xz jk = v j + w k q jk (p * (v, w ))

where qjk(p*(v,w)) is the derived demand for the input package (xj,zk). There remains the issue of how this profit should be shared between the members of the alliance.

We assume that the upstream

suppliers share the profit equally, the Nash bargaining solution for this symmetric game. In a network of alliances, upstream supplier U jx forms alliances with both of the upstream suppliers who produce inputs complementary to xj.

Hence, the maximization problem facing the

upstream supplier U jx is to choose an input price vj to maximize its overall return from being a member

We do not consider the formal mechanisms by which supplier networks might be formed, although there is anecdotal evidence that such networks can be difficult and costly to organize (Rosenfeld, Daffner and Meade (1992)). Our analysis does, however, suggest for what kinds of markets they are more likely to succeed and be beneficial for consumers and producers. This may help policy makers to decide whether or not providing assistance in the organization of networks is a policy goal that is worth pursuing. 14

11


of the network of alliances. That is, in stage one, the firm U jx chooses its input price vj to maximize its total profit: (8)

Π xn j =

1 2

2

∑ (v

j

)

+ w k q jk (p * (v, w )).

(j = 1, 2)

k =1

The maximization problems facing the other upstream suppliers can be similarly defined and determine how input prices are chosen in stage one of the game. The reaction functions for the network alliance (see Appendix) are once more such that the input prices of complementary inputs are strategic substitutes while the input prices of competing inputs are strategic complements. Solving the reaction functions gives the Nash equilibrium input prices for the supplier network. These are identical across all inputs and given by: (9)

v ∗n =

γ ( 8 γ + 7) 32γ 2 + 32γ + 3

Substituting in the reaction functions for output prices gives the Nash equilibrium output prices:

(10)

p ∗n =

(

2 γ 96 γ 2 + 116 γ + 27

( 8γ + 3)( 32γ

2

)

)

+ 32 γ + 3

.

Comparison with (5) confirms that for every value of γ the upstream supplier network leads to lower equilibrium input prices and so to lower equilibrium output prices than does the decentralized structure. This is, of course, to be expected given that the network at least partially removes the coordination failures between suppliers of complementary inputs. Again, as expected, ∂v ∗n ∂γ > 0 and ∂p ∗n ∂γ > 0 . The Nash equilibrium input and output prices are both increasing functions of the degree of product differentiation in the downstream market. As γ → 0 it can be seen that we have a similar outcome as in the decentralized case: v ∗n , p ∗n → 0 . By contrast, as γ → ∞, the equilibrium input prices v ∗n → 1 4 and the equilibrium output prices p ∗n → 3 4 .15 Profits of the upstream and downstream producers continue to be affected as γ increases by the trade-off between softer downstream price competition and coordination failures as can be seen from

15

Given our demand system, these are just the prices that would be charged by a monopoly producer of a composite upstream input and a downstream monopolist.

12


Figure 2. However, the critical value of γ at which the latter become dominant is higher than in the decentralized case (γ = 0.24 upstream and γ = 0.55 downstream). (Figure 2 near here) 5.

PARTIAL VERTICAL INTEGRATION: The network alliance goes some way to removing the pecuniary externality between suppliers of

complementary inputs but leaves the problem of double marginalization unsolved. Vertical integration, by contrast, is often viewed as being equivalent to a long-term complete contract that eliminates the problem of double marginalization between upstream suppliers and their downstream customers. In this section we consider a vertical relationship, partial vertical integration, that is intermediate between the network alliance and complete vertical integration. Specifically, by partial vertical integration we mean a situation in which a subset of the downstream firms integrate backwards into their upstream suppliers. If the integrated firms choose to continue to supply the nonintegrated downstream firms that remain, partial vertical integration only partially solves the upstream/downstream coordination problems. Without loss of generality we assume that downstream firm ii integrates with its upstream suppliers U ix and U iz and becomes firm Iii (i = 1,2). The downstream firms 12 and 21 remain nonintegrated and, assuming that there is no vertical foreclosure, buy their inputs from the upstream divisions of the two integrated firms. Firm 12 buys x1 from I11 at price v1, and z2 from I22 at price w2. Downstream firm 21 buys x2 from I22 at price v2 and z1 from I11 at price w1. While we concentrate throughout on the vertical integration interpretation of this arrangement it is important to point out that the resulting Nash equilibrium can be replicated by another market arrangement.

Assume instead that the upstream supplier of x1 forms an alliance solely with the

upstream supplier of z1 and that the upstream supplier of x2 forms an alliance solely with the upstream supplier of z2. Suppose also that these alliances are sufficiently close for the two suppliers to be able to apply two-part pricing to their captive downstream customer, firm 11 or 22. Each upstream supplier retains the right, however, to continue to supply its other downstream buyer at a linear price. The Nash equilibrium prices and aggregate profits generated by this partial non-linear pricing scheme replicate those for partial vertical integration, although the distribution of profits may well differ.

13


We assume initially that there is no strategic attempt at vertical foreclosure by the integrated firms. The upstream divisions of the two integrated firms continue to sell their specialized inputs to the two non-integrated downstream firms. Having identified the equilibrium for this case we show that foreclosure can never be an equilibrium strategy: the Nash equilibrium for an integrated firm is to continue to supply its non-integrated downstream competitors.16 We should also consider how an integrated firm internally organizes itself since, with competition in the downstream market, it is not at all clear that marginal cost pricing of the upstream division’s services will be desirable. The three different forms of partial vertical integration that we consider are: (i)

Centralized integration in which the upstream division is given no autonomy over the pricing of the input services supplied to the downstream unit, in which case they will be supplied at marginal cost.

(ii)

Divisional integration in which the upstream unit is established as a separate division and profit center and sets the prices of its services to maximize its divisional profits.

(iii)

Divisional integration in which the upstream unit is established as a separate division and profit center and sets prices to maximize the integrated firm’s profits. In the first stage of the game, each upstream unit sets the prices vi, wi for the specialized input

goods xi, zi that it sells to non-integrated firms in the downstream market and the price vii at which its combined input is sold to its own downstream unit. These prices will depend upon the way that the integrated firms are internally structured. However, no matter the structure, we should expect to find that the prices the downstream unit of the integrated firm pays for its inputs are different from those paid by a non-integrated downstream firm.

