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. It tells us firm i’s profit maximizing− strategy conditional to its expectation on the behavior of the (n − 1) rival firms.
Fall 2009-2010 – p.22/133 Cournot equilibrium - Existence - Formal approach (5)
Proof of lemma 2 Proof. Let us now define a one-to-one continuous mapping, w(q), of the compact set F on itself,
w(q)= w1(q 1),w2(q 2),...,wn(q n) − − − Let {q i}τ∞=1 a sequence of strategy vectors in F i, such that − τ o − limτ q i = q i. →∞ − − o Since F i is compact, we know that q i ∈F i. − − − τ The sequence {q i}τ∞=1 allows us to obtain a sequence {wi(q i)}τ∞=1 τ − − where wi(q i) ∈ [0, Q]. Let, −
τ τ qi = wi(q i) − o o qi = wi(q i). −
Fall 2009-2010 – p.23/133 Cournot equilibrium - Existence - Formal approach (6)
Proof of lemma 2 (cont’d) τ o Proof. We say that wi is continuous if limτ qi = qi . By definition, →∞
τ τ τ πi wi(q i), q i ≥ π(qi, q i), qi ∈ [0, Q]. − − − Since the profit function is continuous,
τ τ τ o lim πi wi(q ), q = πi lim wi(q ), q τ i i τ i i →∞ − − →∞ − − τ o lim π(qi, q )= π(qi, q ), τ i i →∞ − − so that we can write,
τ o o πi lim wi(q i), q i ≥ π(qi, q i), qi ∈ [0, Q]. τ − − − →∞ Fall 2009-2010 – p.24/133 Cournot equilibrium - Existence - Formal approach (7)
Proof of lemma 2 (cont’d)
Proof. Given that wi(q i) is a single-valued function, − τ o lim wi(q )= wi(q ) τ i i →∞ − − or equivalently, lim qτ = qo , τ i i →∞ − so that wi(q i) is a continuous function. −
Fall 2009-2010 – p.25/133 Cournot equilibrium - Existence - Formal approach (8)
Brower’s fixed point theorem - illustration for X = [0, 1] 1
A C D
B
0 1
Proof. We can apply Brower’s fixed point theorem, given that F is compact and w(q) is continuous. Therefore, there is at least a point q∗ such that w(q∗)= q∗, where q∗ is the Cournot equilibrium.
Fall 2009-2010 – p.26/133 Cournot equilibrium - Uniqueness
Consider a Cournot duopoly with two equilibria (q1∗, q2∗) and (y1∗, y2∗). Therefore,
If firm 2 would choose q2∗ firm 1’s best reply would be q1∗;
If firm 2 would choose y2∗ firm 1’s best reply would be y1∗ But firm 1 does not know firm 2’s decision. It only makes conjectures. It may happen that firm 1 conjectures that firm 2 will choose q2∗ when firm 2’s choice is y2∗. A production plan (q1∗, y2∗) generally will not be an equilibrium vector of strategies. In other words, there may appear a coordination problem. How guarantee that firms will “point at the same equilibrium production plan"? Study the conditions under which there is a unique equilibrium.
Fall 2009-2010 – p.27/133 Cournot equilibrium - Uniqueness - Illustration
G P F
D ′ C G K E F ′ J
K′ E′ J ′
0 q1 q2 q3 q4 D′ Q
D′D: demand; F F ′, GG′,JJ′, KK′ isoprofit curves of firm i. Firm i conjectures firm j produces q1 → E equilibrium. Firm i conjectures firm j produces q3 → E′ equilibrium.
Multiplicity → lack of concavity of DD′.
Fall 2009-2010 – p.28/133 Cournot equilibrium - Uniqueness - Formal approach
We need to restrict assumption A3, introducing Assumption (A4) The profit function πi(q) is continuous, twice continuously differentiable, and ∀q, q ≫ 0, Q< Q satisfies,
2 2 ∂ πi(q) ∂ πi(q) 2 + < 0, or ∂q ∂qi∂qj i j=i X6 ′′ ′′ ′′ (1) 2f ′(Q)+ qif (Q)−Ci (qi) + (n − 1)|f ′(Q)+ qif (Q)| < 0 equivalent to assume Hessian negative semidefinite implies F compact more restrictive than A3
Fall 2009-2010 – p.29/133 Cournot equilibrium - Uniqueness - Formal approach (2)
′′ Consider the case f < 0, so that
′′ ′′ |f ′(Q)+ qif (Q)| = − f ′(Q)+ qif (Q) . Then, (1) can be rewritten as
′′ ′′ −(n − 3)f ′(Q) − (n − 2)qif (Q) For n = 2 always true. For n ≥ 3 left hand side positive. This means that increasing values of n require increasing values of ′′ Ci (qi) to verify (2). Summarizing, assumptions S1, S2, S4 guarantee that w(q) is a con- traction. Fall 2009-2010 – p.30/133 Cournot equilibrium - Uniqueness - Formal approach (3) Definition: Contraction ′ ′′ Consider two vectors q i and q i, i = 1, 2,...,n. It is said that w(q) is a contraction if − − ′ ′′ ′ ′′ wi(q i) − wi(q i) < kq i − q ik − − − − That is, when all the competitor firms vary their strategies in a certain amount, firm i’s best reply varies in a smaller amount. In IR2 ′ this means that wi < 1. We introduce now a theorem without proof: Theorem 3. Let f : Rl → Rl be a contraction. Then, f has a unique fixed point. We can use this theorem to obtain the result we are after: Theorem 4. Assume S1, S2, S4. Then w(q) is a contraction and q∗ is the unique Cournot equilibrium. Fall 2009-2010 – p.31/133 Cournot equilibrium - Existence & Uniqueness Van Long and Soubeyran (2000) Szidarovsky and Yakowicz (1977) Tirole (1988, pp.224-225) Fall 2009-2010 – p.32/133 Strategic complements and substitutes Properties of reaction functions. Consider Duopoly. w1(q2) solution of ∂π1(q1, q2) df dC1(q1) = f(Q)+ q1 − = 0. (3) ∂q1 dQ q1 The strategic nature of the relation between firms is given by the slope of the reaction function. The slope is obtained by differentiating (3): 2 2 ∂ π1(q1, q2) ∂ π1(q1, q2) 2 dq1 + dq2 = 0, or ∂q1 ∂q1∂q2 2 ∂ π1(q1,q2) dq1 ∂q1∂q2 = − 2 . (4) ∂ π1(q1,q2) dq2 foc 2 ∂q1 Fall 2009-2010 – p.33/133 Strategic complements and substitutes (2) 2 ∂ π1(q1, q2) − 2 > 0, from the second order condition. ∂q1 Then, sign of slope = sign of the numerator in (4). From (3) 2 2 ∂ π1(q1, q2) ∂ ∂π1(q1, q2) df d f = = + q1 2 . (5) ∂q1∂q2 ∂q2 ∂q1 dQ dQ h i Concave demand → sign (5)<0 → w1′ < 0. Convex demand → sign (5) ambiguous. Fall 2009-2010 – p.34/133 Strategic complements and substitutes (3) Definition [Bulow, Geanakoplos and Klemperer (1985)] the actions of the two firms are strategic complements if ∂2π (q , q ) 1 1 2 > 0. ∂q1∂q2 the actions of the two firms are strategic substitutes if ∂2π (q , q ) 1 1 2 < 0 ∂q1∂q2 Nature of the strategic relations among competitors to be examined case by case. Prices are often strategic complements. Quantities are often strategic substitutes. Martin (2002, pp. 21-27) counterexamples. Throughout