A Model of Vertical Oligopolistic Competition

Transcription

1 A Model of Vertical Oligopolistic Competition Markus Reisinger and Monika Schnitzer November 2007 Abstract This paper develops a model of successive oligopolies with endogenous market entry. We allow for varying degrees of product differentiation and entry cost to capture how the competitive environment in upstream and downstream markets interact. In particular, we show that the overall market outcome is dominated by the competitive conditions of the downstream market and we characterize conditions under which deregulation is more effective in the downstream market than in the upstream market and vice versa. Furthermore, we analyze how different forms of vertical restraints affect the market structure and provide conditions under which they are welfare reducing. JEL classification: L13, D43, L40, L50 Keywords: Deregulation, Free Entry, Price Competition, Product Differentiation, Successive Oligopolies, Two-Part Tariffs, Vertical Restraints We like to thank Gergely Csorba, Simon Lörtscher, Salvatore Piccolo, Rey Rees, Patrick Rey and participants at the Universities of Mannheim, Melbourne, Munich and Toulouse, at the Swiss IO Day 2007 (Bern), ESEM 2007 (Budapest), and E.A.R.I.E (Valencia) for very helpful comments and suggestions. Both authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft through SFB/TR 15. During part of this project Reisinger visited the IDEI in Toulouse, which he thanks for the hospitality. Department of Economics, University of Munich, Kaulbachstr. 45, Munich, Germany, markus.reisinger@lrz.uni-muenchen.de. Department of Economics, University of Munich, Akademiestr. 1/III, Munich, Germany, schnitzer@lrz.uni-muenchen.de

2 1 Introduction The structure of input and output markets varies a lot between different industries. In some industries, input suppliers operate in a very competitive environment while output producers do not, 1 in others it is the other way round. 2 others have oligopolistic structures with a small number of firms in both the input and output market. 3 Still Yet, there is little knowledge about how these competitive structures evolve and, in particular, how the competitive environment in output markets affects competition in the input market and vice versa. Understanding this interaction between upstream and downstream markets is important for a variety of policy issues. Consider for example a policy maker who is concerned about the overall market outcome and who needs to decide which market to deregulate, input or output market, in order to best promote competition. 4 The answer seems straightforward if the competitive structures upstream and downstream differ a lot, but it is much less obvious if both input and output markets are oligopolistically structured. While there are a number of studies about the effects of deregulation in the product market, none of these studies addresses explicitly the question at which market level deregulation would be most effective. Similarly, there is concern about the welfare implications of different forms of vertical restraints. Yet, the existing literature studies this issue focussing on a given market structure neglecting potential feedback effects these practices may 1 Examples are the automobile and aviation industry where for almost all components a large number of suppliers exist that are confronted with only a small number of final good producers. 2 An example is the market for micro processors of personal computers. Almost all micro processors that are used in personal computers are produced by only two firms, Intel Corporation and Advanced Micro Devices, Inc. (AMD), while there is a much larger number of producers of personal computers. Another example is the market for refined platinum. Due to the highly concentrated location of platinum in South Africa (more that 79% of world s supply), the two biggest suppliers of refined platinum have together a world market share of more than 60% (see Competition Tribunal, Republic of South Africa, Case Nr: 55/LM/Aug02). On the demand side, over 40% of refined platinum are sold to jewelry manufacturers that are widespread in every country. 3 Examples here are the electricity or gas markets. After deregulation of these markets consumers in many US and European regions can choose between a few electricity or gas suppliers. These suppliers in turn buy from the big electricity producing companies. 4 Recent empirical studies find that governments in all OECD countries pursue a policy of deregulation during the last decade (see e.g. Nicoletti and Scarpetta (2003)). Also, the World Bank report "Doing Business 2007" (available at: documents that from 2003 till 2007, in 175 countries 238 market reforms were taken under way out of which 213 facilitate doing business. 1

3 have on market entry. Policy recommendations that do not take these feedback effects into account may be misguided. Thus, for these and other related questions a model is required that looks at the larger picture of both upstream and downstream markets and that allows to study how different competitive environments, pricing strategies and policy choices affect the overall market outcome when the market structures upstream and downstream evolve endogenously. In this paper we provide such a model of successive oligopolies with endogenous market entry. We allow for varying degrees of product differentiation as well as different entry cost, reflecting different competitive environments in both markets. Thus, we can use the model to explore the endogenous two-tier market structure as a function of both product differentiation and entry cost in both markets for different forms of vertical contracts. An advantage of our model is that despite being more general in this respect than existing models, it can still be solved analytically rather than having to resort to numerical solutions. The idea of our theoretical approach is to model each market like a Salop circle with free entry, where transportation cost reflect different degrees of product differentiation in each market. Such a model is straightforward to analyze if market demand can be assumed to be continuous. For downstream firms selling to a large number of consumers this assumption is innocuous. Yet, the analysis is less straightforward for the upstream market, where demand is generated only by a finite number of input buyers, as this may give rise to potential discontinuities in the demand function. The theoretical innovation of our model is to find a specification that makes demand in the upstream market continuous. We achieve this goal by introducing uncertainty about the location of buyers. An upstream firm sells to every downstream firm with positive probability, and this probability is continuously decreasing in its price. Although solving the model analytically is complicated, the solutions are remarkably simple and intuitive. With this model we are able to investigate how the interaction between upstream and downstream market affects the overall two tier market structure. Prima facie, this interaction could give rise to multiple equilibria, because the free entry condition can be fulfilled with both a large and a small number of firms in both 2