Consequently, the partially vertically integrated market structure

introduces an important asymmetry into the second-stage price game played by the downstream firms. In order to solve for the Nash perfect equilibrium we begin by looking at the maximization problem facing each firm in stage two of the game. The problem facing both the integrated and nonintegrated firms is to choose the profit-maximizing prices p ∗jk for their products sold in the downstream market to satisfy the conditions:

It will emerge below, however, that in some circumstances the non-integrated firms will be eliminated from the market when γ is “low” by demand for their products falling to zero. 16

14


(11a)

p ii∗ = arg max{q ii (p ii , p )(p ii − v ii )}

(11b)

p ij∗ = arg max q ij pij , p p ij − v i − w j

{ (

(i = 1,2)

)(

)}

(i,j = 1,2; i ≠ j)

The solution to each firm's maximization problem leads to a best response function in prices from which the set of equilibrium downstream prices in stage two of the game can be found. The equilibrium prices

{

}

p * ( v, w ) = p ∗jk ( v, w ) are derived in the Appendix, where now v = (v11, v22, v1, v2).17 We first present the Nash equilibria for these three organizational structures and then compare the prices and profits that they generate.18 5.1

CENTRALIZED PARTIAL VERTICAL INTEGRATION With this internal structure of the partially vertically integrated firms we have vii = 0 (i = 1, 2), so

that in the first stage of the game the integrated firm Iii chooses input prices vi and wi to maximize profit ∗ Π Ic ii = p ii (v, w )q ii (p * (v, w )) + v i q ij (p * (v, w )) + w i q ji (p * (v, w )) (i, j = 1,2; i ≠. j).

The resulting reaction functions indicate that the input prices vi and wi (i = 1,2) for a pair of complementary inputs produced by an integrated firm are now strategic complements.

For

complementary input prices set by different firms, (v1,w2 ) and (v2,w1), the relationship of strategic substitutes still holds. Solving the reaction functions gives the equilibrium prices for the traded inputs with centralized partial vertical integration: (12)

v i*cp = w i*cp =

(

2 γ(7 + 8γ ) 27 + 88 γ + 64 γ 2

)

153 + 1758 γ + 5840 γ + 7296 γ + 3072 γ 4 2

3

(i = 1, 2)

Substituting (12) into the reaction functions for output prices yields the following Nash equilibrium prices which hold for γ > 0.029674: (13a)

p ii∗cp =

(

4 γ (7 + 8 γ ) 15 + 62γ + 48 γ 2

)

153 + 1758 γ + 5840 γ + 7296 γ + 3072 γ 4 2

3

17

It makes no difference to the final outcome whether we incorporate the new internal prices internally set by the partially integrated firms into the vector v or w . 18 There remains the question about which of these organizational forms is more likely to emerge. This question raises a series of interesting but complicated issues that we hope to pursue in subsequent analysis.

15


(13b)

p ij∗cp =

(

8 γ 93 + 463 γ + 688 γ 2 + 320 γ 3

)

153 + 1758 γ + 5840 γ + 7296 γ + 3072 γ 4 2

3

For γ < 0.029674 (implying tough downstream competition) the prices charged by the integrated firms for the inputs used by the non-integrated downstream firms are sufficiently high that the nonintegrated firms are unable to sell positive quantities at their Nash equilibrium downstream prices. They exit from the market.

The exit of these high-cost firms further toughens price competition in the

downstream market, reducing prices and profitability of the integrated firms. The Nash equilibrium output prices for the two integrated firms are now: (13c)

p ii∗cp =

2γ 1 + 4γ

for γ < 0.029674.

It is easy to show that ∂v ∗cp ∂γ>0 and ∂p ∗cp ∂γ>0 for both upstream and downstream producers. Once again, the Nash equilibrium input and output prices are both increasing functions of the degree of product differentiation in the downstream market. Because the integrated firms acquire their specialized inputs at marginal cost equal to zero, whereas the non-integrated firms do not, there is of course an asymmetry in downstream prices with p ii∗cp < p ij∗cp . As γ → ∞, the equilibrium input price v*cp → 1/3 as in the decentralized case, while the equilibrium output prices p ii∗cp →1 2; p ij∗cp → 5 6 . 5.2

AN UPSTREAM PROFIT CENTER MAXIMIZING UPSTREAM DIVISIONAL PROFIT This organizational form of an integrated firm establishes the upstream unit as a divisional profit

center, charged with the responsibility of maximizing its own, rather than the firm’s, profit. Thus in the first stage of the game the integrated firm Iii chooses input prices vii, vi and wi to maximize profit Π Id ii = v ii (v, w )q ii (p * (v, w )) + v i q ij (p * (v, w )) + w i q ji (p * (v, w )) (i, j = 1,2; i ≠ j). Solving the resulting reaction functions gives the Nash equilibrium upstream prices: (14)

v ii*dp =

4 γ (7 + 8 γ )

(3 + 60γ + 64γ ) 2

; v i*dp = w i* dp =

(

8 γ (7 + 8 γ )

3 3 + 60 γ + 64 γ 2

)

(i = 1, 2).

As can be seen, v ii*dp < v i* dp + w i*dp , indicating that the upstream division of each integrated firm discriminates against its non-integrated downstream buyers. Moreover, the upstream division will never

16


adopt marginal cost pricing internally unless γ = 0, while v ii*dp → 1 2 and v i*dp , w i*dp → 1 3 as γ → ∞. These outcomes make sense. With γ “low”, competition in the downstream market is fierce, severely constraining the price-setting power of an upstream division. As γ increases, the upstream profit center behaves increasingly like an upstream monopolist with respect to its captive downstream customer. Substituting (14) into the downstream price reaction functions gives the Nash equilibrium downstream prices: (15a)

p ii* dp =

(15b)

p ij* dp =

(

8γ 39 + 172 γ + 144 γ 2

(

)

3(3 + 8γ ) 3 + 60 γ + 64 γ 2

(

)

4 γ 87 + 380 γ + 320 γ 2

(

3(3 + 8γ ) 3 + 60 γ + 64 γ

2

)

(i = 1, 2)

)

(i, j = 1, 2; i ≠ j)

In this case, the downstream prices are such that the non-integrated firms sell a positive amount no matter the degree of product differentiation. As we would expect, p ii∗dp < p ij∗dp . The input cost advantage enjoyed by the downstream units translates into lower downstream prices. As γ → ∞ we have p ii* dp → 3 4 ; p ij*dp → 5 6 . 5.3

