Static Models of Oligopoly

Transcription

1 Static Models of Oligopoly Cournot and Bertrand Models Mateusz Szetela 1 1 Collegium of Economic Analysis Warsaw School of Economics 3 March 2016

2 Outline 1 Introduction Game Theory and Oligopolies 2 The Bertrand Model Basic Model N frims model Diversied product 3 The Cournot Model Basic Model Numerical Example N rms setting

3 Outline 1 Introduction Game Theory and Oligopolies 2 The Bertrand Model Basic Model N frims model Diversied product 3 The Cournot Model Basic Model Numerical Example N rms setting

4 Oligopoly - denition Frame subtitles are optional. Use upper- or lowercase letters. More than one rm, but not too many

5 Oligopoly - denition Frame subtitles are optional. Use upper- or lowercase letters. More than one rm, but not too many Similar products (or the same)

6 Oligopoly - denition Frame subtitles are optional. Use upper- or lowercase letters. More than one rm, but not too many Similar products (or the same) Static oligopoly

7 Why Game Theory? Frame subtitles are optional. Use upper- or lowercase letters. Non continuous prot functions

8 Why Game Theory? Frame subtitles are optional. Use upper- or lowercase letters. Non continuous prot functions Leaders and Followers

9 Why Game Theory? Frame subtitles are optional. Use upper- or lowercase letters. Non continuous prot functions Leaders and Followers Cartels

10 Outline 1 Introduction Game Theory and Oligopolies 2 The Bertrand Model Basic Model N frims model Diversied product 3 The Cournot Model Basic Model Numerical Example N rms setting

11 Basic Bertrand Model Frame subtitles are optional. Use upper- or lowercase letters. Two Players (duopoly)

12 Basic Bertrand Model Frame subtitles are optional. Use upper- or lowercase letters. Two Players (duopoly) Players choose price

13 Basic Bertrand Model Frame subtitles are optional. Use upper- or lowercase letters. Two Players (duopoly) Players choose price Symetric production (prot) functions

14 Basic Bertrand Model Frame subtitles are optional. Use upper- or lowercase letters. Two Players (duopoly) Players choose price Symetric production (prot) functions Constant marginal cost

15 Basic Bertrand Model Frame subtitles are optional. Use upper- or lowercase letters. Two Players (duopoly) Players choose price Symetric production (prot) functions Constant marginal cost Demand linear, decrising with p: x(p)

16 Mathematical representation Demand function

17 Mathematical representation x(p j ) if p j < p k x j (p j,p k ) = 1 2 x(p j) if p j = p k (1) 0 if p j > p k Demand function

18 Mathematical representation x(p j ) if p j < p k x j (p j,p k ) = 1 2 x(p j) if p j = p k (1) 0 if p j > p k Demand function Prot function

19 Mathematical representation x(p j ) if p j < p k x j (p j,p k ) = 1 2 x(p j) if p j = p k (1) 0 if p j > p k Demand function Prot function π j (p j,p k ) = x j (p j,p k )(p j c) (2)

20 Solution Proposition 1 There is an unique Nash equilibrium (p j,p k ) in the Bertrand duopoly model. In this equilibrium, both rms set their prices equal to cost: p j = p k = c.

21 Solution Proposition 1 There is an unique Nash equilibrium (pj,p k ) in the Bertrand duopoly model. In this equilibrium, both rms set their prices equal to cost: pj = pk = c. Proof

22 Outline 1 Introduction Game Theory and Oligopolies 2 The Bertrand Model Basic Model N frims model Diversied product 3 The Cournot Model Basic Model Numerical Example N rms setting

23 Bertrand with N players Firms problems are exactly the same

24 Bertrand with N players Firms problems are exactly the same This time N players compete (N > 2)

25 Bertrand with N players Firms problems are exactly the same This time N players compete (N > 2) Corollary In setting with N > 2 rms Bertrand model of oligopoly produces exactly same results as with N = 2 rms

26 Outline 1 Introduction Game Theory and Oligopolies 2 The Bertrand Model Basic Model N frims model Diversied product 3 The Cournot Model Basic Model Numerical Example N rms setting

27 Bertrand with diversied product Two Players. Products are not perfect subsidies Example: spatial model Reservation Price V > c t - cost of 'traveling' N - number of customers

28 Outline 1 Introduction Game Theory and Oligopolies 2 The Bertrand Model Basic Model N frims model Diversied product 3 The Cournot Model Basic Model Numerical Example N rms setting

29 Basic Model Two Players (duopoly)

30 Basic Model Two Players (duopoly) Players chose quantity

31 Basic Model Two Players (duopoly) Players chose quantity Symetric prot functions

32 Basic Model Two Players (duopoly) Players chose quantity Symetric prot functions Constant marginal cost

33 Basic Model Two Players (duopoly) Players chose quantity Symetric prot functions Constant marginal cost Inverse demand function p(q) linear, decrising with q = N j=1 q j

