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Cournot Equilibrium - Illustration

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Cournot Equilibrium - Illustration A course on Industrial Organization Xavier Martinez-Giralt Universitat Autonoma` de Barcelona [email protected] Fall 2009-2010 – p.1/133 Static Oligopoly Pricing - Homogeneous Product Quantity competition Price competition Price-Quantity competition Fall 2009-2010 – p.2/133 Homogeneous Product Definition: Two products are homogeneous if at the eyes of the consumer they provide exactly the same service. Illustration: Assume consumers’ preferences are rough enough so that all chairs are perceived exactly alike regardless of whether they have arms, wheels, made of wood, metal, etc. In such a case, consumers will simply demand chairs. Thus, there can only be a single demand function for chairs. Examples: Difficult. Sulphuric acid, electricity Fall 2009-2010 – p.3/133 Oligopoly Definition: An industry is said to be oligopolistic whenever the decision of one firm affects and is affected by the decisions of the other firms in the industry. [STRATEGIC INTERACTION]. Features: Typically, such situation is associated with a limited number of firms in the industry Decision variables: prices or quantities; entry; R&D, ... Decision “timing" across firms: simultaneous, sequential (commitment) Decision “timing" across decisions: simultaneous, sequential Equilibrium concept: Nash (and some variations) Starting point: Cournot oligopoly model (modern version); (Original 1838) Fall 2009-2010 – p.4/133 Cournot model - Assumptions Structural: Static model Technology: cost function Ci(qi) Aggregate demand function Q = F (p) Homogeneous product market Large number of consumers There are n firms in the industry, i = 1, 2,...,n No entry, no exit of firms in the market Strategic variable of the firms: production levels qi Fall 2009-2010 – p.5/133 Cournot model - Assumptions (2) Behavioral: Firms choose production levels to maximize profits. Each firm knows that its production decision depends on its expectation over the rivals’ decisions. Also, every rival’s decision depend of what each of them expects all the other competitors will decide. All firms take simultaneously their respective production decisions. Consumers choose a consumption bundle to maximize utility. Fall 2009-2010 – p.6/133 Cournot model - Assumptions (3) On demand Q = F (p) (A1) 1. f : R+ → R+ > 0 if Q< Q, 2. ∃Q s.t. f(Q) (= 0 if Q ≥ Q 3. ∃p< ∞ s.t. f(0) = p 4. f(Q) is continuous and C2 in [0, Q] 5. f ′(Q) < 0 for Q ∈ (0, Q) Implication: qi ∈ [0, Q] ∀i Fall 2009-2010 – p.7/133 Cournot model - Assumptions (4) On technology Ci(qi) (A2) For all i, 1. Ci : R+ → R+ 2. Ci is continuous and continously differentiable ∀qi > 0 3. Ci(qi) > 0 ∀qi > 0 4. Ci(0) ≥ 0 ′ 5. Ci (qi) > 0 ∀qi ≥ 0 Note: Symmetry across firms means Ci(·)= Cj(·), ∀i,j, i 6= j Fall 2009-2010 – p.8/133 Cournot model - Profits Definitions: q = (q1, q2, q3,...,qn) is a production plan. Πi(q)= qif(Q) − Ci(qi) is firm i’s profit function. Π(q) = (Π1(q), Π2(q), Π3(q),..., Πn(q)) is a distribution of profits in the industry. Assumptions (A3) 1. Πi : R+ → R+ 2 2. Πi is continuous and C ∀qi > 0 3. Πi(q) is strictly concave in qi, ∀q s.t. qi > 0,Q< Q. Fall 2009-2010 – p.9/133 Cournot model - More definitions Feasibility: qi is a feasible output for firm i if qi ∈ [0, Q]. def The set F ∈ Rn defined as F = [0, Q]× n. times . ×[0, Q], is the set of all feasible production plans in the industry. F is a compact set. Space of outcomes: The space of outcomes is the set of all possible distribution of def profits in the industry: {Π(q)|q ∈ F} = Π(F). Π(F) is also a compact set. Pareto optimal outcomes: PO = {Π(q)|q ∈F s.t. ∀q′ ∈F, Π(q) > Π(q′)}, where Π(q) > Π(q′) means Πi(q) ≥ Πi(q′) ∀i, and ∃j s.t. Πj(q) > Πj(q′). Fall 2009-2010 – p.10/133 Cournot model - Equilibrium Definitions: Cournot-Nash equilibrium A production plan qc is a C-N equilibrium if no firm can unilaterally improve upon its profit level by modifying its production decision. A production plan qc is a C-N equilibrium if no firm has any profitable unilateral deviation. c def c c c c c Let q i = (q1, q2,...,qi 1, qi+1,...,qn). We say that a production− plan qc is a− C-N equilibrium if c c Πi(q ) = maxqi Πi(qi, q i) ∀i. − A production plan qc is a C-N equilibrium if def c c c c c c 6∃q = (q1, q2,...,qi 1, qi, qi+1,...,qn) s.t. Πi(q ) ≤ Πi(q) ∀i. − A production plan qc is a C-N equilibrium if ce ce e qi = argmaxqi Πi(qi, q i) ∀i. − Fall 2009-2010 – p.11/133 Cournot equilibrium - Illustration Duopoly: Firms 1 and 2 p f(Q) Firm 1's residual demand f (Q ) − q2 ′ C1 0 c q1(q2) q2 Q Firm 1's marginal revenue Firm 1's expectation on Firm 2 Fall 2009-2010 – p.12/133 Cournot model - Game theory The Cournot model is a one-shot, simultaneous move, non-cooperative game [in pure strategies]. Extensive form F1 q1 0 Q F2 0 0 q2 Q q2 Q Π1(Q, 0) . Π1(Q, Q) Π1(Q, 0) . Π1(Q, Q) Π2(Q, 0) . Π2(Q, Q) Π2(Q, 0) . Π2(Q, Q) Fall 2009-2010 – p.13/133 Cournot model - Game theory (2) Normal form: (N, F, Π) N = 1, 2,...,n (set of firms) def F = [0, Q]× n. times . ×[0, Q] (strategy space) Π(q) = (Π1(q),..., Πn(q)) (payoff vector) Payoff matrix (duopoly). 2/1 0 ...q1 ... Q 0 Π1(0, 0), Π2(0, 0) Π1(q1, 0), Π2(q1, 0) Π1(Q, 0), Π2(Q, 0) . q2 Π1(0, q2), Π2(0, q2) Π1(q1, q2), Π2(q1, q2) Π1(Q, q2), Π2(Q, q2) . Q Π1(0, Q), Π2(0, Q) Π1(q1, Q), Π2(q2, Q) Π1(Q, Q), Π2(Q, Q) Fall 2009-2010 – p.14/133 Cournot equilibrium and Pareto optimality Proposition 1. Let qc ≫ 0 be a Cournot equilibrium production plan. Then Π(qc) is not Pareto optimum. c def c Proof. Show Q = i qi < Q. c P ∂Πi As q ≫ 0 it satisfies FOCs. Also, = qif ′(Q) < 0 ∀i,j i 6= j. ∂qj Hence a simultaneous reduction of the output levels of any two firms, qi and qj would improve their profits. Thus, ∃q such that c Πi(q) > Πi(q ) ∀i. c Although qi < qi , ∀i has a negative impact on firm i’s profits, it is second order effect and offset by previous (first order) effect. Fall 2009-2010 – p.15/133 Cournot equilibrium and Pareto optimality (2) Intuition Firm i when deciding qi considers adverse effect of price on its output but ignores the effect on aggregate production. Impact of variation of qi on price f(Q) is given by ∂f df ∂Q df ∂Q ∂q df ∂q = + j = 1+ j ∂qi dQ ∂qi dQ ∂qi ∂qi dQ ∂qi j=i j=i X6 X6 First term: impact on price of additional unit of qi Second term: externality on rivals: conjectural variation. Cournot assumes it away. Thus, in equilibrium firms produce beyond the optimal industry level. Fall 2009-2010 – p.16/133 Cournot equilibrium - Existence - Illustration p D B K′ H I K ′ E J F G J 0 Q q1 q2 q3 q4 R q5 q6 D′ qa ∈ [q4, q6], qb ∈ [q2, q3] and [q4, q6] ∩ [q2, q3]= ∅ Fall 2009-2010 – p.17/133 Cournot equilibrium - Existence - Illustration (2) p D B J ′ F J ′ J 0 ′ Q q1 q5 R D Fall 2009-2010 – p.18/133 Cournot equilibrium - Existence - Formal approach (1) Concavity of Πi: 2 ∂ Πi means 2 ≤ 0 ∂qi (implies ∃qi maximizing profits for any j=i qj 6 P e 2 ∂ Πi(q) ′′ ′′ 2 = 2f ′(Q)+ qif (Q) − Ci (qi) ∂qi ′′ ′′ 2 f < 0 and Ci > 0 ∂ Π (q) ′′ ′′ ′′ ′′ i if 2 < 0 f > 0,Ci > 0 and |2f ′| > qif − Ci ∂q ′′ ′′ ′′ ′′ i f < 0,Ci < 0 and |2f ′ + qif | > |Ci | Sufficient conditionsfor existence, not necessary. See Vives (1999), Mas Colell et al. (1995), Friedman (1977), Okuguchi (1976). Fall 2009-2010 – p.19/133 Cournot equilibrium - Existence - Formal approach (2) Theorem 1. Whenever assumptions S1 to S3 are fulfilled, there is an interior Cournot equilibrium, We will divide the proof in three parts. First, we will show that assumptions S1 to S3 ensure well-defined reaction functions (lemma 1); next, we will show that these reaction functions are con- tinuous (lemma 2); finally we will verify that we can apply Brower’s fixed point theorem. Fall 2009-2010 – p.20/133 Cournot equilibrium - Existence - Formal approach (3) Lemma 1. Whenever assumptions S1 to S3 are fulfilled, there will be a well-defined reaction function for every firm. Lemma 2. wi(q i) is a continuous function. − Theorem 2 (Brower). Let X be a convex and compact set in Rn. Let f : X → X be a continuous application associating a point f(x) in X to each point x in X. Then there exists a fixed point x = f(X). On fixed point theorems see Border (1992). b b Fall 2009-2010 – p.21/133 Cournot equilibrium - Existence - Formal approach (4) Proof of lemma 1 def (n-1) times Proof. Let F i = [0, Q] × [0, Q] × [0, Q]× . ×[0, Q]. − Consider an arbitrary production plan q i ∈F i. − − Given that πi(qi, q i) is continuous in qi, qi ∈ [0, Q] and strictly concave, − and given that [0, Q] is compact we can write the first order condition of the profit maximization problem ∂πi(qi, q i) ′ − = f(Q)+ qif ′(Q) − Ci (qi) = 0, ∂qi as a function qi = wi(q i) called firm i’s reaction function.
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