AN UPSTREAM PROFIT CENTER MAXIMIZING GROUP PROFIT Under this form of internal organization, we assume that the upstream division of an integrated

firm is established as a profit center and allocated a profit share determined by the profitability of the integrated firm as a whole. As a result, in the first stage of the game the integrated firm Iii now chooses the

input

prices

vii,

vi

and

wi

to

maximize

profit

∗ Π Ig ii = p ii (v, w )q ii (p * (v, w )) + v i q ij (p * (v, w )) + w i q ji (p * (v, w )) (i, j = 1, 2; i ≠ j). Solving the resulting

reaction functions gives the Nash equilibrium upstream prices: (16)

v ii*gp =

4 γ (7 + 8γ )(21 + 16 γ ) ; (3 + 4γ )(51 + 896γ + 2304 γ 2 + 1536 γ 3 )

v i* gp = w i*gp =

32γ (7 + 8 γ )(1 + 2γ )

(51 + 896γ + 2304 γ

2

+ 1536 γ 3

)

This form of divisionalization also leads to price discrimination against the non-integrated downstream buyers. By contrast with the previous cases, however, we find that there is a non-monotonic relationship between the internal input price and the degree of product differentiation. The internal input

17


price initially increases with γ, but v ii* gp → 0 as γ → ∞. In other words, when the upstream unit seeks to maximize the integrated firm’s profit rather than its own divisional profit, a high degree of product differentiation in the downstream market will lead it to set its internal price “close” to marginal cost. It should be emphasized, however, that it is only when the downstream firms are effectively monopolists that this form of vertical integration leads to marginal cost pricing of the internally traded input. When there is competition in the downstream market, it is better for the upstream unit of an integrated firm to incorporate some margin in the pricing of its product to the downstream division even though this reintroduces double marginalization between the divisions (see Tirole, 1987).19 Substituting (16) into the downstream price reaction functions gives the Nash equilibrium downstream prices: (17)

p ii* gp =

48 γ (7 + 8 γ )(1 + 2γ )

(51 + 896γ + 2304 γ

2

+ 1536 γ

3

)

; p ij* gp =

(

4 γ 111 + 408 γ + 320 γ 2

(51 + 896 γ + 2304 γ

2

)

+ 1536 γ 3

).

As in the centralized form of partial vertical integration, there is a degree of product differentiation, in this case γ = 0.0175, below which the non-integrated firms will choose to exit the market. In this range of γ the Nash equilibrium price vii for the input is: (18)

v ii*g =

2γ

(1 + 2γ )(1 + 12γ + 16γ 2 )

for γ < 0.0175

(i = 1, 2).

The Nash equilibrium output prices for the two integrated firms are now: 4 γ (1 + 2γ )

(19)

p ii*g =

5.4

RELATIVE PRICES AND PROFITABILITY WITH PARTIAL VERTICAL INTEGRATION

(1 + 12γ + 16 γ ) 2

for γ < 0.0175

(i = 1, 2).

Prices under the three different forms of partial vertical integration are illustrated in Figure 3 and point to a simple relationship. Centralized integration gives the lowest prices whereas the divisional form in which the upstream unit is motivated only by its own profit gives the highest prices. The intuition behind this outcome is straightforward. Supplying the input internally at marginal cost toughens price competition in the downstream market and so constrains the prices of upstream services that can be

If we think of this as a two-part pricing story between non-integrated firms, we have the same result that the unit price of the input should be set to marginal cost only if the downstream market is monopolized. In any other situation (other than γ = 0) the unit price 19

18


charged to non-integrated firms. By contrast, when an upstream division is concerned purely with its own profits, it has the incentive to maintain reasonably high internal prices for its services, which in turn allows it to charge higher external prices for its services. The monopoly power of the autonomous upstream division is, of course, constrained by competition in the downstream market, but this is felt much more indirectly than is the case with either of the other two organizational forms. (Figure 3 near here) Figure 4(a) shows, not surprisingly, that profit of each non-integrated firm increases with the degree of product differentiation in the downstream market. Note however that there is the same ranking between organizational form of the integrated firm and a non-integrated firm’s profitability as there is between organizational form and input prices. This ranking reflects a trade-off between two forces. On the one hand, an organizational form that leads to lower external input prices benefits the non-integrated firms by reducing their costs. On the other hand, lower internal input prices harm the non-integrated firms by making them less price competitive with their integrated competitors. What is quite clear from Figure 4(a) is that the detrimental price competitive effect more than offsets the beneficial cost effect. No such simple relationship characterizes profits for the integrated firms as can be seen from Figure 4(b).

No matter the form that partial vertical integration takes, profitability is affected as γ

increases by the trade-off between softer downstream price competition and the coordination failures that remain under partial vertical integration. Of much more interest, however, is the result that the relative profitability of the different ways of internally organizing the integrated firm also varies with the degree of product differentiation. (Figure 4 near here) With the exception of the polar cases where γ = 0 or γ = ∞, centralized integration, or the traditional U-form, is always less profitable than divisional integration in which the upstream division’s concern is with group profits. Divisional integration reintroduces the efficiency losses associated with double marginalization. However, setting a higher internal input price softens price competition among the integrated firms and therefore increases the external input price that can be charged to non-

should be greater than marginal cost.

19


integrated firms. The ability of divisional form of integration to soften downstream competition more than offsets the efficiency loss of double marginalization that it introduces. However, divisional integration in which the upstream unit is concerned with group profits is not always more profitable than divisional integration in which the upstream unit is a profit center concerned solely with its own profits. Specifically, if the degree of product differentiation in the downstream market is less (greater) than γ = 0.459 an integrated firm will earn greater aggregate profits if the upstream division aims to maximize the upstream division’s (the integrated firm’s) profit. In other words, while a divisional organizational structure is preferable to a centralized structure, the form that divisionalization takes “matters”. The intuition behind this initially surprising result lies in issues that we have already discussed. If an integrated firm establishes its upstream division as a divisional profit center it reintroduces all of the efficiency losses associated with double marginalization. On the other hand, such a structure insulates the upstream division (which, recall, is a monopoly) from downstream price competition much more effectively than if the upstream division aims to maximize group profits.