investigation of games with strategic complementarities (supermodular games): Amir (1996, 2005) and Vives (1999, 2005a, 2005b). Fall 2009-2010 – p.35/133 Geometry of the Cournot model Assume duopoly with linear demand and costs: p = a − b(q1 + q2), Ci(qi)= c0 + cqi, i = 1, 2. Isoprofit curves Fix a level of profits Πi(q)= Π. Then, Π= qi(a − b(qi + qj)) − c0 − ciqi, that is 1 Π+ c0 qj = − + c − a − qi b qi Fall 2009-2010 – p.36/133 Geometry of the Cournot model (2) Properties of isoprofit functions ∂qj 1 Π+c0 Slope: = 2 − 1. ∂qi b qi Π Critical point: 1 Π+ c 2 q = 0 . (6) i b Critical point is increasing in Π: 1 ∂q 1 Π+ c 2 i = 0 − > 0. (7) ∂Π 2b b Isoprofit curve strictly concave in the space (qi, qj ): 2 ∂ qj 2(Π+ c0) 2 = − 3 < 0. ∂qi bqi Fall 2009-2010 – p.37/133 Geometry of the Cournot model (3) Extreme isoprofit curves m Maximum Π at monopoly (qi , 0) a − c (a − c)2 qm = , and Πm = − c . i 2b i 4b 0 Hence, isoprofit curve tangent to qi axis from below. Minimum Π when (0, qj ) maximizes firm i’s profits ∂Πi Compute qj such that = 0 when qi = 0 e∂qi a − c This is q = . j e b e Fall 2009-2010 – p.38/133 Geometry of the Cournot model (4) Lemma 3. Under linear demand and costs, the function linking all maxima of the isoprofit family of curves is linear Proof. Consider an isoprofit curve Π. Its maximum wrt qi is given by (6). Substituting it in the isoprofit curve we obtain the associated value qj: a − c Π+ c0 1 q = − 2( ) 2 . (8) j b b Compute now, −1 ∂q −1 Π+ c 2 j = 0 . (9) ∂Π b b Comparing (7) and (9) we see, ∂q 1 ∂q i = − j . ∂Π 2 ∂Π Fall 2009-2010 – p.39/133 Geometry of the Cournot model (5) When there is a variation in the level of profits, the effect on qi is half the effect on qj regardless of the actual value of profits. This implies a linear relation between the set of maximum points of the family of isoprofit curves. This linear function has 1 slope − 2 . Finally, to identify the expression of this linear function we substitute (6) in (8) to obtain, a − c 1 q = − q . (10) i 2b 2 j Fall 2009-2010 – p.40/133 Geometry of the Cournot model (6) qj q j ! qj Π! < Π < Π m Π! < Π < Π m i i i j j j ! m Π i q j Π i Π m j m qi qi Π q!i q i m j Π! Π i j a − c 1 a − c 1 q = − q qj = − qi i 2b 2 j 2b 2 Fall 2009-2010 – p.41/133 Geometry of the Cournot model (7) Combining both isoprofit maps, Cournot eq. characterized by intersection of the two functions linking max profits. Tangency point between any two isoprofit curves: distribution of profits such that any alternative share of profits cannot make both firms better off simultaneously. Loci of tangency points: set of Pareto optimal production plans. Set of tangency points: m m extremes: (qi , 0), (0, qj ). solution of maxqi,qj (Πi + Πj)= Q(a + bQ) − 2c0 − cQ : a − c linear function q = − q , with slope −1 i 2b j Fall 2009-2010 – p.42/133 Geometry of the Cournot model (8) qj Πc Note : j C !∈ P O m qj C c Πi m P O qi qi Fall 2009-2010 – p.43/133 Geometry of the Cournot model (9) Reaction functions Locus of profit maximizing production plans conditional to the expectation on the behavior of the rival firms. Solution of FOC of its profit maximization program: Πi(q) = (a − bQ)qi − c0 − cqi, ∂Πi(q) = a − c − bqj − 2bqi = 0, ∂qi a − c 1 q = − q . i 2b 2 j Same as (10). This means that firm i’s reaction function is precisely the function linking the maximum points of its family of isoprofit curves. Fall 2009-2010 – p.44/133 Comparative statics on n What if n variable? Two questions: Quasi-competitiveness of Cournot equilibrium: ∂Qc > 0? ∂n Convergence to competitive equilibrium: lim Qc = Qcompet? n →∞ Interest of questions: Approximation effect of oligopolistic markets on welfare Relevance of oligopoly theory Fall 2009-2010 – p.45/133 Comparative statics on n (2) Illustration 1: Martin (2002, pp: 18-19) Symmetric duopoly, C′ = 0 c c c Cournot eq: Q → qi = Q /2. Then, FOC: d d Q f(Q )+ f ′(Q) ≡ 0, or 2 d d 2f(Q )= −Q f ′(Q) In general, with n symmetric firms we obtain nf(Q)= −Qf ′(Q). Intersection of two curves. Generally, price falls as n increases. lim f(Qc) = 0 n →∞ Fall 2009-2010 – p.46/133 Comparative statics on n (3) 3f(Q) 2f(Q) f(Q) ′ −Qf (Q) f(Q) Fall 2009-2010 – p.47/133 Comparative statics on n (4) Illustration 2: Shubik (1959) - efficient point Definition The efficient point is a production plan resulting from equating price to marginal cost for all firms simultaneously, i.e. qe ∈ Rn is an ′ efficient point if it solves the equation f(Q)= Ci (qi), ∀i. Intuition: To study “how far" is the Cournot equilibrium from the competitive equilibrium, we will assume that all firms behave competitively so that they adjust their production levels to the point where price equals marginal cost. Then we will compare the resulting outcome with the Cournot outcome. Fall 2009-2010 – p.48/133 Comparative statics on n (5) Illustration 2 (cont’d) n-firm industry Demand and technology: Ci(qi)= cqi, c> 0, i = 1, 2,...,n. n f(Q)= a − bQ, a,b > 0, Q = qi. Xi=1 Cournot equilibrium: 2 c a c c n(a c) c (a c) qi = b(n−+1) , Q = b(n+1)− , Πi(q )= b(n−+1)2 . Efficient point: e a c e a c a − bQ = c, Q = −b , qi = nb− . Fall 2009-2010 – p.49/133 Comparative statics on n (6) Illustration 2 (cont’d) Comparing both equilibria we should note that, c e c e c e c a c e qi ≤ qi , Q ≤ Q , P ≥ P . and, limn Q = − = Q . →∞ b can assume competitive behavior on oligopolistic firms without losing much? The answer is NO. Assume Ci(qi)= k + cqi, k,c> 0. Then, 2 c (a c) a c Profits: Π (q )= − 2 − k ≥ 0 ⇒ n ≤ − − 1 i b(n+1) √bk finite number of firms nonsense for limn →∞ ∂n ∂Π note (a) < 0, and (b) i < 0. ∂k ∂n Fall 2009-2010 – p.50/133 Comparative statics on n (7) Quasi-competitiveness of Cournot equilibrium (Telser, 1988) Let Ci(qi)= cqi, i = 1, 2,...,n and p = f(Q), f ′ < 0 0 0 Eqbm: q1 = · · · = qn = q∗, Q = nq∗, and Πi∗ = (f(Q) − c)q∗. How does Πi∗ vary with n? ∂Πi∗ ∂f ∂Q ∂q∗ = q∗ − . ∂n ∂Q ∂n ∂n ∂Π ∂Q ∂q f < 0 ⇒ sgn i∗ = −sgn − ∗ . ′ ∂n ∂n ∂n ′′ ∂Q ∂q∗ q∗(2f ′ + q∗f ) − = ′′ . ∂n ∂n f ′(n +1)+ Qf ′′ ∂Q ∂q ∂Π If f < 0 then − ∗ > 0, and i∗ < 0. ∂n ∂n ∂n Fall 2009-2010 – p.51/133 Comparative statics on n (7) Convergence of Cournot equilibrium (Telser, 1988) Assume demand cuts axes at p and Q. Ci(qi) increasing, differentiable in q : i ∈ [0, Q], ∀i. Ci(0) = 0 and Ci(qi)= Cj(qj), ∀i,j; i 6= j Proposition 2. Consider a homogeneous product industry satisfying assumptions above. Then the Cournot equilibrium converges towards the long run competitive equilibrium if ′ Ci (0) = min ACi(qi), ∀i. ′ Proposition 3. If Ci(qi) is U-shaped, and ∃i s.t. Ci (0) > ACi(qi), then the Cournot equilibrium does not converge towards the competitive equilibrium. Fall 2009-2010 – p.52/133 Comparative statics on n (8) Illustration Let Ci(qi)= cqi, c> 0, i = 1, 2,...,n. a − b q if q ≤ a = Q, a,b> 0, f(Q)= i i i i b 0 if q > Q ( P Pi i Eqbm: P 2 a c n(a c) a+nc (a c) qi∗ = (n+1)− b , Q∗ = (n+1)− b , f(Q∗)= n+1 , Πi∗(q∗)= (n+1)− 2b , ∀i Differentiate these equilibrium values with respect to n: ∂q ∂Q ∂P ∂Π i∗ < 0, ∗ > 0, ∗ < 0, i∗ < 0. ∂n ∂n ∂n ∂n Thus, the Cournot equilibrium is quasi-competitive. Fall 2009-2010 – p.53/133 Comparative statics on n (9) Illustration (cont’d) Moreover, a c limn qi∗ = 0, limn Q∗ = − (Q), limn P ∗ = →∞ →∞ b →∞ c, limn Πi∗ = 0. →∞ ′ Note also Ci (qi)= c = ACi(qi), so that , ′ ′ c = lim P ∗ = lim Ci (qi∗) = lim ACi (qi∗) = min ACi(qi) ∀i. n qi 0 qi 0 qi →∞ → → That is the Cournot equilibrium converges towards the competitive equilibrium. Finally, note that assumptions above hold: a ′ p = a< ∞, Q = < ∞, C (0) = 0, C (q ) > 0. b i i i Fall 2009-2010 – p.54/133 Cournot vs. Monopoly vs. Competitive solutions ′ ′ Proposition 4. Consider a symmetric duopoly where C1 = C2 = c. Then, the equilibrium Nash-Cournot price, pN is greater than the competitive price, c, and smaller than the monopoly price pm. Proof. See lecture notes pp. 61-62 In the space of production plans, we can represent the reaction functions and the combinations of output volumes that together give rise to the monopoly (QM ) and competitive (QC ) output levels. N N N Proposition says that the aggregate Cournot output (q1 + q2 = Q ) is an intermediate value between the competitive and monopoly equilibrium outputs. Fall 2009-2010 – p.55/133 Cournot vs. Monopoly vs. Competitive solutions (2) q1 c Q c q1 + q2 = Q QN M M Q q1 + q2 = Q N C q2 0 N M N c q1 Q Q Q q2 Fall 2009-2010 – p.56/133 Stability analysis Stability in static model? Confusing. Dynamic assumption ad hoc: Hahn (1962), Seade (1977) Assumption: Fictitious time. In every period t, t = 1, 2, 3,... each firm recalls the decisions taken by itself and its rival in the previous period t − 1. In period t, firm j expects that its rival, firm i will maintain the e same output as in the previous period, qit = qit 1,i = 1, 2 − i Now, reaction functions : qit = wi(qt− 1), i = 1, 2,...,n where i − qt− 1 denotes a n − 1 dimensional production plans of all firms except− firm i in t − 1. Fall 2009-2010 – p.57/133 Stability analysis (2) Definition Let qc ∈ Rn be a static Cournot equilibrium production plan. Let q0 = (q10, q20,...,qn0) be an arbitrary production plan. We say that qc is a stable equilibrium production plan if the sequence of production plans {qt}t∞=1, qt = (q1t, q2t,...,qnt) c c converges towards q . In other words, if limt qt = q . →∞ A sufficient condition to guarantee the stability of a Cournot equilibrium is that all reaction functions wi, i = 1, 2,...,n be contractions. Example: Lecture notes pp.64-65. Fall 2009-2010 – p.58/133 Stability analysis (3) Definition Let f be a continuous function defined on [a,b]. Consider two arbitrary points x, y ∈ [a,b]. We say that f is a contraction if f(x) − f(y) ≤ c x − y ∀x, y ∈ [a,b], c< 1 In words, f is a contraction if given two arbitrary points in the domain of the function, the distance between their images is smaller than the distance between the points. If f is linear this simply means that the slope has to be smaller than one. Fall 2009-2010 – p.59/133 Stability analysis (4) f(y) f(x) ◦ f(x) 45 45 ◦ f(y) a x y b a x y b Examples where f is a contraction q1 q1 α α1 1 45 ◦ 45 ◦ α α2 0 2 0 q2 q2 Stable equilibrium Unstable equilibrium Fall 2009-2010 – p.60/133 Stability analysis (5) Two objections linked with the construction of the system of reaction functions. It does not make any sense to assume that firms are so myopic to ignore the flow of future profits when deciding today’s production level. It does not make sense to assume that a firm expects that its rivals will not vary their decisions from yesterday, in particular when our firm is changing its decision in every period (see example). Note that this objections refer to the formation of expectations, i.e. to the construction of the reaction functions, but not to the concept of Cournot equilibrium. A more general analysis of the stability of the Cournot equilibrium can be found in Okuguchi (1976), pp. 9-17. Fall 2009-2010 – p.61/133 Price competition - Bertrand (1883) First critique to Cournot model - 45 years later!! Critique: the obvious outcome of Cournot’s analysis is that oligopolists will end up colluding in prices, a behavior ruled out by Cournot. Variation a Cournot’s model with prices as strategic variable. In a scenario with perfect and complete information, homogeneous product, without transport costs, and constant marginal costs, every consumer will decide to buy at the outlet with the lowest price. Fall 2009-2010 – p.62/133 Price competition (2) Bertrand’s point goes beyond. If firms choose quantities, it not specified in Cournot’s model what mechanism determines prices. In a perfectly competitive market, it is irrelevant what variables is decided upon because Smith’s “invisible hand" makes the markets clear. In oligopoly, there is no such device. A different mechanism is needed to determine the price that, given the production of the firms