4 markets. Interestingly, we find that there exists a unique equilibrium under any form of vertical contracting, i.e. the overall two-tier market structure is uniquely determined by the degree of product differentiation, the entry costs and the respective pricing strategies. This allows us to give clear comparative static predictions, and, most importantly, it enables us to provide welfare comparisons between linear pricing and vertical restraints. We start by exploring the comparative statics of this endogenous two-tier market structure under linear pricing 5 and find that upstream and downstream markets interact asymmetrically. For example, a higher number of upstream firms (caused by a reduction of market entry cost) leads to a higher number of downstream firms because costs of buying inputs decrease while the reverse does not hold true. We also find that one tends to misjudge the impact of an exogenous parameter change on the overall market structure when ignoring the interaction between the two markets, the more so when competition is low in the upstream market and high in the downstream market. We show further that the overall market prices are dominated by the competitive conditions in the downstream market. The intuition is that the degree of downstream competition confines the upstream prices to a larger extent than the other way round. If downstream competition is fierce, an upstream firm that charges a higher price than its rivals can attract only little demand since the competitive disadvantage of a downstream firm is severe if it buys the input at a high price. On the other hand, fierce competition upstream does not prevent high final good prices if the degree of competition downstream is low. Finally, we use the model to study two policy issues, namely the welfare implications of different vertical restraints and the relative effectiveness of different deregulations strategies. For this purpose, we compare the market outcome under two different vertical contracts, two-part tariffs and resale price maintenance, to that under linear pricing. Our analysis shows that the welfare effects under exoge- 5 There are two reasons to analyze the case of linear pricing in more detail. First, in many vertical structures linear pricing is a prevalent praxis. For example, Smith and Thanassoulis (2007) in a study of the industry for liquid milk in the UK report that suppliers only charge a per-unit price without a fixed element. Second, linear pricing is a useful benchmark against which to compare the welfare consequences of more elaborate vertical contracts. 3

5 nously given market structures can differ substantially from those under endogenously evolving ones. In particular, we find that in our model with endogenous market entry welfare under linear pricing can be higher than under two-part tariffs, even though, as is commonly known (see e.g. Villas-Boas (2007)), the latter avoids double marginalization. This is due to the fact that entry in the downstream market is larger under linear pricing and so downstream competition is fiercer. Moreover, we show that welfare under two-part tariffs is always larger than under resale price maintenance although the latter induces more market entry. Here, the effect of avoiding double marginalization dominates. For our second policy issue, the relative effectiveness of different deregulation strategies, we find clear cut policy results, even if both upstream and downstream markets have rather similar competitive conditions. Our analysis reveals that if the overall number of firms is small, deregulation downstream is more effective. This is due to the fact that reducing entry barriers downstream has a positive effect on the number of upstream firms if the number of downstream firms is small, thereby reinforcing the decrease in final good prices via reduced input prices. If, on the other hand, the overall number of firms is large, upstream deregulation is preferable because upstream deregulation decreases the input price and this decrease carries over more strongly to final good prices if downstream competition is fiercer, i.e. if the number of firms is larger. The existing literature that deals with vertical market structures is mostly interested in the question under which conditions different forms of vertical relations, like vertical integration or different forms of vertical contracts, arise and whether they are welfare enhancing or reducing. To this end a realistic description of successive markets is needed. Yet, in the vast majority of this literature the basic markets are modelled in a simplified way, with an exogenous number of firms in each market, often only at most two firms upstream and downstream, and homogeneous products in at least one market. For example, Greenhut and Ohta (1979), Salinger (1988), Gaudet and van Long (1996), or Abiru, Nahata, Raychaudhuri, and Waterson (1998) analyze markets with homogeneous goods and competition à la Cournot to assess the implications of vertical mergers. Yet, due to 4