When the degree of

downstream product differentiation is low it is more profitable for the firm to adopt the structure that softens downstream price competition than the structure that increases pricing efficiency. 5.5

VERTICAL FORECLOSURE We now consider whether it is in the interest of either integrated firm to foreclose on one or other

or both of the nonintegrated downstream firms. Note that, since there is no other source of supply of the various inputs, the refusal by firm 11, for example, to supply upstream product x1 to downstream firm 12 forces this firm to exit the market. Repeating the analysis above for the various possible ways in which the integrated firms might foreclose gives the pay-off matrix of Table 1.20 Each of the strategy combinations on the main diagonal constitutes a Nash equilibrium. However, no foreclosure is a weakly dominant strategy for both firms. We can conclude, therefore, that no matter the value of γ, no-foreclosure is a Nash equilibrium and profit-maximizing strategy for firms 11 and 22. The intuition behind this result is straightforward. Foreclosure causes the exit of the foreclosed

20

We present specific results only for centralized partial integration. A sketch of the calculations is presented in the Appendix. The same conclusions apply to either of the divisional forms of partial integration. Further details are available from the authors on request.

20


firm(s) and so intensifies competition between the integrated firms in the downstream market, reducing their downstream prices and profitability. It is, therefore, in the interests of the partially integrated firms to keep the non-integrated, high-cost downstream firms in operation as a means of softening price competition in the downstream market. (Table 1 near here)

6.

AN UPSTREAM CONSORTIUM The final vertical arrangement that we wish to consider is one in which the upstream suppliers

are able fully to coordinate their prices by forming an upstream consortium.

We could think of

professional organizations in, for example, the health sector as an upstream consortium.

This

arrangement permits the upstream suppliers to implement a two-part tariff of the form Tij + vijqij where vij is the unit price charged for the combined input i, j used by downstream firm ij. Given the symmetry of our model, we know that the upstream consortium will charge the same unit price v for the combined input to each of the downstream firms. The problem facing the downstream firms in the second stage price subgame is to choose the profit-maximizing prices p ∗jk for their products sold in the downstream market to satisfy the conditions:

(20)

{ (

)(

p ij∗ = arg max q ij p ij , p p j − v

{

)}

(i, j = 1,2)

}

The equilibrium prices p * (v ) = p ∗jk (v ) are detailed in the Appendix. In the first stage of the game the upstream consortium sets v to maximize aggregate downstream profits since this will enable them to negotiate the highest fixed charge. This leads to the Nash equilibrium combined input price: (21)

v* =

3 . 2(3 + 4 γ )

Substituting (21) into the downstream price reaction functions gives the Nash equilibrium prices: (22)

p ij*uc = 1 2 . As we might expect, the input price falls from the monopoly price v* = 1/2 when γ = 0, in which

case the downstream market is effectively perfectly competitive, to v* = 0 when γ = ∞, in which case each downstream firm is effectively a monopolist. Moreover, the ability of the upstream consortium to

21


apply a two-part tariff allows the consortium to extend its monopoly power into the downstream market: the equilibrium prices (21) and (22) generate an aggregate profit of Π a = 1 no matter the degree of product differentiation in the downstream market.21 Considering how the upstream suppliers are now able through the consortium are able to exploit their monopoly position this result should not be too surprising. 7. COMPARISON OF UPSTREAM AND DOWNSTREAM MARKET STRUCTURES: Our analysis has made clear that there are important differences in prices and profitability among the three different methods of transacting business between the upstream suppliers and their downstream customers, not all of which are intuitively obvious. The main elements of this comparison are summarized in Table 2 and illustrated in Figure 5. Consider first the impact of different market structures on prices. Simply put, coordination between the upstream suppliers of specialized services and downstream producers benefits consumers through lower downstream prices. However, whether downstream prices are lower under the upstream network or under partial vertical integration depends in an important way upon how the integrated firms are internally structured. When the upstream divisions are established as divisional profit centers, the input prices they charge will lead to prices in the downstream market, set by both the integrated and non integrated firms, that are higher than the prices set by downstream firms which buy from an upstream network of suppliers. Under the other forms of partial vertical integration, the lower input costs of the integrated firms intensify price competition in the downstream market, and lead to the result that the downstream prices are, on average, lower than under a supplier network.

Finally, the upstream

consortium is better for consumers than decentralized markets, the network of alliances or the divisional profit center only when the degree of product differentiation downstream is “large enough”. If on the other hand the products marketed downstream are not too differentiated then the upstream consortium’s ability to exploit its monopoly power leads to higher prices downstream. Profitability is important for understanding which method of doing business is likely to emerge in a particular upstream downstream sector. For example, an upstream network of alliances is unlikely to be formed unless the upstream suppliers perceive a profit incentive for creating one. Similarly, partial

21

If the upstream consortium did not apply two-part pricing it would set an upstream price for the combined input of v* = 1/2. 22


vertical integration is unlikely to emerge unless at least one of the firms involved can make an acceptable offer to purchase the assets of the others. Consider first the profitability of the downstream firms. Their profits are always greater with an upstream supplier network than with decentralization because of the lower input prices. This is not the case, however, for non-integrated firms under any of the three forms of partial vertical integration. The profitability of these downstream firms is adversely affected by their having to compete with low-cost integrated firms and this more than offsets the benefit they gain from reduced input prices. When considering the profits of the upstream suppliers, clearly the upstream consortium, which confers upon them the greatest monopoly power, is the most profitable. It may however be the most difficult to implement. Compare next upstream profits under the decentralized benchmark case, the network alliance and the divisional profit center. The divisional profit center generates the greatest upstream profits of these three, reflecting the power that this arrangement gives to the upstream suppliers to maintain reasonably high prices in the downstream market. It might be thought that the same would be true when we compare the network alliance and the decentralized benchmark. However, profits of the upstream suppliers in a network of alliances are not always greater than the profits of the upstream suppliers in the decentralized benchmark case: ΠU∗n < ΠU∗d for γ < 0.551. The intuition for what at first sight may seem a surprising result is as follows. Upstream suppliers compete indirectly through the price competition of their customers in the downstream market. What the network of alliances does is to increase the degree to which the upstream and downstream markets are connected. This benefits the upstream suppliers at reasonably high levels of downstream product differentiation by reducing the profit dissipation effects of coordination failures and double marginalization. But it harms the upstream firms at low levels of downstream product differentiation by increasing the extent to which the upstream firms are directly subject to competitive forces downstream in setting their prices. In order to compare the profitability of the integrated firms 11 and 22 with that achieved under either the decentralized case or the network alliance, we compare Π ∗ii p with Π D∗d + 2Π U∗d

and

Π D∗n + 2Π U∗n , that is with the combined profits of one downstream and two upstream firms in either organizational form. Given our discussion of the relative profitability of different forms of partial vertical