allow the markets to clear. Accordingly, it may be more reasonable to assume that firm decide prices and production is either sold in the market or stocked. Fall 2009-2010 – p.63/133 Price competition (2) Bertrand’s model solves one institutional difficulty, but rises another difficulty. In the real world it is difficult to find homogeneous product markets. Often we observe markets where firms sell their products at different prices and all of them obtain positive market shares. In these markets slight variations of prices generate slight modifications of market shares rather than the bankruptcy of the firm quoting the highest price. Oligopoly models of homogeneous product seem to contain a dilemma: Cournot’s model behaves in a reasonable way but uses the wrong strategic variable; in Bertrand’s model the “good" strategic variable is chosen but, as we will see below, behaves in a degenerated way. This is the so-called Bertrand paradox. Fall 2009-2010 – p.64/133 Bertrand model - Assumptions n firm industry of a homogeneous product same constant marginal cost technology, Ci(qi)= cqi ∀i. strategic variable: prices Consumers behavior described by a (direct) demand function, Q = f(P ) satisfying all the necessary properties. Fall 2009-2010 – p.65/133 Bertrand model - Assumptions (2) Sharing rule: the firm deciding the lowest price, gets all the demand (Pi < P i =⇒ Dj(Pi, P i) = 0, j 6= i) − − if all firms decide the same price, they share demand evenly (Pi = Pj, ∀j 6= i =⇒ Di(Pi, P i)= Dj(Pi, P i), j 6= i); This is a particular sharing rule− based on the− symmetry of the model. A possible alternative sharing rule could be to decide randomly which firm gets all the market (Hoernig, 2007, Vives, 1998, ch. 5). consumers have reservation prices sufficiently high so that they are all served regardless of the prices decided by firms. To ease computations, wlog, we normalize the size of the n market to the unit, that is i=1 Di(Pi, P i) = 1. − P Fall 2009-2010 – p.66/133 Bertrand model - Demand & profits Contingent demand functions 0 if Pi > Pj, ∀j 6= i, 1 Di(Pi, P i)= if Pi = Pj, ∀j 6= i, − n 1 if Pi < Pj, ∀j 6= i. Contingent profit functions 0 if Pi > Pj, ∀j 6= i, 1 Πi(Pi, P i)= (Pi − c) if Pi = Pj, ∀j 6= i, − n (Pi − c) if Pi < Pj, ∀j 6= i. Fall 2009-2010 – p.67/133 Bertrand model - Illustration P i Πi P − c P j j 1 c (P j − c) 2 ◦ 0 45 1 2 1 0 / Di(Pi, P j) c P j Pi Figure illustrates firm i’s contingent demand and profits for a duopolistic market, where P i reduces to P j the expectation on the − behavior of the rival firm. Fall 2009-2010 – p.68/133 Bertrand model - Equilibrium Definition 1. A n-dimensional vector of prices (Pi, P i) is a Bertrand (Nash) equilibrium if and only if − ∀i, ∀Pi Πi(Pi∗, P ∗i) ≥ Πi(Pi, P ∗i) − − Proposition 5. Let us consider a n firm industry where firms produce a homogeneous product using the same constant marginal cost technology, Ci(qi)= cqi ∀i. Let us normalize the size of the market to the unit and assume consumers have sufficiently high reservation prices. Then, there is a unique Bertrand equilibrium given by Pi∗ = c ∀i. Fall 2009-2010 – p.69/133 Bertrand model - Equilibrium (2) Proof. Duopoly. Consider an alternative price vector (Pi, Pj). if P < P ⇒ D (P , P ) = 0 and, Π (P , P ) = 0. Firm j can i j j i j j i j e e improve profits with Pj < Pi. Therefore, (Pi, Pj ) not equilibrium. e e e e e e if P > P ⇒ D (P , P ) = 0 and Π (P , P ) = 0. Firm i can i j i i j e i i ej e improve profits with Pi < Pj. Therefore, (Pi, Pj ) not equilibrium. e e e e e e Then, in equilibrium, P = P . Consider P = P >c. i e j ie e j 1 1 Now, Di(P )= Dj(P )= 2 and Πi(P ) = Πj(P )= 2 (Pi − c). e e 1 e e But Πi(Pi − ε, Pj)= Pi − ε − c> (Pi − c). Same for firm j. e e e2 e e Price war so that Pi = Pj >c not equilibrium. e e e e Let P = P = c. Now, D (P )= D (P )= 1 and i j e e i j 2 Πi(P ) = Πj(P ) = 0. No firm has a profitable unilateral deviation. e e e e A price vector P = P = c is the only Bertrand equilibrium. e i e j e e Fall 2009-2010 – p.70/133 Bertrand model - Equilibrium (3) Reaction functions. Let P m denote the monopoly price. m If Pj > P , firm i’s best reply is to choose the monopoly price to obtain monopoly profits. If Pj If Pi > Pj, then Πi = 0; 1 If Pi = Pj, then Πi = (Pi − c) 2 ; If Pi < Pj, then Πi = (Pi − c). In this case the profit function is increasing in Pi, so that firm i’s best reply is the highest possible price, that is Pi = Pj − ε, for ε arbitrarily small. Fall 2009-2010 – p.71/133 Bertrand model - Equilibrium (4) Summarizing, firm i’s reaction function is, m m P if Pj > P m Pi∗(Pj)= Pj − ε if c < Pj ≤ P c if Pj ≤ c By symmetry, firm j has a similar reaction function exchanging the subindices adequately. Fall 2009-2010 – p.72/133 Bertrand model - Equilibrium (5) Pi P ∗ P j ( i) P m P ∗ P i ( j) c 45 ◦ c m P Pj Reaction fncts intersect only at Pi = Pj = c: the Bertrand eqbm. Fall 2009-2010 – p.73/133 Cournot vs. Bertrand Why different behavior? ⇒ Different residual demand. Illustration: Duopoly with q1 + q2 same aggregate output in C and B. P B −1 ∗ D (q2 ) A C P ∗ B B CM C IM 0 ∗ ∗ ∗ 1 q1 q1 + q2 q Fall 2009-2010 – p.74/133 Bertrand and “proper" behavior Attempts to obtain “normal" behavior in Bertrand models: 6 variations Capacity constraints Contestability Price rigidities Commitment Conjectural variations Dynamic models Fall 2009-2010 – p.75/133 Variation 1. Capacity constraints Assumption on technology: decreasing returns to scale/ strictly convex costs. Models: exogenous: Edgeworth (1897), endogenous: Kreps-Scheinkman (1983) Preliminaries: Rationing rules Efficient rationing rule Proportional rationing rule Consider a duopoly with P1 < P2 and q¯1 ≡ S(P1) < D(P1). Fall 2009-2010 – p.76/133 Variation 1. Capacity constraints. Rationing (1) Efficient rationing rule “first come, first served" rule. Firm 1 serves the most eager consumers; firm 2 serves the rest. D1(P1)= q1 D(P2) − q1 if D(P2) > q1 D2(P2)= (0 otherwise Firm 