A Model of Vertical Oligopolistic Competition. Markus Reisinger & Monika Schnitzer University of Munich University of Munich

Transcription

1 A Model of Vertical Oligopolistic Competition Markus Reisinger & Monika Schnitzer University of Munich University of Munich

2 1 Motivation How does an industry with successive oligopolies work? How do upstream and downstream markets interact? (Multiple equilibria? Asymmetry? Which market dominates?) Important for Deregulation Policies (Electricity Market in Germany vs. gas market in the U.S.) Important for judging welfare effects of vertical restraints

3 2 Contribution of this paper Provide an analytically solvable model with oligopolistic price competition upstream and downstream endogenous market entry varying degrees of competition in both markets different pricing regimes upstream

4 3 Main results Unique equilibrium, allowing for comparative statics and welfare analysis. Downstream market tends to have stronger impact on overall market performance than upstream market. Deregulation is more effective downstream than upstream, if overall number of firms is small and vice versa. Welfare under two-part tariffs can be higher or lower than under linear pricing.

5 4 Existing literature Salinger (1988), Ordover, Saloner & Salop (1990), Chen (2001): Vertical integration with an exogenous number of firms and homogeneous products upstream Hart & Tirole (1990), McAfee & Schwartz (1992), White (2007): Non-linear tariffs with an exogenous number of firms, often upstream monopoly. Hendricks & McAfee (2007): Vertical relationship under Cournot competition

6 5 Outline 1. Introduction 2. The Model and Equilibrium 3. Interplay between the two Markets and Implications for Deregulation 4. Two-Part Tariffs 5. Conclusion

7 6 2. The Model Upstream market with m firms and downstream market with n firms. Free entry in both markets; set-up costs F u and F d, respectively. Each upstream firm produces intermediate goods at zero cost. Each downstream firms buys intermediate goods and transform them on a one-to-one basis into output goods at zero costs. Upstream firms set simultaneously prices r i, i (1,..., m), downstream firms set simultaneously prices p j, j (1,..., n).

8 7 Downstream market Consumers uniformly distributed on Salop circle with density one. Quadratic disutility of travel: transportation cost parameter t d Gross utility is sufficiently high such that in every price equilibrium all consumers buy. Downstream firms are distributed with equal distance. Marginal consumer between firm j and j + 1: z m = 1 2n + n(p j+1 p j ) 2t d

9 8 Upstream market Upstream firms are distributed on a Salop circle with equal distance. Downstream firms are allocated along the upstream circle as buyers of the intermediate good. Problem: finite number of buyers in the upstream market Potential discontinuity in the demand function of upstream firms Resolution: When an upstream firm decides about its price, it does not know the locations of the downstream firms in the upstream market. Instead, it expects that the location of a downstream firm is uniformly distributed over the upstream circle. Continuous demand function

10 9 Implication: Location of a downstream firm on the upstream market is independent from its location in the downstream market Idea: Different upstream locations reflect different technologies while different downstream locations reflect different varieties. Cost specification: When buying from upstream firm i, a downstream firm j faces per unit cost of r i and a fixed cost of t u (x j x i ) 2. Downstream firms know their own location in the upstream market but do not observe the location of other downstream firms. (Expectation is uniformly distributed over the upstream circle.)

11 10 Downstream firm j buys from upstream firm i if 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. Probability that firm j buys its intermediate good from firm i is given by q i = ( 1 m + m(2e[πi j ] E[Π i 1 j ] E[Π i+1 j ]) 2t u ).

12 11 Three Stage Game 1. Firms enter in the upstream and in the downstream market. 2. Upstream firms simultaneously set r i without knowing the position of downstream firms on the upstream circle. Then downstream firms learn their position in the upstream market and choose their preferred supplier. 3. Downstream firms simultaneously set p j.

13 12 2. Equilibrium Game is solved by backward induction. Downstream market: Profit function of downstream firm j when it buys from upstream firm i: E[Π i j (p j, r i, p j 1, p j+1 )] = (p j r i ) ( 1 n + n(e[p j 1] + E[p j+1 ] 2p j ) 2t d t u (x j x i ) 2 F d Maximizing with respect to p j and plugging back in the profit function gives the expected profits dependent on the upstream prices r i, i {1,..., m}. From that we can calculate q i and y i, the probability of buying and the respective quantity. )

14 13 Upstream Market: Profit function of upstream firm i can be written as E[P i (r i, r i )] = r i y (( n) i qi (1 q i ) n ( n) q i (1 q i ) n (n 1) ( n ) q n 1 n 1 i (1 q i ) + nqi n ) = ri y i nq i. Maximizing with respect to r i and solving the system of equations gives the equilibrium upstream prices.