23


integration, it should not be surprising to find that here again divisional form “matters”. Under either centralized integration or the group profit center, there is a trade off between the profit-increasing effect of removing, at least partially, pecuniary externalities and double marginalization, and the profit-reducing effect of making the upstream suppliers more directly subject to competitive pressures in the downstream market on the other. There is in fact a direct relationship between the degree of product differentiation and the relative profitability of alternative market structures determined by the degree of coordination between upstream and downstream markets exhibited by these market structures. When γ is “low” the decentralized benchmark structure gives the greatest profitability, for intermediate values of γ there is greatest profit potential under a network alliance. Only for γ “high” is it the case that partial vertical integration offers greater profitability than the alternative market structures we have considered. A very different picture emerges when partial vertical integration establishes upstream divisional profit centers. In this case, partial vertical integration gives the greatest profits for “low” values of γ and the network alliance for “high” values. arrangement.

The decentralized structure is never the most profitable

The reasoning behind this ranking has already been discussed. This form of partial

vertical integration reintroduces the efficiency losses of double marginalization but also gives the upstream divisions the ability to soften downstream competition. The competitive effect dominates the efficiency loss so long as the downstream market is not too highly differentiated. (Table 2 near here) (Figure 5 near here) 8.

POLICY CONSIDERATIONS AND CONCLUSIONS: Market transactions between upstream suppliers of specialized services and their downstream

customers are affected by two types of market failure, the first being the well-known problem of double marginalization, and the second being the lack of price coordination between suppliers of complementary inputs. These market failures are greatest in a fully decentralized market setting. A move to more coordinated pricing between the upstream and downstream markets generally benefits consumers through lower downstream prices. The benefit is not always there for producers, however, with the result that there will not always be a strategic incentive either to form a supplier network or to integrate vertically.

24


Downstream firms do always benefit if their complementary upstream suppliers form network alliances, but the upstream suppliers will not necessarily do so. The profitability of the upstream sellers in a supplier network is affected by the trade-off between two forces. Improved coordination among upstream suppliers and between upstream and downstream markets increases profits by partially mitigating the market failures noted above but reduces upstream profitability by making the upstream suppliers more directly subject to price competition downstream. As a result, upstream suppliers will have a profit incentive to form a network of alliances only if the degree of product differentiation in the downstream market is “sufficiently great”. This would appear to give us a sufficient condition for the formation of network alliances. The same forces are at work in understanding the profit incentive to move towards partial vertical integration.

While we have shown that there will be no attempt by the integrated firms to

foreclose on their non-integrated rivals, it is the case that partial vertical integration significantly lowers the profits of non-integrated downstream firms by putting them at a cost disadvantage with respect to their integrated competitors. Whether integrated producers are more profitable than they would be with a decentralized market or a network alliance depends upon two factors: the organizational form that integration takes and the degree of product differentiation in the downstream market.

When the

integrated firm is structured with a focus on the firm’s aggregate profit, partial vertical integration offers benefits provided that there is a “sufficient” degree of product differentiation. By contrast, if the upstream division of an integrated firm is established as a divisional profit center, partial vertical integration increases profits provided that the degree of product differentiation is “low”. This raises a number of important strategic issues to which we hope to return in subsequent analysis. It is tempting to believe that partial vertical integration, or its non-linear pricing interpretation, will always be attractive provided that the integrating firms can be flexible with respect to the internal structure they adopt. What is not clear is first, whether competing integrated firms will adopt the same internal structure and second, whether firms will be driven to integrate even though this could lead to profit reducing downstream competition. The final important lesson that we draw from our analysis is that the way in which upstream suppliers supply their specialized inputs to downstream firms will be characterized by a considerable

25


degree of diversity across different industries. This occurs because the relative profitability of different relationships between the upstream and downstream sectors depends in a non-trivial and not necessarily obvious way upon the nature and strength of competition in the downstream market.

The recent

attention in the press on a project-based approach to work is suggestive of a trend toward flexible specialization upstream and a move away from complete vertical integration. Harrison (1994) questions whether this trend is, in fact, evident across a wide range of industries. ambiguity in the evidence is to be expected.

26

This paper suggests why


BIBLIOGRAPHY Beath, J. and Y. Katsoulacos, [1991], The Economic Theory of Product Differentiation, Cambridge University Press. Bonanno. Giacomo and John Vickers, [1988]. “Vertical separation,” Journal of Industrial Economcs, 36, 257-265. Bolton, Patrick and Michael Whinston, [1993], “Incomplete contracts, vertical integration and supply assurance”, Review of Economic Studies, 60, 121-148. Cournot, Augustin, [1838], Researches into the Mathematical Principles of the Theory of Wealth, Macmillan, New York, original published in French in 1827. Economides, Nicholas, [1994], "The economics of networks", Discussion Paper EC-94-24, Stern School of Business, New York University. __________ and Steven Salop, [1992], "Competition and integration among complements and network market structure", Journal of Industrial Economics, 105-123. Gaudet, Gerard and Ngo Van Long, [1996], “Vertical integration, foreclosure and profits in the presence of double marginalization”, Journal of Economics and Management Strategy, 5, 409-432. Goodman, Edward and Julia Bamford, (eds.), [1989], Small Firms and Industrial Districts in Italy, Routledge, London. Harrison, Bennett, [1994], Lean and Mean, Basic Books, Harper Collins Inc. Hart, Oliver and Jean Tirole, [1990] “Vertical integration and market foreclosure”, Brookings Papers: Microeconomics, 205-286. Ordover, Janusz A., Garth Saloner and Steven Salop, [1990] “Equilibrium vertical foreclosure”, American Economic Review, 82, 698-703. Piore, Michael and Charles Sabel, [1984], The Second Industrial Divide, Basic Books Inc., New York. Pyke, Frank, Becattini, G. and W. Sengenberger, (eds.), [1990], Industrial Districts and Inter-firm Cooperation in Italy, International Institute for Labour Studies, Geneva. Rosenfeld, Stuart, R. Daffner and William Meade, [1992], Manufacturing Networks and State Policy in North Carolina, Southern Technology Council, North Carolina.