2’s residual demand: shift market demand inwards by q1. This rule is efficient because it maximizes consumer surplus. Fall 2009-2010 – p.77/133 Variation 1. Capacity constraints. Rationing (2) Efficient rationing rule P q = D(P ) q = D(P ) − q1 P2 P1 q1 q2 q q1 + q2 Fall 2009-2010 – p.78/133 Variation 1. Capacity constraints. Rationing (3) Proportional rationing rule Randomized rationing rule. Any consumer same prob. of rationed. Probability of not being able to buy from firm 1 is D(P ) − q 1 1 . D(P1) Firm 2’s residual demand rotates inwards: slope of the residual demand is modified by the probability of buying at firm 2. D(P1) − q1 D2(P2)= D(P2) . D(P1) This rule is not efficient. Consumers with valuations below P2 may buy the commodity because they find it at a bargain price P1. Fall 2009-2010 – p.79/133 Variation 1. Capacity constraints. Rationing (4) Proportional rationing rule P D(P ) D(P1) − q1 D(P2) ! D(P1) " P2 P1 q1 q2 q q1 + q2 Fall 2009-2010 – p.80/133 Variation 1. Capacity constraints. Rationing (5) Efficient rationing rule Proportional rationing rule vs. At any price the PRR yields higher residual demand to firm 2. More consumers are served under PRR although consumer surplus is not maximized. P D(P ) − q1 D(P1) − q1 D(P2) ! D(P1) " P2 P1 D(P ) q1 ef pr q2 q2 q Fall 2009-2010 – p.81/133 Variation 1. Capacity constraints. Edgeworth (1) Assumptions Demand: P = 1 − q n firms CRS up to Ki (Ki = K) Note: K exogenous 1 nK < 1 Thus, K < n 1 K > 2n Starting point: Full collusion m 1 m 1 m 1 q = 2 ; P = 2 ; Π = 4 1 1 qi = 2n ; Πi = 4n Collusive agreement stable? NO Fall 2009-2010 – p.82/133 Variation 1. Capacity constraints. Edgeworth (2) Price war! m Firm i undercutting Pi = P − ε, sells K and obtains K 1 1 1 Πi = ( 2 − ε)K > 2 2n for ε sufficiently small. Firm j may undercut firm i’s price to obtain profits K Πj = (Pi − ε)K. And so on Two points to note: How far will this undercutting arrive? Assume duopoly If undercutting, rival is monopolist over residual demand Fall 2009-2010 – p.83/133 Variation 1. Capacity constraints. Edgeworth (3) 1 m P = 1 /2 m P − ε 0 1/4 K 1/2 1 Fall 2009-2010 – p.84/133 Variation 1. Capacity constraints. Edgeworth (4) Other features Residual demand left to undercut firm: RD = 1 − K 2 mK 1 K Monopoly profits over RD : Πj = −2 1 K 1 K 1 Output level −2 must be feasible, i.e. −2 < K or K > 3 . Range of feasible values for the K to be meaningful: 1 1 K ∈ ( 3 , 2 ). Fall 2009-2010 – p.85/133 Variation 1. Capacity constraints. Edgeworth (5) P 1 1 − K 1 − K 2 0 q 0 K 1 − K 1 − K 2 Fall 2009-2010 – p.86/133 Variation 1. Capacity constraints. Edgeworth (6) Other features (cont’d) Both firms full capacity, qi = K, q = 2K and minimum feasible market price is P min = 1 − 2K Undercut firm as two options: undercut its rival, or max profits over residual demand Define P as the price yielding the same profits in both 2 situations: 1 K b P K = −2 b 1 K P ∈ 1 − 2K, −2 b Fall 2009-2010 – p.87/133 Variation 1. Capacity constraints. Edgeworth (7) Summarizing P 1 1 2 1 − K 2 1 − 2K 0 1 K 1 1 − K 2K 1 q 3 2 2 Fall 2009-2010 – p.88/133 Variation 1. Capacity constraints. Edgeworth (8) Behavior Consider a fictitious time span where rivals decide in alternate time periods Assume it is firm i’s turn Compares profits if undercutting or monopoly over RD If undercutting more profitable price war goes on At some point in the mutual undercutting process where the corresponding firm will be indifferent, i.e. firm will hit price P Next firm, undercut on P yield less profit than monopoly over b RD. Therefore, will give up price war and will jump up to 1 K monopoly price −2 . b Fall 2009-2010 – p.89/133 Variation 1. Capacity constraints. Edgeworth (9) Conclusion Price cycle non-stop ⇒ No equilibrium. P 1/2 1 − K 2 P! 1 − 2K 0 Fall 2009-2010 – p.90/133 Variation 1. Capacity constraints. Kreps-Scheinkman (1) Assumptions two-stage game: production-then-prices duopoly capacity levels q¯i technology: constant mg cost (=0) up to q¯i, then ∞. efficient rationing rule concave market demand Conclusion Pure-strategy equilibrium results in the production and the price that would have resulted in a one-shot Cournot game. Fall 2009-2010 – p.91/133 Variation 1. Capacity constraints. Kreps-Scheinkman (2) Tirole’s illustration q1 + q2 = 1 − P Price game assume q¯i was bought in the previous stage at a unit cost 3 C0 ∈ [ 4 , 1] m 1 monopoly profit Π = P (1 − P ); maxP Π= 4 1 firm i’s total profit is at most 4 − C0q¯i 1 firm i’s total profit is negative for q¯i > 3 . 1 Assume q¯i ≤ 3 (i.e. q¯i not too large) 1 Note that qi = 3 is the Cournot symmetric equilibrium 1 Assume qi ∈ [0, 3 ], i = 1, 2 Fall 2009-2010 – p.92/133 Variation 1. Capacity constraints. Kreps-Scheinkman (3) Tirole’s illustration Result Lemma 4. In a pure-strategy equilibrium, P1∗ = P2∗ = 1 − (q1 + q2). That is, firms sell up to capacity. “Proof" At P ∗ consumers are not rationed Hence, no incentives to lower price because firms are already selling their full capacity Incentives to increase price? Πi(Pi, P ∗) = (1 − qi − q¯j)qi, where qi ≤ q¯i because Pi ≥ P ∗ Πi(Pi, P ∗) is profit function of a firm deciding qi given expectation q¯j Hence, Πi(Pi, P ∗) is Cournot profit function. Fall 2009-2010 – p.93/133 Variation 1. Capacity constraints. Kreps-Scheinkman (4) Tirole’s illustration Result (cont’d) Incentives to increase price? (cont’d) Πi(Pi, P ∗) is concave in qi ∂Πi | = 1 − 2¯q − q¯ > 0 because q¯ < 1 ∂qi q¯i i j z 3 Hence, lowering output below q¯i (i.e. raising the price above P ∗) is not optimal. Capacity game Profits are Πi(¯qi, q¯j ) = (1 − q¯i − q¯j − C0)¯qi But these are the profits that would have obtained should firms had decided on output levels (¯qi, q¯j ) and market clearing would have set the price. Conclusion Cournot equilibrium is the eq. of first stage of the game. Fall 2009-2010 – p.94/133 Variation 1. Price-Quantity competition Simultaneous decisions mixed strategies Dasgupta-Maskin (1986) Maskin (1986) Supply-function equilibria Grossman (1981) Hart (1982) Two-stage capacity-price games Kreps-Scheinkman (1983) Vives (1993) Boccard-Wauthy (2000) Two-stage quantity-price games durable goods: Friedman (1988) Maskin (1986) perishable goods: Judd (1990) Simultaneous vs. sequential decisions: Chowdhury (2005) Dynamic models durable goods: Judd(1990) alternate decisions: Maskin-Tirole (1988) Fall 2009-2010 – p.95/133 Variation 2. Contestable markets Bailey, Baumol, Panzar, Willig (1980s). Martin (2002, supplement) Extend the theory of perfectly competitive markets to situations where scale economies are relevant. A market is contestable when entry is free and exit is costless Potential entrants evaluate the profitability of entry wrt the prices of the incumbents before entry. In other words, potential entrants think that they can undercut incumbents and “steal" all the demand before the incumbents will react. Contestable market if vulnerable to “hit-and-run" entry. In a contestable market the equilibrium production is always efficient regardless of the number of firms, since price always equals marginal cost. No Nash strategies. Fall 2009-2010 – p.96/133 Variation 3. Sticky prices Casual empirical observations of price rigidities downwards. First model: Sweezy (1939) Sweezy’s idea: oligopolistic firm when lowering its price should expect its rivals’ to react in a similar fashion. But when the firm increases its price, its rivals’ should be expected not to react. Sweezy’s construction assumes a more elastic demand for increases than for decreases in prices. Modern treatments of these arguments in Bhaskar (1988), Maskin and Tirole (1988b), or Sen (2004). Fall 2009-2010 – p.97/133 Variation 3. Sticky prices. Sweezy’s model (1) Assumptions Duopoly Constant marginal cost k Market demand: p = A − (qi + qj) Assume firms are producing qi and qj respectively. Conjectures b b Firm i conjectures that firm j will continue producing qj as long as it produces qi ≤ qj (i.e. price increases). Firm i also conjectures that if it changes its productionb to qi > qj (i.e. price decreases),b then firm j will increase its production until level with that of firm i. b Fall 2009-2010 – p.98/133 Variation 3. Sticky prices. Sweezy’s model (2) Given these conjectures, the only consistent production plans are vectors of the type qi = qj. If qi < qj then, firm j’s conjectures say that firm i will increase its production till qj. b b Mutatisb b mutandis in the symmetric case qi > qj. b Restrict analysis to situations where both firms decide the same production levels qi = qj = q. b b Firm i faces a demand function showing a kink at qj = q: b b b A − qi − qj if qi < qj, p = b (A − 2qi if qi > qj. Marginal revenue function is discontinuous at that point qj = q: A − qj − 2qi if qi < qj, IMi = b (A − 4qi if qi > qj. Fall 2009-2010 – p.99/133 Variation 3. Sticky prices. Sweezy’s model (3) P P A − qj A − qj A − 2qj A − 2qj k A − 3qj A − 3qj A − 4qj A − 4qj k 0 qj A/ 2 qi 0 qj A/ 2 qi Fall 2009-2010 – p.100/133 Variation 3. Sticky prices. Sweezy’s model (4) Equilibrium Assume now that firm j produces qj = q. Firm i’s problem is maxqi (A − qi − yj(qi, q) − k)qi where yj(qi, q)= max{qi, q}. b yj represents firm j’s reply: b b b to produce q if qi ≤ q and to produce qi if qi > q Reaction function.b b b If qj is large enough (with respect to k), the equality between marginal revenue and marginal cost appears in the lower segment of the marginal revenue curve; otherwise marginal revenue and marginal cost intersect in the upper part of the marginal revenue curve Fall 2009-2010 – p.101/133 Variation 3. Sticky prices. Sweezy’s model (5) Reaction functions 0 if qj ≥ A − k, A k qi A k −2− if −3 ≤ qj ≤ A − k, q∗(qj )= i q if A k ≤ q ≤ A k , j −4 j −3 A k A k −4 if qj ≤ −4 . A k A k Both curves intersect in the interval qi = qj = q ∈ [ −4 , −3 ]. Therefore, there is a continuum of equilibria. Note though that in all those equilibria the aggregateb production lies A k 2(A k) in the interval 2q ∈ [ −2 , 3− ], that is from the monopoly output to the Cournot output. b Fall 2009-2010 – p.102/133 Variation 3. Sticky prices. Sweezy’s model (6) qj A − k ∗ qi (qj) A − k 2 A − k 3 A − k ∗ qj (qi) 4 A − k A − k A − k A − k qi 4 3 2 Fall 2009-2010 – p.103/133 Variation 4. Commitment (1) Commitment Situation of strategic interaction were one agent may restrict in a credible way its choice set to gain an advantage over a competitor. Stackelberg (1934) first to propose a model to capture commitment in oligopoly pricing. Stackelberg’s model A firm acts as leader and several other firms (followers) conditional on the behavior of the leader, choose their actions. Leader aware of the behavior of the followers. Leader’s profit maximizing decision is conditional on the reaction of the followers, and the followers’ profit maximizing decisions are conditional on the choice of the leader. Fall 2009-2010 – p.104/133 Variation 4. Commitment (2) Assumptions duopoly P = a − b(q1 + q2) Ci(qi)= c0 + cqi Firm 1: leader; Firm 2: follower 2-stage model. Firm 1 chooses q1 first. Next, firm observes q1 and chooses q2 Commitment: Firm 1 cannot modify q1. Equilibrium: Subgame perfect eq: profit maximizing production plan (q1∗, q2(q1∗)). Fall 2009-2010 – p.105/133 Variation 4. Commitment (3) Follower’s optimal choice maxq2 Π2(q1, q2)= a − b(q1 + q2) q2 − c0 − cq2, yielding, a c 1 q2 = 2−b − 2 q1. Leader’s optimal choice Choose a profit maximizing output level anticipating the impact of this decision on the follower: a − c 1 max Π1(q1, q2) s.t. q2 = − q1. q1 2b 2 a c Accordingly, q1∗ = 2−b . a c Thus, follower’s optimal decision is q2∗ = 4−b Fall 2009-2010 – p.106/133 Variation 4. Commitment (4) Intuition Leader chooses a point on a isoprofit curve on firm 2’s reaction function. Follower plugs in the leader’s decision q1∗ in reaction function, to obtain q2∗. Completing characterization a + 3c P ∗ = , 4 (a − c)2 Π∗ = − c , 1 8b 0 (a − c)2 Π∗ = − c . 