Endogenizing the cost parameter in Cournot oligopoly

Transcription

1 Endogenizing the cost parameter in Cournot oligopoly Stefanos Leonardos 1 and Costis Melolidakis National and Kapodistrian University of Athens Department of Mathematics, Section of Statistics & Operations Research January 9, Supported by the Alexander S. Onassis Public Benefit Foundation. 1/27

2 Overview 1 Introduction 2 Model 3 Complete information 4 Incomplete information 5 Remarks 2/27

3 Outline - section 1 1 Introduction 2 Model 3 Complete information 4 Incomplete information 5 Remarks 3/27

4 Motivation I Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α Q, profit functions are u i (Q i ) = Q i (p h) = Q i (α h Q) 1 Leslie M. Marx and Greg Schaffer, Cournot competition with a common input supplier, Working paper Duke University, /27

5 Motivation I Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α Q, profit functions are u i (Q i ) = Q i (p h) = Q i (α h Q) where (cost input) h < α is assumed to be given. But 2 1 Leslie M. Marx and Greg Schaffer, Cournot competition with a common input supplier, Working paper Duke University, /27

6 Motivation I Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α Q, profit functions are u i (Q i ) = Q i (p h) = Q i (α h Q) where (cost input) h < α is assumed to be given. But 2 where does cost input h come from? 1 Leslie M. Marx and Greg Schaffer, Cournot competition with a common input supplier, Working paper Duke University, /27

7 Motivation I Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α Q, profit functions are u i (Q i ) = Q i (p h) = Q i (α h Q) where (cost input) h < α is assumed to be given. But 2 where does cost input h come from? do firms produce the inputs themselves or do they purchase their inputs from a third-party supplier? 1 Leslie M. Marx and Greg Schaffer, Cournot competition with a common input supplier, Working paper Duke University, /27

8 Motivation I Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α Q, profit functions are u i (Q i ) = Q i (p h) = Q i (α h Q) where (cost input) h < α is assumed to be given. But 2 where does cost input h come from? do firms produce the inputs themselves or do they purchase their inputs from a third-party supplier? Concerns about the robustness of the results insights obtained by the prevailing approach. 1 Leslie M. Marx and Greg Schaffer, Cournot competition with a common input supplier, Working paper Duke University, /27

9 Motivation II Strategic considerations come into play 5/27

10 Motivation II Strategic considerations come into play 1 do firms have own production capacities? firms may produce limited quantities need to place orders. 5/27

11 Motivation II Strategic considerations come into play 1 do firms have own production capacities? firms may produce limited quantities need to place orders. 2 does the supplier (third party) know the actual demand? if he asks for a too high price no transactions take place. 5/27

12 Motivation II Strategic considerations come into play 1 do firms have own production capacities? firms may produce limited quantities need to place orders. 2 does the supplier (third party) know the actual demand? if he asks for a too high price no transactions take place. 3 the supplier becomes a player in a 2-stage sequential game. what is his equilibrium strategy? 5/27

13 Motivation II Strategic considerations come into play 1 do firms have own production capacities? firms may produce limited quantities need to place orders. 2 does the supplier (third party) know the actual demand? if he asks for a too high price no transactions take place. 3 the supplier becomes a player in a 2-stage sequential game. what is his equilibrium strategy? Key in studying the effects of an exogenous source of supply for Cournot oligopolists: relation between demand and various costs. 5/27

14 Objective It is the aim of this paper is to: 1 address these questions: complete/ incomplete information market structure 6/27

15 Objective It is the aim of this paper is to: 1 address these questions: complete/ incomplete information market structure 2 extend classic Cournot theory to oligopolies that may purchase additional quantities from a supplier 6/27

16 Objective It is the aim of this paper is to: 1 address these questions: complete/ incomplete information market structure 2 extend classic Cournot theory to oligopolies that may purchase additional quantities from a supplier 3 determine the equilibrium strategies of the Cournot oligopolists and the supplier. 6/27

17 Objective It is the aim of this paper is to: 1 address these questions: complete/ incomplete information market structure 2 extend classic Cournot theory to oligopolies that may purchase additional quantities from a supplier 3 determine the equilibrium strategies of the Cournot oligopolists and the How: supplier. 6/27