27


Salinger, Michael, [1988], “Vertical mergers and market foreclosure”, Quarterly Journal of Economics, 103, 345-356. Storper,

M.

and

S.

Christopherson,

[1987],

"Flexible

specialization

and

regional

industrial

agglomerations: the case of the U.S. motion picture industry.", Annals of Assoc. of American Geographers, 77, 104-117. Thisse, Jacques -F. and George Norman (1995) The Economics of Product Differentiation vols I and II, Edward Elgar Publishing, Aldershot, England.

28


MATHEMATICAL APPENDIX. 1. A Generalization Assume the basic model presented in the body of the paper with the generalization that that demand in the downstream market for product ij is Dij(p), where p = (p11, p12, p21, p22) and where we have ∂D ij ∂pkl < 0. Then profit to downstream firm ij is: (A.1)

(

)

π ij = p ij − v i − w j D ij (p ) .

The price reaction function for firm ij is: (A.2)

(

)

∂π ij ∂D ij = p ij − v i − w j + D ij (p ) = 0. ∂p ij ∂p ij

R ij :

The comparative statics of the solution to these reaction functions are then:

(i)

∂p ij ∂p kl

=−

∂R ij ∂pkl ∂R ij ∂pij

=−

(pij − v i − w j )∂ 2 D ij

∂p ij ∂p kl + ∂D ij ∂p kl

∂ 2 π ij ∂p ij2

.

The denominator is negative so long as second order conditions are satisfied. The numerator is also negative provided that demand for product ij is a decreasing function of its own and the other product prices. We can expect, therefore, for the price reaction functions to be upward sloping.

(ii)

∂D ij ∂p ij ∂p ij > 0. = 2 ij ∂v i ∂ π ∂pij2

Any arrangement upstream that reduces the prices charged to the downstream firms will result in lower downstream prices. Solving the downstream reaction functions gives upstream demand for input i as a function of the upstream prices: DUij (p(v, w )) . We can then investigate the maximizing problem for an upstream firm in

the first-stage price game. The profit function for an upstream firm is: πUi = v i DUii (p (v, w )) + DUij (p(v, w )) . (A.3)

(

)

We can then compare the decentralized case with the upstream alliance. 1.1 The Decentralized Case The upstream price reaction function is:  ∂πUi (A.4) Ri : = DUii + DUij + v i   ∂v i 

(

)

 ∂D ii ∂p ∂DUij ∂p ik   ik  U + = 0. k =1  ∂p ∂v i ∂p ik ∂v i   ik  

∑

4

1.2 The Network Case Now the profit function for an upstream firm is: 1 (A.5) πUi = (v i + w i )DUii (p(v, w )) + v i + w j DUij (p(v, w )) . 2 This gives the upstream price reaction function:

(

(

)

)

29


ij ii  4   ∂πUi  (v + w ) ∂DU ∂p ik + v + w ∂DU ∂p ik   = 0. = DUii + DUij +  i i j  k =1  i ∂v i ∂p ik ∂v i ∂p ik ∂v i      Comparison with the decentralized case confirms that (A.6) will be negative at the solution for the decentralized case. In other words, the network alliance always leads to lower input prices than the decentralized case and so to lower final product prices.

(A.6)

(

Ri :

) ∑

(

)

No similar general comparison is possible for partial vertical integration since such an arrangement is likely to have asymmetric effects on integrated and non-integrated downstream firms. 2.

The Linear Demand Case

2.1 Downstream prices for decentralized markets or an upstream network alliance: The downstream prices are the solutions to the following problem:22 (A.7)

{ (

)(

p ∗jk = argmax q jk p jk , p p jk − v j − w k

(

)

where: q jk = 1 − p jk − p jk − p γ and p = The solution is: (A.8a)

∗ p 11 =

(A.8b)

∗ p 12 =

(A.8c)

∗ p 21

(A.8d)

∗ p 22

=

=

)}

(∑ p )/ 4 .

(jk = 11, 12, 21, 22)

jk

(

)

28γ + 32γ 2 + 15 + 44 γ + 32 γ 2 (v 1 + w 1 ) + ( 6 + 8 γ )(v 2 + w 2 )

(21 + 80γ + 64γ ) + (15 + 44 γ + 32 γ )(v + w ) + ( 6 + 8 γ )(v + w ) (21 + 80γ + 64γ ) + (15 + 44γ + 32 γ )(v + w ) + ( 6 + 8γ )(v + w ) (21 + 80γ + 64γ ) + (15 + 44 γ + 32γ )(v + w ) + ( 6 + 8 γ )(v + w ) . (21 + 80γ + 64γ ) 2

28γ + 32γ 2

2

1

21

2

1

2

28γ + 32γ 2

2

2

1

1

2

2

28 γ + 32γ 2

2

2

2

1

1

2

2.2 Upstream prices for the decentralized case: The reaction functions corresponding to the stage one game played in input prices are then: (A.9a) v 1∗ =

( 4γ + 3)v 2 − γ ( 8γ + 7)(w 1 + w 2 − 2)

)

(

)

(

)

(

)

( 4γ + 3)w 2 − γ ( 8γ + 7)(v 1 + v 2 − 2)

(A.9b) w 1∗ = (A.9c) v 2∗ =

(

2 16 γ 2 + 18 γ + 3

2 16 γ 2 + 18γ + 3

( 4γ + 3)v 1 − γ ( 8γ + 7)(w 1 + w 2 − 2)

(A.9d) w 2∗ =

2 16γ 2 + 18 γ + 3

( 4γ + 3)w 1 − γ ( 8γ + 7)(v 1 + v 2 − 2) 2 16γ 2 + 18 γ + 3

.

It follows immediately that ∂v 1∗ ∂v 2 = ∂v 2∗ ∂v 1 > 0 and ∂v ∗j ∂ w k < 0 . 2.3 22

Upstream prices for the network alliance

These and all other calculations were performed using Mathematica. Details are available from the authors on request.