2 16b 0 Fall 2009-2010 – p.107/133 Variation 4. Commitment (5) q2 c Π2 ∗ Π2 C c q2 ∗ Πc S q2 1 ∗ Π1 0 c ∗ q1 q1 q1 Fall 2009-2010 – p.108/133 Variation 4. Stackelberg vs. Cournot Cournot eqbrm a − c a + 2c qc = qc = , P c = , 1 2 3b 3 (a − c)2 Πc = Πc = − c . 1 2 9b 0 Stackelberg vs.Cournot c c q1∗ > q1; q2∗ < q2; c c Π1∗ > Π1; Π2∗ < Π2; c c Q∗ > Q ; P ∗ < P . Leader has a “first-mover advantage" over the follower. Fall 2009-2010 – p.109/133 Variation 4. SGPE vs. Nash eq. SGP E means that empty (non-credible) threats by the follower are ruled-out. SGP E requires the follower’s strategy to be optimal in front of any decision of the leader q1, and not only against the equilibrium output q1∗. In contrast, a Nash equilibrium only requires optimality along the equilibrium path. In our two-stage game, it only imposes production levels for the leader that do not generate loses. For C0 = 0, any output in [0, (a − c)/b] is sustainable as a Nash equilibrium of the two-stage game. Fall 2009-2010 – p.110/133 Variation 5. Conjectural variations Definition Firms aware of the impact of their decisions on rivals Incorporate that impact in profit maximization decision process CV: assumptions on how the behavior of a firm impacts on rivals’ decision processes CV: set of conjectures (expectations) of every firm on the sequence of moves of rivals Bowley (1924); Boyer and Moreaux (1983), Bresnahan (1981), Perry (1982). Models: Bowley; Consistent CV; Marschack-Selten Fall 2009-2010 – p.111/133 Variation 5. Conjectural variations - Bowley (1) Assumptions Duopoly Same assumptions on profits as Cournot FOC on profits are ∂Π ∂Π dq i + i j = 0, i 6= j ∂qi ∂qj dqi dq where j represents firm i’s conjecture on the behavior of firm j, dqi after a marginal variation of its production level, dqi. This is the content of the Conjectural variation Fall 2009-2010 – p.112/133 Variation 5. Conjectural variations - Bowley (2) Illustration linear demand p = a − b(q1 + q2) common constant marginal cost c Reaction function: a − bq − c q (q )= j i j dq 2b + b j dqi Fall 2009-2010 – p.113/133 Variation 5. Conjectural variations - Bowley (3) Cooperative CV Assume firms coordinate to adjust the aggregate production to the monopoly level by equally adjusting their individual outputs. dq dq This translates in a conjectural variation 2 = 1 = 1. Then, dq1 dq2 a − bq − c q (q )= 2 , 1 2 3b a − bq − c q (q )= 1 . 2 1 3b and a − c q + q = . 1 2 2b that corresponds to the monopoly solution. Fall 2009-2010 – p.114/133 Variation 5. Conjectural variations - Bowley (4) Competitive CV Assume that every firm conjectures that if it reduces production in one unit, the rival will increase its production in one unit so that the aggregate output remains constant. dq dq This translates in a conjectural variation 2 = 1 = −1. Then, dq1 dq2 a − bq − c a − bq − c q (q )= 2 , q (q )= 1 . 1 2 b 2 1 b and a − c q + q = . 1 2 b This is precisely the competitive solution. Fall 2009-2010 – p.115/133 Variation 5. Conjectural variations - Bowley (5) Cournot CV Assume each firm takes the output of the rival as given. dq dq This translates in a conjectural variation 2 = 1 = 0. Then, dq1 dq2 a − bq − c a − bq − c q (q )= 2 , q (q )= 1 . 1 2 2b 2 1 2b and 2(a − c) q + q = . 1 2 3b This is the Cournot solution. CV ∈ [−1, 1] generate the perfect collusion, perfect competition, and Cournot solutions. Fall 2009-2010 – p.116/133 Variation 5. Conjectural variations - Bowley (6) Other CV Bertrand: infinite conjectural variations; Stackelberg: zero conjectural variation for the follower finite conjectural variation for the leader Sweezy collusive conjectural variations when output expands Cournot conjectural variations when output contracts. Fall 2009-2010 – p.117/133 Variation 5. Consistent Conjectural variations (1) So far, values of the conjectural variations yielding the equilibrium of some particular models. Definition Find an equilibrium set of conjectural variations. In this equilibrium no firm would have incentives neither to modify its behavior nor to change its conjectural variations. This approach tries to identify a Nash equilibrium in conjectural variations. This equilibrium is called a set of consistent conjectural variations. Fall 2009-2010 – p.118/133 Variation 5. Consistent Conjectural variations (2) Assume duopoly, linear demand, constant mg. cost and let, dq2 dq1 k1 ≡ ; k2 ≡ dq1 dq2 Then, a − bq2 − c a − bq1 − c q1 = , and q2 = 2b + bk1 2b + bk2 Therefore, dq1 1 dq2 1 = k2 = − , and = k1 = − . dq2 2+ k1 dq1 2+ k2 Solving, k2 = −1, and k1 = −1 Fall 2009-2010 – p.119/133 Variation 5. Consistent Conjectural variations (3) Comments In example, equilibrium in consistent conjectural variations, both firms use conjectural variations -1 giving rise to the competitive equilibrium. Suggests possibility that the behavior of the firms producing a homogeneous product may be approximated by the competitive behavior even with few firms in the market. If simultaneous decision model, it does not make sense qi being a function of qj or viceversa. Such a situation implies that firm i observes firm j’s decision, and according to its conjectural variation, determines qi. Therefore, conjectural variations is a device to understand (not to explain) the decision process of firms aware of their strategic dependency. Fall 2009-2010 – p.120/133 Variation 5. Conjectural variations - Marschack-Selten (1) Static model, 1977 Firms simultaneously announce prices → public info. Recursive process to revise prices Firm 1 revises price → (n − 1) rivals adjust Firm 1 re-revises price → (n − 1) rivals adjust ... until Firm 1 does not want to revise its price. Firm 2 revises its price → (n − 1) rivals adjust, etc, etc Consistent conjectural variations A non-coop equilibrium: price vector satisfying no profitable deviation for any firm reaction function maximizes profits Fall 2009-2010 – p.121/133 Variation 5. Conjectural