6 the Cournot assumption they are not able to provide welfare implications about more ornate contracts than linear pricing schemes. 6 Ordover, Saloner and Salop (1990) and Chen (2001) allow for price competition in a framework with vertical integration. To keep the model simple, they firstly assume homogeneous products in the upstream market resulting in perfect Bertrand competition and secondly restrict the number of downstream firms to two. They show that in this framework it is possible to generate asymmetric equilibria in which one firm is integrated and the other one not. There are several papers that deal with vertical contracting issues and their welfare implications, e.g. Hart and Tirole (1990), McAfee and Schwartz (1992) or White (2007). These studies mainly assume a monopolistic upstream supplier and analyze if it is able to extract monopoly rents on the downstream market where a given number of firms compete. In contrast to these papers, our focus is on endogenously determining the market structure in a vertical relation in order to show how the possibility of different forms of contracts affects the overall structure and to give policy implications whether these forms of contracts should be banned. A recent paper that analyzes a similar market structure as ours, with oligopolistic competition upstream and downstream, is Hendricks and McAfee (2007). They assume Cournot competition with homogeneous goods and allow for a general number of firms. Additionally, due to uncertainty about cost and demand realizations, firms in the downstream market can exert market power both upstream and downstream. In order to get explicit results, they employ a supply function approach, where upstream firms can overstate their costs and downstream firms can understate their valuations. In contrast to our paper, Hendricks and McAfee (2007) allow for oligopsonistic power of the downstream firms in the input market but, due to their assumption of homogeneous products and Cournot competition, they cannot determine how the market structure evolves endogenously under varying degrees of substitutability and differing entry costs. It is also not possible in their framework to compare the welfare implications of different forms of vertical con- 6 For a paper that is not interested in vertical integration but also assumes Cournot competition and considers different brands in the upstream market, see Belleflamme and Toulemonde (2003). 5

7 tracts. 7 To sum up, the literature about vertical market structure so far usually supposes very specific market conditions and often quantity competition with homogenous goods. There does not exist a general framework to deal with the question how vertically interrelated oligopolies work and to assess the welfare consequences of vertical restraints that can change the market structure. One of the reasons for this lack of more generality is that such models quickly become very complicated. The main contribution of this paper is to provide such a general model that can serve as a framework to address a variety of questions. In this paper we use it to compare the welfare implications of different pricing regimes. Other applications could be the analysis of supply chains or vertical integration. 8 The remainder of the paper is organized as follows. The next Section sets out the model. In Section 3 we solve for the subgame perfect equilibrium of the model with linear pricing. Section 4 studies the interplay of the upstream and the downstream market, providing comparative statics for the two-tier market structure and exploring the relative importance of the upstream and downstream market for prices and overall welfare. In Section 5 we study the model under two forms of vertical restraints, two-part tariffs and resale price maintenance, focussing in particular on the welfare implications of these regimes. Section 6 concludes. 2 The Model Consider an industry with two successive oligopolies, an upstream and a downstream market. In the upstream market each firm produces an intermediate good at marginal cost that is normalized to zero. The upstream firms sell the intermedi- 7 Ghosh and Morita (2007) also analyze a model with homogenous goods and Cournot competition. They study endogenous entry in the upstream sector and compare it with the socially optimal number of firms. They find that insufficient entry may arise, because part of the profit of an upstream firm is captured by the downstream firms. Eső, Nocke and White (2007) consider a different but related question. They analyze a downstream industry in which firms compete for a scarce input good. They show that if the supply of this input good is large enough, the resulting industry structure is asymmetric with one firm being larger than the others. In contrast to our paper, they do not model the upstream market explicitly but rather assume that the input is allocated efficiently from the perspective of the downstream firms. As a consequence, in their model the interrelation between the two markets is not present. 8 See the conclusion for a discussion of these issues. 6

8 ate goods to the downstream firms, the producers of the final good. Downstream firms transform the intermediate goods on a one-to-one basis into output at zero costs. 9 There is free entry in both markets but all firms that enter in the upstream market must incur a fixed set-up cost of F u > 0 while firms entering in the downstream market must incur a fixed set-up cost F d > 0. We will determine the number of firms that enter in each market and denote the number of upstream firms by m and the number of downstream firms by n. For simplicity, we treat n and m as continuous variables. This implies that in equilibrium the profit of each firm net of set-up cost is zero. An upstream firm i sells its intermediate good to the downstream firms at a price per unit that is denoted by r i, i (1,..., m). 10 Similarly, we denote a downstream firm j s final good price by p j, j (1,..., n). First we describe the downstream market. We model the downstream market in a similar way as Salop (1979). There exists a continuum of consumers of mass 1 that is uniformly distributed on a circle with unit circumference. A consumer who is located at z and purchases the good from firm j located at z j incurs total cost of p j + t d (z z j ) 2. We assume that the gross utility of consuming this good is sufficiently high, such that all consumers buy for the range of prices considered. t d (z z j ) 2 is the disutility that a consumer incurs if she does not buy her most preferred variety. This disutility is assumed to be quadratic in the distance between the consumer and the firm. 11 The n downstream firms are equidistantly distributed. So the marginal consumer between firm j and j + 1 lies at distance z m from firm j, with z m = 1 2n + n(p j+1 p j ) 2t d. 9 The normalization of zero marginal cost in the upstream and the downstream sector is without loss of generality. If we assume constant marginal costs of c u > 0 and/or c d > 0 instead, the qualitative results remain unchanged. 10 In Section 5 we extend the model and allow for more elaborate pricing schemes. 11 With this assumption, we avoid the well-known problem that the profit function of firm j becomes discontinuous if its price is low enough such that the consumer located exactly at the position of its neighboring firm j 1 or j + 1 prefers to buy from firm j rather than from firm j 1 or j + 1. All our results also hold for the case of linear transport costs under the additional assumption that t d is sufficiently high. 7