34 Players' Problems Maximization problem j-th player faces problem: Max q j 0 p(q j + q k )q j cq j (3)

35 Players' Problems Maximization problem j-th player faces problem: Max q j 0 p(q j + q k )q j cq j (3) in this setting rm j faces inverse demand function, like monopolist: p (q j + q k )

36 Players' Problems Maximization problem j-th player faces problem: Max q j 0 p(q j + q k )q j cq j (3) FOC: in this setting rm j faces inverse demand function, like monopolist: p (q j + q k ) Solution p (q j + q k )q j + p (q j + q k ) c with equality if q j > 0 (4)

37 Players' Problems Maximization problem j-th player faces problem: Max q j 0 p(q j + q k )q j cq j (3) FOC: in this setting rm j faces inverse demand function, like monopolist: p (q j + q k ) Solution p (q j + q k )q j + p (q j + q k ) c with equality if q j > 0 (4) Let b j ( q k ) denote set of optimal responses of player j, given strategy (quantity) of player k

38 Nash Equilibrium NE if and only if: p ( q j + q k) q j + p ( q j + q k) c (5) and p ( q j + q k) q k + p ( q j + q k) c (6)

39 Nash Equilibrium NE if and only if: p ( q j + q k) q j + p ( q j + q k) c (5) and p ( q j + q k) q k + p ( q j + q k) c (6) Let's add (5) and (6): ( ) p ( qj + ) q q j + q k k + p ( qj 2 + k) q = c (7) Proposition 2 In any Nash equilibria of the Cournot duopoly model with costs c > 0 per unit forthe two rms and an inverse demand function p( ) satisfying p (q) < 0 for all q 0 and p(0) > c, the market price is greater than c (the competitive and smaller than monopoly price.

40 Outline 1 Introduction Game Theory and Oligopolies 2 The Bertrand Model Basic Model N frims model Diversied product 3 The Cournot Model Basic Model Numerical Example N rms setting

41 Problem Let q = N j=1 q j and N = 2, inverse demand is given by: Cost functions are linear, that is c j (q j ) = cq j p(q) = a bq (8)

42 Problem Let q = N j=1 q j and N = 2, inverse demand is given by: Cost functions are linear, that is c j (q j ) = cq j p(q) = a bq (8) Solution solve for monopolist (N = 1) (nd q m );

43 Problem Let q = N j=1 q j and N = 2, inverse demand is given by: Cost functions are linear, that is c j (q j ) = cq j p(q) = a bq (8) Solution solve for monopolist (N = 1) (nd q m ); solve for competetive markets (p = c) (nd q );

44 Problem Let q = N j=1 q j and N = 2, inverse demand is given by: Cost functions are linear, that is c j (q j ) = cq j p(q) = a bq (8) Solution solve for monopolist (N = 1) (nd q m ); solve for competetive markets (p = c) (nd q ); nd best response functions for each rm;

45 Problem Let q = N j=1 q j and N = 2, inverse demand is given by: Cost functions are linear, that is c j (q j ) = cq j p(q) = a bq (8) Solution solve for monopolist (N = 1) (nd q m ); solve for competetive markets (p = c) (nd q ); nd best response functions for each rm;...

46 Problem Let q = N j=1 q j and N = 2, inverse demand is given by: Cost functions are linear, that is c j (q j ) = cq j p(q) = a bq (8) Solution solve for monopolist (N = 1) (nd q m ); solve for competetive markets (p = c) (nd q ); nd best response functions for each rm;... prot!

47 Outline 1 Introduction Game Theory and Oligopolies 2 The Bertrand Model Basic Model N frims model Diversied product 3 The Cournot Model Basic Model Numerical Example N rms setting

48 Cournot with N players Firms face exact same problems N rms compete

49 Cournot with N players Firms face exact same problems N rms compete Equation (7) takes following form with N rms: p (q ) (q ) N + p (q ) = c,where q is aggregate output in NE (9)

50 Cournot with N players Firms face exact same problems N rms compete Equation (7) takes following form with N rms: p (q ) (q ) N + p (q ) = c,where q is aggregate output in NE (9) if N equation (9) takes form: p (q ) = c = p (10)

51 Cournot with N players Firms face exact same problems N rms compete Equation (7) takes following form with N rms: p (q ) (q ) N + p (q ) = c,where q is aggregate output in NE (9) if N equation (9) takes form: p (q ) = c = p (10) if N = 1 equation (9) takes form: p (q )q + p (q ) = c (11)

52 Cournot with N players Firms face exact same problems N rms compete Equation (7) takes following form with N rms: p (q ) (q ) N + p (q ) = c,where q is aggregate output in NE (9) if N equation (9) takes form: p (q ) = c = p (10) if N = 1 equation (9) takes form: which is monopolist solution p (q )q + p (q ) = c (11)

53 Summary The Bertrand Model The Cournot Model Strategical Supbstitutes vs Strategical Complements Outlook Dynamic Games