Endogenizing the cost parameter in Cournot oligopoly Stefanos - - PowerPoint PPT Presentation

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Endogenizing the cost parameter in Cournot oligopoly

Stefanos Leonardos1 and Costis Melolidakis

National and Kapodistrian University of Athens Department of Mathematics, Section of Statistics & Operations Research

January 9, 2018

1Supported by the Alexander S. Onassis Public Benefit Foundation.

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Overview

1

Introduction

2

Model

3

Complete information

4

Incomplete information

5

Remarks

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Outline - section 1

1

Introduction

2

Model

3

Complete information

4

Incomplete information

5

Remarks

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Motivation I

Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α − Q, profit functions are ui (Qi) = Qi (p − h) = Qi (α − h − Q)

1Leslie M. Marx and Greg Schaffer, “Cournot competition with a common input

supplier”, Working paper Duke University, 2015.

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Motivation I

Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α − Q, profit functions are ui (Qi) = Qi (p − h) = Qi (α − h − Q) where (cost input) h < α is assumed to be given. But2

1Leslie M. Marx and Greg Schaffer, “Cournot competition with a common input

supplier”, Working paper Duke University, 2015.

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Motivation I

Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α − Q, profit functions are ui (Qi) = Qi (p − h) = Qi (α − h − Q) where (cost input) h < α is assumed to be given. But2 where does cost input h come from?

1Leslie M. Marx and Greg Schaffer, “Cournot competition with a common input

supplier”, Working paper Duke University, 2015.

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Motivation I

Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α − Q, profit functions are ui (Qi) = Qi (p − h) = Qi (α − h − Q) where (cost input) h < α is assumed to be given. But2 where does cost input h come from? do firms produce the inputs themselves or do they purchase their inputs from a third-party supplier?

1Leslie M. Marx and Greg Schaffer, “Cournot competition with a common input

supplier”, Working paper Duke University, 2015.

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Motivation I

Cournot market: basic model for oligopoly theory (quantity). For affine demand p = α − Q, profit functions are ui (Qi) = Qi (p − h) = Qi (α − h − Q) where (cost input) h < α is assumed to be given. But2 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.

1Leslie M. Marx and Greg Schaffer, “Cournot competition with a common input

supplier”, Working paper Duke University, 2015.

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Motivation II

Strategic considerations come into play

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Motivation II

Strategic considerations come into play

1 do firms have own production capacities?

firms may produce limited quantities − → need to place orders.

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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.

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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?

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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

• ligopolists: relation between demand and various costs.

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Objective

It is the aim of this paper is to:

1 address these questions: complete/ incomplete information market

structure

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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 addi-

tional quantities from a supplier

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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 addi-

tional quantities from a supplier

3 determine the equilibrium strategies of the Cournot oligopolists and the

supplier.

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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 addi-

tional quantities from a supplier

3 determine the equilibrium strategies of the Cournot oligopolists and the

supplier. How:

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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 addi-

tional 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.

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Outline - section 2

1

Introduction

2

Model

3

Complete information

4

Incomplete information

5

Remarks

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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 Qi and α is the demand parameter.

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Model - Notation II

With following differences

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Model - Notation II

With following differences

1 capacity constraints: firms may produce limited quantities ti

up to Ti at a common fixed marginal h cost (normalized to 0);

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Model - Notation II

With following differences

1 capacity constraints: firms may produce limited quantities ti

up to Ti at a common fixed marginal h cost (normalized to 0);

2 external supplier: firms may order additional quantities qi

at cost w > 0 set by the supplier;

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Model - Notation II

With following differences

1 capacity constraints: firms may produce limited quantities ti

up to Ti at a common fixed marginal h cost (normalized to 0);

2 external supplier: firms may order additional quantities qi

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.

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Model - Notation II

With following differences

1 capacity constraints: firms may produce limited quantities ti

up to Ti at a common fixed marginal h cost (normalized to 0);

2 external supplier: firms may order additional quantities qi

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) < +∞.

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Model - Notation II

With following differences

1 capacity constraints: firms may produce limited quantities ti

up to Ti at a common fixed marginal h cost (normalized to 0);

2 external supplier: firms may order additional quantities qi

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.

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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

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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. Demand parameter α observed by the supplier and the retailers. 1st Stage. Supplier sets price w = r + c. 2nd Stage. Retailers set quantities Qi(w) = ti(w) + qi(w).

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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. Demand parameter α observed by the supplier and the retailers. 1st Stage. Supplier sets price w = r + c. 2nd Stage. Retailers set quantities Qi(w) = ti(w) + qi(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 Qi(w).

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Outline - section 3

1

Introduction

2

Model

3

Complete information

4

Incomplete information

5

Remarks

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Complete information – Equilibria

Case I duopoly with T1 > T2 (big and small firm); complete information: supplier knows demand parameter α.

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Complete information – Equilibria

Case I duopoly with T1 > T2 (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 R1 and R2 for all possible values of T1 ≥ T2 are given by

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Complete information – Equilibria

Case I duopoly with T1 > T2 (big and small firm); complete information: supplier knows demand parameter α.