15 14 Equilibrium Formulas for the equilibrium prices are remarkably simple. Proposition 1 In the unique subgame perfect equilibrium of the three stage game upstream firms charge a price of r 2t u t = d m n t u (n ) 3 (m 1) + 2(m ) 3 t d while downstream firms charge a price of p = t d(2(m ) 2 t d + t u (n ) 2 (3m 1)) (n ) 2 (2(m ) 3 t d + t u (n ) 3 (m 1)).

16 15 The number of entering upstream and downstream firms, n and m, is implicitly defined by and t d (n ) 3 t u 6(m ) 2 = F d 2t u t d n t u (n ) 3 (m 1) + 2(m ) 3 t d = F u.

17 16 Graphical representation: Equilibrium number of firms, n and m n I D n I U m m

18 17 Intuition: Profit of a downstream firm is increasing in m More upstream firms reduce expected transport costs for downstream firms Profit of an upstream firm can either be increasing or decreasing in n More downstream firms imply downward pressure on the downstream prices and this carries over to the upstream market. Larger n means more potential buyers and this increases the possibility of self cannibalization Upstream prices increase First effect dominates if n is relatively large. Second effect dominates for relatively small n.

19 18 3. Interplay between the Two Markets Proposition 2 (i) The equilibrium number of upstream firms, is decreasing in F u and increasing in t u. (ii) The equilibrium number of downstream firms is decreasing in F u and decreasing in t u.

20 Proposition 3 (i) The equilibrium number of downstream firms is increasing in t d and decreasing in F d. (ii) The equilibrium number of upstream firms is increasing in t d. There exists an F d, such that the equilibrium number of upstream firms is increasing in F d if F d < F d, and decreasing in F d if F d > F d. 19

21 20 Graphical representation: Increase in t d (left) and increase in F d (right) n n n n I D I D n I U I D I U n I D I U m m m m m m

22 21 3. Deregulation Issues Policy maker can increase consumer surplus via reducing entry barriers (decrease in F u or F d ) or via spurring competition (decrease in t u or t d ). Consumer surplus is given by CS = K p t d 12n 2. Comparison of the effectiveness of reducing entry barriers upstream or downstream dcs df dcs d df u.

23 22 Focus on case where market conditions are similar, t d t u and n m. Proposition 3 If competitive conditions in both markets are equal, t d = t u and n = m, then W F F W F d F > 0 u if and only if m < m, where m is implicitly defined by 429( m) 2 37( m) m 157 = 0.

24 23 Intuition: If number of downstream firms is small, an increase in n causes also an increase in the number of upstream firms. This reinforces the effect on downstream firms because fixed costs decrease. The opposite holds true for a large number of downstream firms (non-monotonicity of the I U -curve).

25 24 4. Two-Part Tariffs Upstream firm i charges per unit price r i, i {1,..., m} and a fixed fee T i, i {1,..., m}, where T i is independent from the quantity that a downstream buys Proposition 4 There exists a unique symmetric subgame perfect equilibrium of the three stage game with two-part tariffs. Both curves, I D and I U, are increasing. n n tp I D I U m tp m

26 25 Proposition 5 (i) Concerning the entering number of firms we get n > n tp and m < m tp. (ii) Welfare under linear prices can either be higher or lower than under two-part tariffs. Intuition: Upstream firms can extract higher revenues from downstream firms via the fixed fee. Under two-part tariffs no double marginalization arises but n > n tp. Thus, welfare result is ambiguous. Welfare under linear pricing tends to be larger if e.g. downstream competition is high.

27 26 5. Conclusion Paper provided a model of successive vertical oligopolies with free entry and varying degrees of competition. Market outcome tends to be dominated by the competitive conditions on the downstream market. Deregulation in the downstream market is more effective than in the upstream market if the overall number of firms is small. Two-part tariffs can increase and decrease welfare depending on the competitive conditions upstream and downstream.