18 Objective It is the aim of this paper is to: 1 address these questions: complete/ incomplete information market structure 2 extend classic Cournot theory to oligopolies that may purchase additional quantities from a supplier 3 determine the equilibrium strategies of the Cournot oligopolists and the supplier. How: endogenize the oligopolists cost parameter(s) in a 2-stage sequential game. 6/27

19 Outline - section 2 1 Introduction 2 Model 3 Complete information 4 Incomplete information 5 Remarks 7/27

20 Model - Notation I We follow the classic Cournot oligopoly 1 one homogenous good; 2 fixed number of n 2 profit maximizing firms; 3 competition over quantity; 4 quantity choices are simultaneous and independent; 5 affine inverse demand function: p = α Q where Q := n i=1 Q i and α is the demand parameter. 8/27

21 Model - Notation II With following differences 9/27

22 Model - Notation II With following differences 1 capacity constraints: firms may produce limited quantities t i up to T i at a common fixed marginal h cost (normalized to 0); 9/27

23 Model - Notation II With following differences 1 capacity constraints: firms may produce limited quantities t i up to T i at a common fixed marginal h cost (normalized to 0); 2 external supplier: firms may order additional quantities q i at cost w > 0 set by the supplier; 9/27

24 Model - Notation II With following differences 1 capacity constraints: firms may produce limited quantities t i up to T i at a common fixed marginal h cost (normalized to 0); 2 external supplier: firms may order additional quantities q i at cost w > 0 set by the supplier; external supplier: may produce unlimited quantities but at a higher cost c > 0 with profit r := w c. 9/27

25 Model - Notation II With following differences 1 capacity constraints: firms may produce limited quantities t i up to T i at a common fixed marginal h cost (normalized to 0); 2 external supplier: firms may order additional quantities q i at cost w > 0 set by the supplier; external supplier: may produce unlimited quantities but at a higher cost c > 0 with profit r := w c. 3 demand uncertainty: 0 α random variable for the supplier, with non-atomic distribution and finite expectation E (α) < +. 9/27

26 Model - Notation II With following differences 1 capacity constraints: firms may produce limited quantities t i up to T i at a common fixed marginal h cost (normalized to 0); 2 external supplier: firms may order additional quantities q i at cost w > 0 set by the supplier; external supplier: may produce unlimited quantities but at a higher cost c > 0 with profit r := w c. 3 demand uncertainty: 0 α random variable for the supplier, with non-atomic distribution and finite expectation E (α) < +. All the above are common knowledge among the market participants. 9/27

27 Formal setting: 2 variations Market structure represented by a sequential 2-stage game. We examine 2 variations depending on the timing of the demand realization 10/27

28 Formal setting: 2 variations Market structure represented by a sequential 2-stage game. We examine 2 variations depending on the timing of the demand realization Complete information Demand realization. and the retailers. 1st Stage. Supplier sets price w = r + c. Demand parameter α observed by the supplier 2nd Stage. Retailers set quantities Q i (w) = t i (w) + q i (w). 10/27

29 Formal setting: 2 variations Market structure represented by a sequential 2-stage game. We examine 2 variations depending on the timing of the demand realization Complete information Demand realization. and the retailers. 1st Stage. Supplier sets price w = r + c. Demand parameter α observed by the supplier 2nd Stage. Retailers set quantities Q i (w) = t i (w) + q i (w). Incomplete information 1st Stage. Supplier sets price w, based on his belief about α. Demand realization. Demand parameter α observed by the retailers. 2nd Stage. Retailers set quantities Q i (w). 10/27

30 Outline - section 3 1 Introduction 2 Model 3 Complete information 4 Incomplete information 5 Remarks 11/27

31 Complete information Equilibria Case I duopoly with T 1 > T 2 (big and small firm); complete information: supplier knows demand parameter α. 12/27

32 Complete information Equilibria Case I duopoly with T 1 > T 2 (big and small firm); complete information: supplier knows demand parameter α. Result: Proposition. Given the values of α and w, the second stage equilibrium strategies between retailers R 1 and R 2 for all possible values of T 1 T 2 are given by 12/27