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Complete information – Equilibria II

Case II duopoly with identical firms T1 = T2 = T;

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Complete information – Equilibria II

Case II duopoly with identical firms T1 = T2 = 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: t∗

i (w) = T − 1 3 (3T − α)+

and orders: q∗

i (w) = 1 3 (α − 3T − w)+

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Complete information – Equilibria II

Case II duopoly with identical firms T1 = T2 = 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: t∗

i (w) = T − 1 3 (3T − α)+

and orders: q∗

i (w) = 1 3 (α − 3T − w)+

For α “big enough”: Q∗

i = 1 3 (α − w) as in the classic Cournot.

Theorem 3.5 generalizes to n > 2 firms.

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Outline - section 4

1

Introduction

2

Model

3

Complete information

4

Incomplete information

5

Remarks

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Incomplete information – Equilibria I

Case III duopoly with identical firms T1 = T2 = T; incomplete information: supplier’s belief (distribution) about demand parameter α is continuous (non-atomic).

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Incomplete information – Equilibria I

Case III duopoly with identical firms T1 = T2 = 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 m (t) : =

    

E (α − t | α > t) = E (α − t)+ 1 − F (t) if P (α > t) > 0

• therwise

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Incomplete information – Equilibria I

Case III duopoly with identical firms T1 = T2 = 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 m (t) : =

    

E (α − t | α > t) = E (α − t)+ 1 − F (t) if P (α > t) > 0

• therwise

Used in actuaries/reliability: not (to our knowledge) in a Cournot context.

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Incomplete information – Equilibria II

Case III (continued) duopoly with identical firms T1 = T2 = 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.

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Incomplete information – Equilibria III

Case III (continued) duopoly with identical firms T1 = T2 = T; incomplete information: supplier’s belief about demand parameter α is continuous.

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Incomplete information – Equilibria III

Case III (continued) duopoly with identical firms T1 = T2 = 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.

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Incomplete information – Equilibria III

Case III (continued) duopoly with identical firms T1 = T2 = 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.

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Incomplete information – Equilibria IV

Case III (continued) duopoly with identical firms T1 = T2 = T; incomplete information: supplier’s belief about demand parameter α is continuous.

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Incomplete information – Equilibria IV

Case III (continued) duopoly with identical firms T1 = T2 = 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)

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Incomplete information – Equilibria IV

Case III (continued) duopoly with identical firms T1 = T2 = 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.

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Classic Cournot – Equilibria V

Special case T = 0: classic Cournot duopoly with an external supplier; duopolists’ cost = price set by the supplier under incomplete informa- tion about the demand parameter α;

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Classic Cournot – Equilibria V

Special case T = 0: classic Cournot duopoly with an external supplier; duopolists’ cost = price set by the supplier under incomplete informa- tion 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)+

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Outline - section 5

1

Introduction

2

Model

3

Complete information

4

Incomplete information

5

Remarks

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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.

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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.

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Remarks III

Lastly, examine possible relaxations of the DMRL property – mainly via the increasing generalized failure rate (IGFR) property.

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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 αk+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 us (r) = 2 3 (k − 1)r (r + c + 3T)1−k → +∞ as r → +∞.

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Selected References I

[1] Bagnoli M., Bergstrom T. 2005, Log-concave probability and its applications, Eco- nomic Theory, Vol. 26 (2), 445-469. [2] Bernstein F. & Federgruen A., 2005, Decentralized supply chains with competing retailers under demand uncertainty, Management Science, (51) No.1, 18-29. [3] Bradley D. & Gupta R., 2003, Limiting behavior of the mean residual life, Annals of the Institute of Statistical Mathematics, (55) No. 1, 217-226. [4] Einy E., Haimanko O., Moreno D. & Shitovitz B., 2010, On the existence of Bayesian Cournot equilibrium, Games and Economic Behavior, (68) 77-94. [5] Guess F. & Proschan F., 1988, Mean residual life: theory and applications, Handbook

• f Statistics: Quality Control and Reliability, (7), 215-224.

[6]

• W. Hall & Jon Wellner, 1981, Mean residual life, In M. Cs´
• rg´
• , D. A. Dawson, J. N.
• K. Rao, and A. K. Md. E. Saleh, editors, Proceedings of the International Symposium
• n Statistics and Related Topics, pages 169-184. North Holland Amsterdam.

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Selected References II

[7] Lagerl¨

• f J., 2006, Equilibrium uniqueness in a Cournot model with demand uncer-

tainty, Topics in Theoretical Economics, Vol. (6), Issue 1, Article 19. [8] Lariviere M. A., 2006. A note on probability distributions with increasing generalized failure rates, Operations Research, Vol. (54), No 3, 602-604. [9] Leslie Marx & Greg Schaffer. Cournot competition with a common input supplier, Working paper Duke University, 2015. Availabe at: https://faculty.fuqua.duke.edu/ marx/bio/papers/MarxShafferCournot2015.pdf. [10] Myerson R., Satterthwaite M., Efficient Mechanisms for Bilateral Trading, 1983, Journal of Economic Theory, 29(2), 265-281. [11] Vives X., 2001, Oligopoly Pricing: Old ideas and new tools, The MIT Press, Cam- bridge MA.

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Source

The complete paper (with many more interesting results!) may be found

• nline at: https://arxiv.org/abs/1601.07365.

Thank you for your attention!