9 We now turn to the upstream market. Here again, we consider a Salop circle on which the upstream firms are located with equal distance from each other. The buyers in the upstream market are the downstream firms. We first specify the costs a downstream firm incurs when buying its intermediate goods. Then we describe the location of the downstream firms as customers in the upstream market. When buying from upstream firm i, a downstream firm j faces per unit cost of r i for the intermediate good and additionally a fixed cost that is given by t u (x j x i ) 2, where t u is transportation cost in the upstream market and (x j x i ) 2 is the shortest arc length between the location of firm j and the location of firm i in the upstream market. These fixed cost reflects how well the intermediate goods fit the particular needs of the final good producer. For instance, the characteristics of the intermediate good provided by firm i may not exactly fit the technology of firm j and so firm j must costly reposition its machine. One can also imagine that firm i is a producer from abroad, and so firm j needs some instructions to correctly handle the input that results in additional costs for firm j. In contrast to the downstream market there is only a finite amount of buyers in the upstream market instead of a continuum. This could potentially lead to discontinuities in the demand curve of an upstream firm. 12 In order to deal with this problem, we suppose that at the time when upstream firm i decides about its price for the intermediate good, it does not know the locations of the downstream firms in the upstream market. Instead, it expects that each location of a downstream firm along the upstream circle is equally likely, i.e. it is uniformly distributed over the upstream circle. With this specification demand functions are continuous. We also assume that when choosing its input supplier each downstream firm knows its own location in the upstream market but does not observe the location of other downstream firms. Instead, like an upstream firm, it expects the location of every other downstream firm to be uniformly distributed on the upstream circle. An implication of this assumption is that the location of a downstream firm on the upstream market is independent from its location in the downstream market. This reflects the idea that different locations of downstream firms in the upstream 12 For an in-depth discussion of that problem, see e.g. Gabszewicz and Thisse (1986). 8

10 market stem from different technologies while a difference in the locations in the downstream market emerges due to production of different varieties or represents a geographic distance. 13 As a consequence of this assumption, a downstream firm cannot observe the input suppliers from which its rival downstream firms are buying and so it does not observe their input prices. Thus, after having observed the upstream price vector r downstream firm j forms expectations about the profit it earns in the product market, E[Π j (r)], dependent on the input supplier it chooses. As a consequence, firm j buys from upstream firm i if 14 E[Π i j(r i, r i )] t u (x j x i ) 2 E[Π k j (r k, r k )] t u (x j x k ) 2 k i. It follows that the probability that firm j buys its intermediate good from firm i is given by 15 ( 1 qi = m + m(2e[πi j] E[Π i 1 j ] E[Π i+1 ) j ]). (1) 2t u To make the problem interesting, we finally assume that the fixed set-up costs are low enough such that in both sectors at least two firms enter. It turns out that this is fulfilled if 16 F d < 1 8 ( td t ) u 6 and F u < t u 3(t u + 12F d ) 3 48(48F d + t u ) 2. (2) We consider the following three stage game. In the first stage a large number of firms can enter either in the upstream or in the downstream market at the respective entry costs F u or F d. After entry, both upstream and downstream firms are symmetrically distributed in their respective markets. The location of the downstream firms as customers in the upstream market is uncertain and each down- 13 For example, in the market for batteries Duracell and Valance Technologies are close competitors. Yet, Valance Technologies is doing R & D to improve the performance of its batteries and uses a completely different technology than Duracell which does not engage in R & D. Therefore, the two firms are not closely located to each other in the input market. 14 Here and in the following r i denotes the prices of all upstream firms but firm i, {r 1,..., r i 1, r i+1,..., r m }. 15 For the sake of notation we abbreviate E[Π i j (p j, r i, r i )] by E[Π i j ]. 16 Footnote 14 explains how these conditions are derived. 9

11 stream firm can be distributed over the whole circle with equal probability. In the second stage, upstream firms set their prices r i. Afterwards downstream firms learn their position in the upstream market and choose their preferred supplier of the intermediate good but they do not learn the positions of all rival downstream firms. In stage three, downstream firms set prices in the downstream market. 3 Equilibrium to the Model In this section we describe the solution of the three stage game. We solve the game by backward induction. We determine the equilibrium upstream and downstream prices and the number of firms that enter in both markets. A rigorous proof of the results can be found in Appendix A. Downstream Market In the third stage each downstream firm decides about its final good price knowing the number of its competitors and the upstream price vector r. When setting its price p j, firm j does not observe from which upstream firms its two neighboring downstream firms buy the intermediate good, and so it does not observe their input prices. Since input prices influence final good prices, downstream firms that buy from different upstream firms might set different final good prices. However, in equilibrium firm j knows the expected input price of its rival. Anticipating that in equilibrium all competitors with the same input price will charge the same output price, it also knows the expected output price. 17 As a consequence, the expected profit of firm j (net of fixed costs) when buying its input from upstream firm i can be written as ( 1 E[Π i j(p j, r i, p j 1, p j+1 )] = (p j r i ) n + n(e[p ) j 1] + E[p j+1 ] 2p j ). 2t d This maximization problem is identical for all downstream firms with the exception that they potentially buy their intermediate goods at different prices. Thus, there can be at most m different downstream prices. Since its expectation about 17 For models with a similar structure, see Raith (2003), Aghion and Schankerman (2004), or Syverson (2004). 10