33 Complete information Equilibria Case I duopoly with T 1 > T 2 (big and small firm); complete information: supplier knows demand parameter α. 13/27

34 Complete information Equilibria II Case II duopoly with identical firms T 1 = T 2 = T ; 14/27

35 Complete information Equilibria II Case II duopoly with identical firms T 1 = T 2 = T ; Result: Theorem 3.5 The 2-stage game has a unique subgame perfect Nash equilibrium under which 1st Stage. supplier s profit margin: r (α) = 1 2 (α 3T c)+ 2nd Stage. each firm produces: ti (w) = T 1 3 (3T α)+ and orders: qi (w) = 1 3 (α 3T w)+ 14/27

36 Complete information Equilibria II Case II duopoly with identical firms T 1 = T 2 = T ; Result: Theorem 3.5 The 2-stage game has a unique subgame perfect Nash equilibrium under which 1st Stage. supplier s profit margin: r (α) = 1 2 (α 3T c)+ 2nd Stage. each firm produces: ti (w) = T 1 3 (3T α)+ and orders: qi (w) = 1 3 (α 3T w)+ For α big enough : Q i = 1 3 Theorem 3.5 generalizes to n > 2 firms. (α w) as in the classic Cournot. 14/27

37 Outline - section 4 1 Introduction 2 Model 3 Complete information 4 Incomplete information 5 Remarks 15/27

38 Incomplete information Equilibria I Case III duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief (distribution) about demand parameter α is continuous (non-atomic). 16/27

39 Incomplete information Equilibria I Case III duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief (distribution) about demand parameter α is continuous (non-atomic). Result: A. necessary condition. Expressed in terms of the mean residual lifetime (MRL) function m ( ) of the supplier s belief E (α t)+ E (α t α > t) = if P (α > t) > 0 m (t) : = 1 F (t) 0 otherwise 16/27

40 Incomplete information Equilibria I Case III duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief (distribution) about demand parameter α is continuous (non-atomic). Result: A. necessary condition. Expressed in terms of the mean residual lifetime (MRL) function m ( ) of the supplier s belief E (α t)+ E (α t α > t) = if P (α > t) > 0 m (t) : = 1 F (t) 0 otherwise Used in actuaries/reliability: not (to our knowledge) in a Cournot context. 16/27

41 Incomplete information Equilibria II Case III (continued) duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief about demand parameter α is continuous. Formally: Theorem 4.3 A. necessary condition If a Bayes - Nash equilibrium exists, i.e. if the optimal profit margin r of the supplier exists when the firms follow their second stage equilibrium strategies, then it satisfies the fixed point equation r = m (r + 3T + c) Hence r is a fixed point of a translation of the MRL function. 17/27

42 Incomplete information Equilibria III Case III (continued) duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief about demand parameter α is continuous. 18/27

43 Incomplete information Equilibria III Case III (continued) duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief about demand parameter α is continuous. Result: B. sufficient conditions. If the belief has bounded support, then an equilibrium exists. If the MRL function is decreasing, then an equilibrium exists and is unique. 18/27

44 Incomplete information Equilibria III Case III (continued) duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief about demand parameter α is continuous. Result: B. sufficient conditions. If the belief has bounded support, then an equilibrium exists. If the MRL function is decreasing, then an equilibrium exists and is unique. Most distributions that are used in oligopoly applications have the DMRL property. 18/27

45 Incomplete information Equilibria IV Case III (continued) duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief about demand parameter α is continuous. 19/27

46 Incomplete information Equilibria IV Case III (continued) duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief about demand parameter α is continuous. Formally: Theorem 4.3 (B. sufficient condition) Under the DMRL property, the optimal profit margin r of the supplier exists under equilibrium and it is the unique solution of the equation r = m (r + 3T + c) 19/27

47 Incomplete information Equilibria IV Case III (continued) duopoly with identical firms T 1 = T 2 = T ; incomplete information: supplier s belief about demand parameter α is continuous. Formally: Theorem 4.3 (B. sufficient condition) Under the DMRL property, the optimal profit margin r of the supplier exists under equilibrium and it is the unique solution of the equation r = m (r + 3T + c) For identical firms the results generalize to n > 2. 19/27