variations - Marschack-Selten (2) Dynamic model, 1978 Firms after choosing their first price face an adjustment cost for any price change. If a firm i varies its price, there is a period of time between the new price is posted and the competitors adjust their prices. The adjustment cost faced by firm i is sufficiently high to offset the extra profits firm i may obtain in the interim period until the rivals react. Accordingly, a price variation is only profitable if it generates more profits in the long run, once the rivals have reacted. Fall 2009-2010 – p.122/133 Variation 6. Dynamic models (1) Two alternative ways of explicitly consider the introduction of time in a model: Repeated games or supergames. These replicate a static Cournot (Bertrand) type of game a finite or infinite number of times (see Friedman, 1977). dynamic strategies of firms are of Markov type and the objective of the model is to characterize a Markov perfect equilibrium (see Maskin and Tirole, 1987, 1988a). An overview of dynamic oligopoly models can be found in Fudenberg and Tirole (1986), Kreps and Spence (1984), Shapiro (1989) or Maskin and Tirole (1987, 1988a, 1988b). Fall 2009-2010 – p.123/133 Variation 6. Dynamic models (2) Repeated games Set-up Game is played a certain number of times (iterations) One iteration is independent of another, players can condition their present or future behavior to the history of moves. room for punishments as (credible) threats to affect players’ future decisions. Fall 2009-2010 – p.124/133 Variation 6. Dynamic models (3) Repeated games (cont’d) Illustration (Tirole, 1988) Symmetric Bertrand duopoly firm lowest price, gets all demand both firms same price, share evenly demand Game repeated T + 1 times (T finite or infinite) Πi(pit,pjt), t = 0, 1,...,T Firm i’s objective: max present value flow of profits T t δ πi(pit,pjt), δ is the discount rate t=0 X Fall 2009-2010 – p.125/133 Variation 6. Dynamic models (4) Repeated games (cont’d) Illustration (Tirole, 1988) (cont’d) In every period t both firms simultaneously choose a price. Firms have perfect recall of all the history of past decisions. Let, Ht = (p10,p20; p11,p21; p12,p22; . . . ; p1,t 1,p2,t 1) be the − − history of prices chosen by both firms up to period t. Firm i’s strategy depends on Ht. Equilibrium Characterize a perfect equilibrium. For any history Ht in period t, firm i’s strategy from t on should maximize the present value of the flow of its future profits conditional on the expectation of firm j’s strategy in period t. Fall 2009-2010 – p.126/133 Variation 6. Dynamic models (5) Repeated games (cont’d) T finite. By backward induction, in period T the model is equivalent to the static version. Accordingly, in equilibrium price equals marginal cost. Decisions in period T are not dependent on T − 1. Therefore, T − 1, as if it would be the last period. Thus, for any HT 1, the equilibrium strategies price = marginal cost. − We can repeat this reasoning until the initial period. Summarizing, if the number of iterations is finite, the only equilibrium of the repeated game is simply the iteration in every period of the equilibrium strategies of the static game. Fall 2009-2010 – p.127/133 Variation 6. Dynamic models (6) Repeated games (cont’d) T infinite. Multiplicity of equilibria: Eq. under T finite Other equilibria: Let p ∈ [pc,pm] and consider symmetric strategies: at t = 0 both firms choose p In t = τ if both chosen p in the past, choose p in t = τ If at t = s deviation, choose p = MC forever Profits 1 2 3 If no deviations, 2 Π(p)(1 + δ + δ + δ + · · · ). If at t = s one deviates, obtains π(p) in s, and zero afterwards: 1 2 3 s 1 s δ 2 Π(p)(1 + δ + δ + δ + · · · + δ − )+ δ π(p)= π(p) 2(1 δ) 1 − If δ ≥ 2 deviating is never optimal Fall 2009-2010 – p.128/133 Variation 6. Dynamic models (6) Repeated games (cont’d) T infinite. Argument true ∀p ∈ [pc,pm] Any price p can be supported as equilibrium → folk theorem Fall 2009-2010 – p.129/133 Variation 6. Dynamic models (7) Markov games Interaction among time periods “Markov perfect equilibria": firms condition their actions to a reduced subset of state variables rather than in the full history of the game. Consider an infinite duopoly à la Cournot. Let πi(qit, qjt), t = 0, 1, 2,... firm i’s profits in period t. Assume profits concave in qit and decreasing in qjt Accordingly, reaction functions are well-defined and are negatively sloped. Every firm aims at maximizing the present value of the flow of s future profits, s∞=0 δ πi(q1,t+s, q2,t+s), δ discount factor P Fall 2009-2010 – p.130/133 Variation 6. Dynamic models (7) Markov games (cont’d) decision process: in odd periods, firm 1 decides a production volume that remains fixed until the next odd period, i.e. until t + 2. In other words, q1,t+1 = q1,t, if t is odd. in even periods firm 2 decides a production volume that remains fixed until the next even period, i.e. q2,t+1 = q2,t, if t is even. Fall 2009-2010 – p.131/133 Variation 6. Dynamic models (8) Markov games (cont’d) In every period, the relevant state variables are the ones involved in the profit functions. In odd periods when firm 1 decides, the relevant information is the production of firm 2, q2,2k+1 = q2,2k. Firm 1’s decision is contingent only on q2,2k, so that its reaction function is of the type q1,2+1 = w1(q2,2k). In even periods when firm 2 chooses its output level, its reaction function is q2,2k+2 = w2(q1,2k+1). We call these Markov strategies dynamic reaction functions Fall 2009-2010 – p.132/133 Variation 6. Dynamic models (9) Markov games (cont’d) The objective of the model is to find a pair (w1,w2) that constitutes a perfect equilibrium. That is, for any period t, the dynamic reaction function of a firm must maximize the present value of the discounted flow of future profits given the dynamic reaction function of the rival firm. This pair (w1,w2) is called a Markov perfect equilibrium. Fall 2009-2010 – p.133/133