# LECTURE 6: OLIGOPOLY. Jan Zouhar. Games and Decisions

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1 LECTURE 6: OLIGOPOLY

2 Market Structures the list of basic market structure types (seller-side types only): Number of sellers Seller entry barriers Deadweight loss Perfect competition Many No None Monopolistic competition Many No None Oligopoly Few Yes Medium Monopoly One Yes High

3 Collusive vs. Non-Collusive Oligopolies 3 note: oligopoly differs from monopoly (allocation-wise) only if there s no collusion collusion: a largely illegal form of cooperation amongst the sellers that includes price fixing, market division, total industry output control, profit division, etc. controlled by competition/anti-trust laws well-known collusion cases: OPEC, telecommunication, drugs, sports, chip dumping (poker) game-theoretical models: cooperative setting (collusive oligopoly) coalition theory games in the characteristic-function form non-cooperative setting (competitive, non-collusive oligopoly) normal form game analysis NE s etc.; however, matrices can t typically be used for payoffs

4 Oligopoly Model Specification 4 to make the analysis simple, we ll make several assumptions: 1. single-product model: oligopolists produce a single type of homogenous product. one strategic variable: firms decide about prices or output levels 3. static model: single-period analysis only in dynamic models, there are more diverse strategic options: elimination of competitors even with contemporary losses etc. 4. single objective: all firms maximize their individual profit Three basic non-cooperative oligopoly models: Bertrand oligopoly firms simultaneously choose prices Cournot oligopoly firms simultaneously choose quantities Stackelberg oligopoly firms choose quantities sequentially note: sequential-move games are typically not modelled as normal-form games. Instead, we use the extensive-form approach (not this lecture).

5 5 Bertrand Duopoly Bertrand duopoly ( oligopolists only) model notation: market demand function: q = D(p) prices charged by the players: p 1, p resulting quantities: q 1, q unit costs: c 1, c (for simplicity: AC = MC = c) homogenous product lower price attracts all the consumers p 1 < p q 1 = D(p 1 ), q = 0 p 1 > p q 1 = 0, q = D(p ) p 1 = p equal market share, q 1 = q = ½ D(p 1 ) = ½ D(p ) as long as the prices are higher than c 1 and c, both oligopolists tend to push prices down (below the other player s price) imagine the prices are equal and above c 1 ; by lowering the price just slightly, player 1 can gain the whole market (if p stays the same) best response of player 1 to p is to choose p 1 = p ε ( just below p ) (until the prices reach c 1 player 1 suffers a loss below)

6 6 Bertrand Duopoly (cont d) NE depends on the MC of the players: c 1 = c p 1 * = p * = c 1 = c zero economic profit for both c 1 < c p 1 * = c ε player 1 wins all, positive profit c 1 > c p * = c 1 ε player wins all, positive profit more precisely: graphical best-response analysis reaction curves: p 1 (p ) 45 monopoly price (no player 1) p p M NE p (p 1 ) p just below p 1 (curve just below 45 ) c c 1 p 1 M p 1

7 Bertrand Duopoly (cont d) 7 price competition leads to fairly efficient allocation Critique of the Bertrand model (or, when Bertrand model fails to work) capacity constraints of production e.g., consider the c 1 < c situation: if player 1 can t supply enough for the whole market, player can still charge p above c and attract some customers (and achieve a positive profit) if c 1 = c and neither player can supply to all customers, either player can raise the output price above c lack of product homogeneity (homogeneity disputable in most cases) transaction/transportation costs: may differ for the specific customer firm interactions e.g., shops at both ends of a street: people tend to pick the closer one if transportation costs are accounted for, the consumer expenditures vary even if prices are equal

8 Cournot Oligopoly Formal Treatment 8 model type normal-form game with the following elements: list of firms: 1,,,N strategy spaces: X 1, X,,X N potential quantities: typically intervals like [0,1000] infinite alternatives! the output level produced by i th player (the strategy adopted) is denoted x i a strategy profile is an N-tuple: (x 1,x,,x N ) (where x i X i ) cost functions: c 1 (x 1 ), c (x ),, c N (x N ) total cost as the function of output level price function (or inverse demand function): p = f(x 1 + x + + x N ) i.e., market price is the function of total industry output profit of i th firm: i( x1,..., xn ) TRi TCi xi f ( x1... xn ) ci ( xi )

9 Nash Equilibrium in Cournot Oligopoly 9 mathematical definition: A strategy profile (x 1 *, x *,, x N *) is a NE if for all i = 1,,N ( x *, x *,..., x,..., x *) ( x *, x *,..., x *,..., x *) holds for all x i X i. i 1 i N i 1 i N finding the NE: best-response approach (again) NE: the strategies have to be the best responses to one another best-response functions: for player 1: r 1 (x,,x N ) is the best-response x 1 chosen by player 1, given that the other player s quantities are x,,x N mathematically: r ( x,... x ) argmax ( x, x,..., x ) 1 N i 1 N x X 1 1 NE: x * r ( x *,..., x *, x *,..., x *) i i 1 i1 i1 N for i = 1,,N