48 Classic Cournot Equilibria V Special case T = 0: classic Cournot duopoly with an external supplier; duopolists cost = price set by the supplier under incomplete information about the demand parameter α; 20/27

49 Classic Cournot Equilibria V Special case T = 0: classic Cournot duopoly with an external supplier; duopolists cost = price set by the supplier under incomplete information about the demand parameter α; Result: For supplier s cost c < range α same as in classic Cournot Corollary 4.6 Under the DMRL property the 2-stage game has a unique subgame perfect Bayes - Nash equilibrium under which 1st stage: the supplier sells at: r = m (r ) 2nd stage: each firm orders: q (r) = 1 3 (α r)+ 20/27

50 Outline - section 5 1 Introduction 2 Model 3 Complete information 4 Incomplete information 5 Remarks 21/27

51 Remarks I Through the fixed point equation study variational properties of the equilibrium Corollary 4.3 Under the DMRL property: 1 If the firms own production capacity T increases, then at equilibrium, the supplier s profit margin and the price he asks, both decrease. 2 Whereas if the cost of the supplier increases, then at equilibrium, the supplier s profit margin decreases while the price he asks increases. 22/27

52 Remarks II Similarly measure the inefficiencies at equilibrium due to incomplete information Theorem 5.2 Under the DMRL property: 1 The conditional probability that a transaction does not occur under equilibrium in the incomplete information case, given that a transaction would have occurred under equilibrium if we had been in the complete information case, can not exceed the bound: 1 e 1. 2 This bound is tight over all DMRL distributions. 23/27

53 Remarks III Lastly, examine possible relaxations of the DMRL property mainly via the increasing generalized failure rate (IGFR) property. 24/27

54 Remarks III Lastly, examine possible relaxations of the DMRL property mainly via the increasing generalized failure rate (IGFR) property. e.g. Pareto distribution: not DMRL / has IGFR property, yet an optimal strategy for the supplier does not exist No Nash Equilibrium. Let α f (α) = k α k+1 for α 1 and 3T > 1. Mean residual lifetime: m (r) = r k 1 Generalized failure rate: rh (r) = c Supplier has no optimal strategy for 1 < k < 2, since as r +. u s (r) = 2 3 (k 1) r (r + c + 3T )1 k + 24/27

55 Selected References I [1] Bagnoli M., Bergstrom T. 2005, Log-concave probability and its applications, Economic Theory, Vol. 26 (2), [2] Bernstein F. & Federgruen A., 2005, Decentralized supply chains with competing retailers under demand uncertainty, Management Science, (51) No.1, [3] Bradley D. & Gupta R., 2003, Limiting behavior of the mean residual life, Annals of the Institute of Statistical Mathematics, (55) No. 1, [4] Einy E., Haimanko O., Moreno D. & Shitovitz B., 2010, On the existence of Bayesian Cournot equilibrium, Games and Economic Behavior, (68) [5] Guess F. & Proschan F., 1988, Mean residual life: theory and applications, Handbook of Statistics: Quality Control and Reliability, (7), [6] W. Hall & Jon Wellner, 1981, Mean residual life, In M. Csórgó, D. A. Dawson, J. N. K. Rao, and A. K. Md. E. Saleh, editors, Proceedings of the International Symposium on Statistics and Related Topics, pages North Holland Amsterdam. 25/27

56 Selected References II [7] Lagerlöf J., 2006, Equilibrium uniqueness in a Cournot model with demand uncertainty, Topics in Theoretical Economics, Vol. (6), Issue 1, Article 19. [8] Lariviere M. A., A note on probability distributions with increasing generalized failure rates, Operations Research, Vol. (54), No 3, [9] Leslie Marx & Greg Schaffer. Cournot competition with a common input supplier, Working paper Duke University, Availabe at: marx/bio/papers/marxshaffercournot2015.pdf. [10] Myerson R., Satterthwaite M., Efficient Mechanisms for Bilateral Trading, 1983, Journal of Economic Theory, 29(2), [11] Vives X., 2001, Oligopoly Pricing: Old ideas and new tools, The MIT Press, Cambridge MA. 26/27

57 Source The complete paper (with many more interesting results!) may be found online at: Thank you for your attention! 27/27