30


The reaction functions corresponding to the stage one game for the network alliance are: (A.10a) v 1∗ =

(

(

)

4 16γ + 18 γ + 3

(

2

)

2( 4 γ + 3)w 2 − 32 γ 2 + 32γ + 3 (v 1 + v 2 ) + 4 γ ( 8γ + 7)

(A.10b) w 1∗ = (A.10c) v 2∗ =

)

2( 4 γ + 3)v 2 − 32 γ 2 + 32γ + 3 (w 1 + w 2 ) + 4 γ ( 8γ + 7)

(

)

4 16γ + 18 γ + 3

(

2

)

2( 4 γ + 3)v 1 − 32γ 2 + 32γ + 3 (w 1 + w 2 ) + 4 γ ( 8 γ + 7)

(

4 16γ + 18 γ + 3

(

2

)

)

2( 4γ + 3)w 1 − 32γ 2 + 32γ + 3 (v 1 + v 2 ) + 4 γ ( 8 γ + 7)

(A.10d) w 2∗ =

(

)

4 16 γ + 18 γ + 3 2

As in the decentralized case we have

∂v 1∗

∂v 2 = ∂v 2∗ ∂v 1 > 0 and ∂v ∗j ∂ w k < 0 .

2.4 Partial Vertical Integration The downstream prices are the solutions to the following: (A.11a) p ii∗ = arg max{q ii (p ii , p )(p ii − v ii )} (i = 1,2)

{ (

)(

(A.11b) p ij∗ = arg max q ij p ij , p p ij − v j − w k This gives: ∗ = (A.12a) p11

∗ = (A.12b) p 22

∗ = (A.12b) p12

∗ (A.12c) p 21 =

(28γ + 32γ ) + 4v 2

11

)}

(i,j = 1,2, i ≠ j).

(1 + 2γ )(3 + 4γ ) + (3 + 4γ )(v 22 + v 1 + v 2 + w 1 + w 2 )

(21 + 80γ + 64γ ) 2

(28γ + 32γ ) + 4v 2

(1 + 2γ )(3 + 4γ ) + (3 + 4γ )(v 11 + v 1 + v 2 + w 1 + w 2 )

(21 + 80 γ + 64γ ) ) + (12 + 40γ + 32γ )(v + w ) + (3 + 4γ )(v (21 + 80γ + 64γ ) ) + (12 + 40γ + 32γ )(v + w ) + (3 + 4γ )(v (21 + 80γ + 64γ ) 22

2

(28γ + 32γ

2

11

+ v 22 + v 2 + w 1 )

11

+ v 22 + v 1 + w 2 )

2

1

2

2

(28γ + 32γ

2

2

2

1

2

.

2.4.1 Centralized integration With centralized partial vertical integration we know that vii = wii = 0. The reaction functions corresponding to the stage one pricing game are then: A( γ ) + B ( γ )v 2 + C ( γ )w 1 − E ( γ )w 2 (A.13a) v 1∗ = D( γ ) (A.13b) w 1∗ = (A.13c) v 2∗ = (A.13d) w 2∗ =

A( γ ) + B ( γ )w 2 + C ( γ )v 1 − E ( γ )v 2 D( γ )

A( γ ) + B ( γ )v 1 + C ( γ )w 2 − E ( γ )w 1 D( γ )

A( γ ) + B ( γ )w 1 + C ( γ )v 2 − E ( γ )v 1

D( γ ) where the positive coefficients are:

A(γ) = 756γ + 3328γ2 + 4608γ3 + 2048γ4; B(γ) = 81 +372γ + 544γ2 + 256γ3; C(γ) = 144 + 696γ + 1056γ2 + 512γ3; D(γ) = 360 + 3072γ + 8864γ2 + 10240γ3 + 4096γ4;

31


E(γ) = 171 + 1512γ + 4416γ2 + 5120γ3 + 2048γ4. 2.4.2 A divisional upstream profit center When the upstream division aims to maximize its own divisional profits we have the upstream reaction functions: 28 γ + 32γ 2 + (3 + 4 γ )(v 22 + 2(v 1 + w 1 ) + v 2 + w 2 ) ∗ (A.14a) v 11 = 2 9 + 40 γ + 32 γ 2

(

)

(A.14d) w 1∗ (A.14e) v 2∗ (A.14f) w 2∗

)

(28γ + 32γ ) + (3 + 4γ )(v + v + w + 2(v + w )) 2(9 + 40 γ + 32γ ) (28γ + 32γ ) + (3 + 4γ )(2v + v + v + 2w ) − (9 + 40 γ + 32γ )w = 2(9 + 40 γ + 32 γ ) (28γ + 32γ ) + (3 + 4γ )(2v + v + 2v + w ) − (9 + 40γ + 32γ )v = 2(9 + 40 γ + 32γ ) (28γ + 32γ ) + (3 + 4γ )(v + 2v + v + 2w ) − (9 + 40 γ + 32γ )w = 2(9 + 40 γ + 32 γ ) (28γ + 32γ ) + (3 + 4γ )(v + 2v + 2v + w ) − (9 + 40γ + 32γ )v = 2(9 + 40 γ + 32γ )

∗ = (A.14b) v 22

(A.14c) v 1∗

(

2

11

1

1

2

2

2

2

2

11

22

2

1

2

2

2

2

11

22

1

2

2

2

2

2

11

22

1

2

1

2

2

2

11

22

2

1

1

2

2.4.3 An upstream profit center maximizing group profits When the upstream division is established as a profit center concerned with the aggregate profits of the integrated firm the upstream reaction functions are: (v + v 2 + w 2 )F (γ ) + (v 1 + w 1 )G(γ ) + H (γ ) ∗ (A.15a) v 11 = 22 J (γ ) ( v + v + w ) F ( γ ) + (v 2 + w 2 )G (γ ) + H (γ ) ∗ 1 1 = 11 (A.15b) v 22 J (γ ) v K (γ ) + (v 22 + v 2 )L(γ ) + w 1M (γ ) + N (γ ) − w 2O (γ ) (A.15c) v 1∗ = 11 P (γ ) v K (γ ) + (v 11 + v 1 )L(γ ) + w 2 M (γ ) + N (γ ) − w 1O (γ ) (A.15d) v 2∗ = 22 P (γ ) v K ( γ ) + ( v + w ) L ( γ ) + v 1M (γ ) + N (γ ) − v 2O (γ ) 22 2 (A.15e) w 1∗ = 11 P (γ ) v K γ + v + w L γ ( ) ( ) ( ) + v 2 M (γ ) + N (γ ) − v 1O(γ ) 11 1 (A.15f) w 2∗ = 22 P (γ ) where the positive coefficients are: F (γ ) = (9 + 12 γ ) ; G (γ ) = 72 + 336 γ + 512 γ 2 + 256 γ 3 ; H (γ ) = 84 γ + 96 γ 2 ;

(

)

J (γ ) = 8 27 + 210 γ + 56 γ 2 + 640 γ 3 + 256 γ 4 ; K (γ ) = 72 + 336 γ + 512 γ 2 + 256 γ 3 ;

L(γ ) = 81 + 372 γ + 544 γ + 256 γ ; M (γ ) = 144 + 696 γ + 1056 γ 2 + 512 γ 3 ; 2

3

N (γ ) = 756 γ + 3320 γ 2 + 4608 γ 3 + 2048 γ 4 ; O (γ ) = 1512 γ + 4416 γ 2 + 5120 γ 3 + 2048 γ 4 ;

(

)

P (γ ) = 8 45 + 384 γ + 1108 γ 2 + 1280 γ 3 + 512 γ 4 .