12 the upstream location of its two neighbors is the same, firm j s expectation about E[p j 1 ] and E[p j+1 ] is the same and is given by m q k p k, where p k is the price that a downstream firm charges when buying the input from upstream firm k. 18 expected profit of firm j when facing an input price of r i can then be written as k=1 ( m ) ( n q 1 k p k p j ) E[Π i j(p j, r i, p j )] = (p j r i ) n + k=1. t d Differentiating the last equation with respect to p j and setting the results equal to zero gives a system of m equations because there can be potentially m different input prices. Solving this system for p j gives The p j = 1 2( ri + m k=1 q k r k + 2t d n 2 ), i {1,...m}. Plugging these prices back into the respective profit functions gives the expected profit of all downstream firms dependent on r i and q i, i {1,..., m}. Since the probabilities are functions of the downstream profits, we can solve for these probabilities via plugging the profits into (1). The general formula for q i is given by (19) in Appendix A for the case of m = 2 and by (18) for the case m 3. Inserting this formula back into the profit function, we can derive the quantity that downstream firm j buys from upstream firm i, denoted by y i (r i, r i ), and the price that downstream firm j charges, p j (r i, r i ). These expressions are given by equations (23) and (21) in Appendix A for the case m = 2 and (22) and (20) for the case m 3. Having solved the third stage of the game, we now proceed to the second stage, the upstream market. Upstream market Since production costs are equal to zero, the profit of an upstream firm i can be written as the probability that it sells to exactly one downstream firm multiplied with the revenue from selling, r i y i (r i, r i ), plus the probability that it sells to exactly two downstream firms multiplied with two times the revenue and so forth. 18 Recall that q k is the probability that a downstream firm buys from upstream firm k. 11

13 This can be written as 19 (( ) ( ) n n E[P i (r i, r i )] = r i y i q i (1 q i ) n qi 2 (1 q i ) n 2 + (3) 1 2 ( ) ) n + + (n 1) qi n 1 (1 q i ) + nqi n. n 1 The intuition behind this equation is probably easiest to grasp when considering the case n = 2. In this case, (1 q i ) 2 is the probability that no downstream firm buys from upstream firm i. The probability that exactly one firm buys is given by 2q i (1 q i ) and the probability that both downstream firms buy is given by q 2 i. In this last case firm i sells two times y i. Extending this line of reasoning to the case of n firms yields the right hand side of (3). Using a modification of the Binomial Theorem, the profit function simplifies to E[P i (r i, r i )] = r i y i nq i. (4) We can now substitute the respective expressions for y i and q i that we determined in the third stage and differentiate (4) with respect to r i and set the result equal to zero. Since the profit function is the same for all upstream firms we get that the equilibrium is symmetric. The formula for the equilibrium upstream prices is remarkably simple and is given by r = 2t u t d mn t u n 3 (m 1) + 2m 3 t d. (5) The equilibrium upstream prices can now be inserted into the formula for the downstream prices to get p = t d(2m 3 t d + t u n 3 (3m 1)) n 2 (2m 3 t d + t u n 3 (m 1)). (6) Having solved for the equilibrium prices in both stages, we can now proceed to the first stage and determine the equilibrium number of firms in both markets. Entry Decision 19 For notational simplicity we suppress the dependence of y i and q i on r i and r i. 12

14 Inserting the equilibrium prices in the profit functions yields that the expected profit of a downstream firm is given by E[Π] = t d /n 3 t u /(12m 2 ) F d and the expected profit of an upstream firm is given by E[P ] = r /m F u = (2t u t d n)/(t u n 3 (m 1) + 2m 3 t d ) F u. As a consequence, the equilibrium number of firms, n and m, simultaneously solve the two equations t d (n ) 3 t u 12(m ) 2 F d = 0 (7) and 2t u t d n t u (n ) 3 (m 1) + 2(m ) 3 t d F u = 0. (8) Given our assumptions on F d and F u above, it is always optimal for at least two firms to enter in each market. Thus, we can be sure that there is always an equilibrium in which at least two firms are active in both markets. It is more interesting to examine whether or not the equilibrium is indeed unique. For this purpose, we use the the iso-profit-lines of a downstream and an upstream firm, as determined by the zero profit conditions (7) and (8). For a downstream firm, the slope of the iso-profit line is given by n m = t u(n) 4 18t d (m) 3 > 0 (9) This means that the equilibrium number of downstream firms increases with the number of upstream firms. The intuition for this is simple. If the number of upstream firm increases, downstream firms benefit from lower input prices and expect to face a lower travel distance to their nearest upstream firm. As a consequence, more firms enter the downstream market. This is depicted by the isoprofit-line I D in Figure 1. It shows that the zero profit condition in the downstream market could be achieved by e.g. many upstream and downstream firms, but also by few upstream and downstream firms. For an upstream firm the slope of the iso-profit-line is given by m n = 2 ( (m) 3 t d (n) 3 (m 1) ). (10) n(t u (n) 3 + 6t d (m) 2 ) 13