10 10 Example 1: Cournot Duopoly price function: other characteristics: p f( x x ) 100 ( x x ) 1 1 X 0, c ( x ) 150 1x , ( ) X c x x profit functions: ( x, x ) x f( x x ) c ( x ) x1 100 ( x1 x) 150 1x x1 x1 x1x 150 ( x1, x) 100x x x1x 100x x x x 150 1x1 best response of player 1 to x : profit-maximizing (π 1 -maximizing) value of x 1 for the given x

11 11 Example 1: Cournot Duopoly profit of player 1 for three different levels of x : 1500 (cont d) best response to x = 10 is x 1 40, best response to x = 5 is x 1 33, best response to x = 50 is x 1 0, x = 10 x = 5 x = x 1 best response for an arbitrary level of x : as the function π 1 (x 1,x ) is strictly concave in x 1 for any x, we can use the first-order condition for a local extreme! 1( x1, x) 88 x1 x 0 x 1

12 Note: first order conditions are generally not sufficient for a maximum, only necessary conditions for extreme (but: concave function global maximum) f( x) x f(x) x

13 13 Example 1: Cournot Duopoly (cont d) we can also write the result in terms of the reaction function r 1 :! 1( x 1, x ) 88 x x 0 x r ( x ) 44 x x similarly, for player, we obtain: ( x 1, x ) 100 x 4 x 0 x r ( x ) 5 x x! altogether, we have linear equations; for NE strategies, both have to hold at the same time in order to find the NE, we just need to solve 1 88 x1* x* 0 x1* r1( x*) x1* 36 or 100 x * 4 x * 0 x * r ( x *) x * Question: What are the equilibrium profits and price?

14 Collusive Oligopoly 14 model framework: as in case of Cournot oligopoly, only that players can form coalitions coalition: a group of firms that coordinate output levels and redistribute profits grand coalition: the coalition of all oligopolists, Q = {1,,,N } other coalitions are denoted by K,L, a single-firm coalition is still called a coalition, e.g. K = {}, and so is the empty coalition { } Question: How many different coalitions can be formed with N firms? characteristic function (of the oligopoly): a function v(k) that assigns to any coalition K the maximum attainable total profit of K payoff function: single player, individual payoff, for a given strategy profile characteristic function: coalition, sum of members profits, max. attainable

15 15 Collusive Oligopoly (cont d) characteristic function for grand coalition: v ( Q ) max ( x,..., x ) N ( x1,..., xn ) i1 i 1 N characteristic function for other coalitions: profit of coalition members depends on the quantity chosen by non-members what will the other players do? (Generally, difficult to say.) 1. minimax characteristic function: assume non-members supply as much as they can (up to their capacity constraints). equilibrium characteristic function: assume the other players choose the NE quantities characteristic function for an arbitrary coalition: v( K ) max i K i ( x 1,..., x N ) ( x ) i ik

16 16 Example : Collusive Duopoly consider the same duopoly as in example 1, only with capacity constraints: price function: other characteristics: profit functions: first, we ll find the equilibrium characteristic function: we already know the NE values: p f( x x ) 100 ( x x ) 1 1 X 0,40 c ( x ) 150 1x ,0 ( ) X c x x 1 x1 x x1 x1 x1x ( x1, x) 100x x x1x (, ) x x * 36 * * 16 * 51 immediately, we have: v(1) max ( x,16) (36,16) 1146 x X v() max (36, x ) (36,16) 51 x X

17 17 Example : Collusive Duopoly (cont d) for grand coalition Q = {1,}, we obtain: v(1,) max ( x, x ) ( x, x ) ( x, x ) 1 ( x, x ) max 88x 100x x x x x 150 function of two variables now, but still concave (see next slide) first-order conditions (both partial derivatives equal zero)! 1,( x1, x) 88 x1 x 0 opt x1 x1 38 opt! opt 1, 1,( x1, x) x x1 4x 0 x 18 v(q) = v(1,) = 18

18 Example : Collusive Duopoly (cont d) 0 x πy 1, - -4 x x x x 1

19 first-order conditions: necessary conditions for local extremes (not sufficient, not for maxima only!) 0.5 y x x

20 0 Example : Collusive Duopoly (cont d) the complete equilibrium characteristic function is as follows: minimax characteristic function: v( ) 0 v(1) 1146 v() 51 v(1,) 18 v( ) and v(1,) are the same as in the equilibrium char. function for v(1) and v(), we calculate the players profits under the condition that the other player produces up to his/her capacity constraint: x X x X v(1) max ( x,0) max 68x1 x 150 (34,0) x X x X v() max (40, x ) max 60x x (40,15) 450

21 1 Example : Collusive Duopoly (cont d) a comparison of the two characteristic functions: equilibrium minimax v( ) 0 0 v(1) v() v(1,) v(1,) v(1) v() core of the oligopoly: a division of payoffs a 1,a such that a a 18, a a 18, 1 1 a a 1146, or a 1006, , a 450.

22 LECTURE 6: OLIGOPOLY

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