2.5 The upstream consortium The problem facing the downstream firms is to choose prices to solve: (A.16) p ij∗ = arg max q ij p ij , p p ij − v (ij = 11, 12, 21, 22).

{ (

)(

)}

The solutions are:

32


3v + 4γ (1 + v ) (ij = 11, 12, 21, 22). 3 + 8γ The upstream consortium chooses v to maximize aggregate profit of the upstream and downstream firms. This is: 4(3 + 4 γ )(1 − v )(3v + 4 γ (1 + v )) . (A.18) Π (v, γ ) = (3 + 8γ )2 Maximizing this with respect to v gives: (A.19) v * = 3 2(3 + 4 γ ).

(A.17)

p ij =

2.6 Vertical Foreclosure We provided the analysis for the centralized partial vertical integration case. Assume that integrated firm 11 forecloses on downstream firm 12 while integrated firm 22 either remains willing to supply both downstream firms or also forecloses on firm 12. In either case, firm 12 is forced to exit the market, The downstream demand functions become: 1 (A.20) q jk p jk , p = 1 − p jk − p jk − p , jk = 11, 21, 22, γ 1 where p = ( p 11 + p 21 + p 22 ) . 3 As above, the downstream prices satisfy: (i = 1,2) (A.21) p ii∗ = argmax q ii ( p ii , p ) p ii

(

(A.22)

)

(

)

{ } ∗ p 21 = argmax{q 21( p 21, p )( p 21 − v 2 − w 1 )} .

The solutions are: ∗ ∗ (A.23a) p 11 = p 22 =

(A.23b)

3γ ( 5 + 6 γ ) + ( 2 + 3γ )(v 2 + w 1 )

( 5 + 6γ )( 2 + 6γ ) 3(γ ( 5 + 6γ ) + ( 2 + 3γ )(1 + 2γ )(v 2 ∗ p 21 = ( 5 + 6γ )( 2 + 6γ )

+ w 1)

).

Substituting these prices into the profit functions for firms 11 and 22, which are: ∗ ∗ (A.24a) Π I11 = p 11 (v 2 ,w 1 )q 11∗ (v 2 ,w 1 ) + w 1q 21 (v 2 ,w 1 )

∗ ∗ (A.24b) Π I22 = p 22 (v 2 ,w 1 )q 22 (v 2 ,w 1 )

deriving the reaction functions for v2 and w1 and solving gives the input prices to firm 21: (A.25) v 2∗ = w 1∗ =

(

3γ ( 5 + 6 γ ) 7 + 24 γ + 18γ 2

)

(4 + 27γ + 27γ )(13 + 48γ + 36γ ) 2

2

.

Substituting (A.25) into (A.23) and (A.24) gives the profits to the upstream firms with this partial foreclosure. Symmetry implies that the same solution applies if either or both integrated firms foreclose on firm 21 but not 12. Finally, foreclosure by the integrated firms on different downstream firms is equivalent to their foreclosing on both downstream firms. This gives the pay-off matrix for the integrated firms of Table 2. Straightforward but tedious analysis confirms that for any value of γ > 0 the only Nash equilibrium for this game is “no foreclosure” by either integrated firm.

_______________________________________________

33

TUFTS UNIVERSITY ECONOMICS DEPARTMENT DISCUSSION PAPERS SERIES 1998

98-18 PEPALL, Lynne and George NORMAN; Product Competition and Upstream Flexible Specialization. 98-17 YU, David; Two Equivalence Theorems For Government Finance. 98-16 YU, David; Rational Bubbles Under Diverse Information. 98-15 MELLOR, Jennifer, and Jeff MILYO; Income Inequality and Health Status in the United States: Evidence from the Current Population Survey. 98-14 DE FRAJA, Gianni, and George Norman; Product Differentiation and the Location of International Production. 98-13 NORMAN, George, and Lynne PEPALL; Mergers in a Cournot Model of Spatial Competition: Urban Sprawl and Product Specialization. 98-12 MCMILLAN, Margaret; A Dynamic Theory of Primary Export Taxation: Evidence From Sub-Saharan Africa. 98-11 MILYO, Jeffrey; The Political Economics of Campaign Finance: Lessons for Reform. 98-10 NORMAN, George; Foreign Direct Investment and International Trade: a Review. 98-09 MILYO, Jeffrey and Samantha SCHOSBERG; Gender Bias and Selection Bias in House Elections. 98-08 NORMAN, George and Jacques-François THISSE; Technology Choice and Market Structure: Strategic Aspects of Flexible Manufacturing. 98-07 MILYO, Jeffrey and Joel WALDFOGEL; The Effect of Price Advertising on Prices: Evidence in the Wake of 44 Liquormart.

98-06 MILYO, Jeffrey; The Electoral Effects of Campaign Spending in House Elections: A Natural Experiment Approach. 98-05 DOWNES, Thomas, and David FIGLIO; School Finance Reforms, Tax Limits, and Student Performance: Do Reforms Level-Up or Dumb Down? 98-04 NORMAN, George and Lynne PEPALL; Horizontal Mergers in Spatially Differentiated NonCooperative Markets: a Comment. 98-03 NORMAN, George and Jacques-François THISSE; Should Pricing Policies be Regulated when Firms may Tacitly Collude? 98-02 BIANCONI, Marcelo; Intertemporal Budget Policies in an Endogenous Growth Model with Nominal Assets. 98-01 METCALF, Gilbert E.; A Distributional Analysis of an Environmental Tax Shift.

Discussion Papers are available on-line at http://www.tufts.edu/as/econ/papers/papers.html