15 Here the sign is ambiguous. Inspection of (10) reveals that the sign of the isoprofit line depends on n. If n 3 (2t d m)/(t u (m 1))m, it is negative, while it is positive if the reverse holds true. The reason for this ambiguity is that a larger number of downstream firms has two effects on the profit of an upstream firm. Since the downstream firms compete against each other, a larger number of downstream firms implies a larger number of marginal consumers in the output market. If an upstream firm i lowers its price r i, each downstream firm that buys from upstream firm i sells a higher quantity. But since there are more marginal consumers, upstream firm i receives a bigger quantity increase if more downstream firms are present. Thus, each upstream firm has a bigger incentive to lower its price and this increased competition effect lowers profits upstream. 20 On the other hand, an increase in the number of downstream firms has also a positive effect on upstream prices. With a larger number of downstream firms each upstream firm faces more potential buyers. Thus, the probability to sell to at least one downstream firm increases. But the profit that an upstream firm receives with its first buyer is larger than the one with its second buyer and so on. The reason is that an upstream firm potentially cannibalizes itself with a price reduction given it already has several buyers. This is due to the fact that with some probability these buyers are neighbors in the product market and so the demand of the upstream firm does not increase. As a consequence, upstream firms compete fiercest to get at least one buyer. If now the number of downstream firms increase, the probability to serve at least one increases and this dampens the pressure on upstream prices. Naturally, this effect is stronger if there are only two downstream firms present compared to the case with already several downstream firms. As a consequence, this second effect dominates if n is small, while the first one dominates for a large n. So m increases in n if n is small and it decreases in n if n is large. This is shown by the iso-profit-line I U -curve in Figure 1. Thus, for the upstream market we find 20 The result that upstream profits decrease in the number of downstream firms also arises in Rey and Tirole (2007). They consider a monpolistic upstream suppliers that is able to use non-linear tariffs. The monopolist faces a commitment problem that gets more severe the more downstream firms are present. Instead, our result does not rely on a commitment effect but on increased competition that carries over to the upstream market. So the results can be seen as complementary to each other. 14

16 n I D n I U m m Figure 1: Equilibrium number of firms, n and m that the same number of upstream firms can be part of an equilibrium with both a low and a high number of downstream firms. 21 This non-monotonicity of the equilibrium number of upstream firms as a function of downstream firms raises the issue of potential multiplicity of equilibria. Multiple equilibria could exist if the two functions crossed more than once. For this to happen the slope of the iso-profit-line of a downstream firm needs to be steeper than the one of an upstream firm in the region in which the latter increases, i.e. n < (2t 3 d m)/(t u (m 1))m. However, it is easy to show that this can never be the case for the relevant parameter range. Thus, the equilibrium is unique. The following Proposition summarizes the equilibrium. Proposition 1 There exists a unique symmetric subgame perfect equilibrium of the three stage game. In this equilibrium, the number of entering upstream and downstream firms, n and m, is implicitly defined by t d (n ) 3 t u 12(m ) 2 = F d 21 The restrictions on F d and F u in (2) are derived from the preceding analysis. Since n is increasing in m, the condition on F d assures that at least two downstream frims enter given that there are only two upstream firms. The condition on F u is derived by solving (7) for n, plugging this into (8) and setting m = 2. 15

17 and 2t u t d n t u (n ) 3 (m 1) + 2(m ) 3 t d = F u. Upstream firms charge a price of r = while downstream firms charge a price of 2t u t d m n t u (n ) 3 (m 1) + 2(m ) 3 t d, p = t d(2(m ) 3 t d + t u (n ) 3 (3m 1)) (n ) 2 (2(m ) 3 t d + t u (n ) 3 (m 1)). Proof: See Appendix A. 4 Interplay between Upstream and Downstream Market In this section we study in more detail the interplay between upstream and downstream market. In particular, we explore the comparative statics of the endogenous two tier market structure and compare this with the comparative statics of a single market model. Furthermore, we determine which market conditions matter more for market prices. Finally, we compare the welfare implications of deregulating upstream versus downstream markets. 4.1 Comparative Static Results for the Two Tier Market Structure In this section we analyze how a change in the set-up costs and in the degree of competition affects the overall market structure. We are particularly interested in finding out whether changes in the upstream and in the downstream market have symmetric or asymmetric effects on each other. Consider first the set-up costs. 16

18 Proposition 2 (i) The equilibrium number of upstream and downstream firms, n and m, is decreasing in F u. (ii) The equilibrium number of downstream firms, n, is decreasing in F d. There exists an F d, such that the equilibrium number of upstream firms, m, is increasing in F d if F d < F d, and decreasing in F d if F d > F d. Proof: See Appendix B. Note that a change in F k, k {u, d}, has a direct effect on the number of firms in market k and because of that an indirect effect on the number of firms in market k. The proposition shows that an increase of the set-up costs in the upstream market reduces the number of firms in both markets while the same does not necessarily hold true for an increase of the set-up costs in the downstream market. Both results are easy to understand given the insights derived for Proposition 1. A change in the set-up cost F u has a symmetric effect on the number of firms entering in both markets. If F u increases, the number of upstream firms decreases and so does the number of downstream firms because their expected travel cost rise. Graphically, an increase in the upstream set-up cost shifts the I U -curve to the left. This is depicted by the left hand side of Figure 2. If F d increases instead, the number of downstream firms decreases, but this can have either a positive or a negative impact on m, the number of upstream firms. This is due to the fact that, as shown in the last section, m does not change monotonically with n. If the number of downstream firms is low, i.e. F d > F d, m decreases as F d increases, and vice versa if the number of downstream firms is high, i.e. F d < F d. The right hand side of Figure 2 displays an increase in the downstream entry costs which results in a downward shift of the I D -curve. In this example, the intersection occurs in the increasing part of the I U curve, which means that the number of upstream firms decreases as a result of F d increasing. We now turn to the case of differing degrees of competition. Proposition 3 17

19 n n I U n n I D n I D I U n I D I U m m m m m m Figure 2: Increase in F u (left) and increase in F d (right) (i) The equilibrium number of upstream and downstream firms, n and m, are both increasing in t d. (ii) The equilibrium number of upstream firms m is increasing in t u while the equilibrium number of downstream firms n is decreasing in t u. Proof: See Appendix B. In contrast to a change in entry costs, a change in transportation costs has a direct effect only on prices and with that indirectly affects the equilibrium number of firms. Consider first an increase in t d, which means less competition in the downstream market. This results in higher prices and hence in a larger equilibrium number of downstream firms. At the same time, upstream firms increase their prices as well, as can be seen from (5). An upstream firm can charge a higher price without losing much demand because the disadvantage for a downstream firm that buys from this upstream firm is less severe if competition in the downstream market is less fierce. With higher upstream prices and higher upstream profits there is more entry in the upstream market. This direct price effect dominates any countervailing effect that the resulting increase of n could have on m. The increase in m, on the other hand, reinforces entry in the downstream market. The result is displayed graphically by the left hand side of Figure 3. 18

20 n n n n I D I D n n I D I D I U I U I U I U m m m m m m Figure 3: Increase in t d (left) and increase in t u (right) On the other hand, less competition in the upstream market leads to an increase in the number of upstream firms but to a decrease in the number of downstream firms. This is intuitive: if competition in the upstream market becomes less fierce higher profits can be reaped in the upstream market and so more firms enter. But if t u increases, transportation costs of a downstream firm increase. This effect is partly offset by the increase in upstream firms that reduces the travel distance. But the direct effect of increased transportation costs dominates and so fewer downstream firms enter the market (see the right hand side of Figure 3). This shows that the degree of competition in the downstream market affects the number of firms in both markets in the same direction while the opposite holds true for the degree of competition in the upstream market. The reason is that the level of downstream competition crosses over from the downstream market to the upstream market and enters the pricing formula of upstream firms positively. Conversely, the level of upstream competition enters the cost function of downstream firms and so reduces their profits. Put differently, the level of downstream competition affects the overall profitability of the two tier market structure and thus affects both markets in a similar way. The level of upstream competition, on the other hand, affects mostly the distribution of the overall profits between upstream and downstream markets. 19

21 4.2 Comparison with the Single Market Model Economic models very often analyze only the downstream market in which the final products are sold to consumers completely ignoring the upstream market. It is therefore of interest to find out how large the mistake is that one makes when ignoring the upstream market. This is characterized in the next proposition. Proposition 4 (i) The magnitude of (dn )/(dt d ) is always undervalued when one ignores the upstream market. (ii) The magnitude of (dn )/(df d ) can either be over- or undervalued when one ignores the upstream market. There exists an F d such that for F d > F d, it is undervalued while for F d < F d it is overvalued. Proof: See Appendix C. The first result shows that the effects of a changing degree of competition is always underestimated when one ignores the upstream market. The intuition is easy to grasp from Proposition 3. Since the degree of competition in the downstream market also affects upstream prices and profits, it changes the number of upstream firms. If downstream competition becomes fiercer, i.e. if t d is reduced, this crosses over to the upstream market in which fewer firms enter. This in turn increases the transportation costs for downstream firms and reinforces the effect of lower entry. Our second result shows that the effect of varying downstream set-up costs can be over- or underestimated when ignoring the downstream market. It is underestimated if we are in the increasing part of the I U -curve in Figure 1, because here the change in m goes in the same direction as the change in n and so reinforces the latter while it is overestimated if m and n change in opposite directions, as they do in the decreasing part of the I U -curve. Concerning the first result, it is interesting to explore under which conditions the underestimate is especially high. The answer is given in the following corollary. 20

22 Corollary 1 The magnitude of the underestimation of (dn )/(dt d ) is increasing in t u and decreasing in t d and m. Proof: See Appendix C. The results with respect to m and t u are intuitive. If competition in the upstream market is low, which means that either m is small or that t u is high, the upstream market matters a lot for the competitive conditions in the downstream market. Thus, the mistake that one makes when ignoring the upstream market is big. On the other hand, this mistake is also big when the degree of downstream competition is high which means a small t d. The reason is that since competition is fierce, a change in the upstream price is passed over almost one by one on the downstream price. To the contrary, if t d is high, the mistake of ignoring the upstream market is small since the equilibrium number of downstream firms is mainly determined by t d and not by the upstream market conditions. The analysis shows that there is generally a problem when one looks only at the downstream markets ignoring previous layers in the production chain. Especially, if the upstream market exhibits a low degree of competition, the feed back effects on the downstream market are large and one should take them into account. 4.3 Competitive Conditions and their Impact on Market Prices We now turn to the question which market conditions matter more for the overall market outcome, those of the upstream or those of the downstream market. For this purpose we examine how equilibrium market prices are affected by the competitive conditions in the two markets. We are interested if the conditions in the upstream or in the downstream market are more important for determining upstream and downstream prices. A first guess would probably be that a market s own condition should matter more for the respective price. We will see that this is true for the downstream market but need not be true for the upstream market. To this end, consider two different scenarios. In scenario (i) the degree of competition, captured by the transport costs, in the downstream market is lower than 21

23 in the upstream market, i.e. t d = t < t = t u, while in situation (ii) the reverse holds, t d = t > t = t u. To focus on the pure effect of the degree of competition we suppose that the equilibrium number of firms is the same in both situations, i.e. n (i) = n (ii) = n and m (i) = m (ii) = m. Implicitly, this means that set-up costs differ between the two situations in such a way that the number of firms is the same. Prima facie, it is not obvious if, for example, the downstream price in one or the other scenario is higher because a high t u results in a high input price and thus also in a high output price while a low t d puts downward pressure on the output prices. A similar logic holds for upstream prices. Yet, we get the following result. Proposition 5 Independent of the equilibrium number of firms, the downstream price is lower in scenario (i) than in scenario (ii). The upstream price is higher in scenario (i) than in scenario (ii) if and only if n 3 2 m 1 m. Proof: See Appendix B. This result shows that the degree of competition in the downstream market matters more for downstream prices than the degree of competition in the upstream market while the reverse is not necessarily true. 22 Intuitively, if t u is low and so upstream prices are close to marginal costs, consumer prices can still be well above marginal costs if the degree of downstream competition is small. In this case, downstream firms demand high margins. Conversely, if t d is low, downstream competition is fierce and this heavily influences upstream prices. A downstream firm that pays a higher input price than its neighbors, must also demand a higher final good price and gets only little demand. But this leaves an upstream firm that charges a high input price also with low demand. Thus, it has an incentive to lower its price. As a consequence, in equilibrium upstream prices are low 22 Note that 3 2/(m 1) 1 for m 3. Thus, the condition is never fulfilled for n > m 3. 22

24 as well. 4.4 Upstream vs. Downstream Deregulation We now turn to the question in which market it is more effective to deregulate in order to spur competition. The answer seems straightforward if upstream and downstream markets are very asymmetric, one being very competitive while the other one being very little competitive. And indeed, one can show in our model that in such cases it is most effective to intervene in the less competitive market. The answer is no longer obvious if both markets are similarly competitive. Previous models have analyzed the effects of deregulations but focussed only on the product market alone due to the lack of a model that is able to deal with an endogenous structure in both markets. The aim of this analysis is to explicitly differentiate between deregulation in the output and in the input market and give policy implications where intervention is more effective. A policy maker can control two variables to foster competition. She can decrease entry barriers, for example, by reducing red tape to start up new businesses or by making access to the home market easier for foreign firms. In our model, this corresponds to a decrease in firms set-up costs, F u or F d. Another possibility is to increase the degree of competition directly, e.g. by the introduction of standardization measures making it easier for consumers to compare products or by investing in infrastructure that decreases transport or communication costs. This corresponds to a decrease in t d or t u. 23 We first look at the case of a decrease in set-up costs. Since profits are zero in equilibrium, welfare and consumer surplus coincide. Let a denote a consumer s 23 t d and t u were formally introduced as preference resp. technology parameters. To think about a decrease in t d or t u as the result of deregulation, one can interpret the specification of consumer utility resp. downstream firms production function as reduced form. There, t d reflects higher product substitutability while t u reflects higher input substitutability. For a similar exercise and a discussion of this interpretation, see Blanchard and Giavazzi (2003). 23