Bertrand Model of Price Competition Advanced Microeconomic Theory 1 Bertrand Model of Price Competition • Consider: – An industry with two firms, 1 and 2, selling a homogeneous product – Firms face market demand 푥(푝), where 푥(푝) is continuous and strictly decreasing in 푝 – There exists a high enough price (choke price) 푝ҧ < ∞ such that 푥(푝) = 0 for all 푝 > 푝ҧ – Both firms are symmetric in their constant marginal cost 푐 > 0, where 푥 푐 ∈ (0, ∞) – Every firm 푗 simultaneously sets a price 푝푗 Advanced Microeconomic Theory 2 Bertrand Model of Price Competition • Firm 푗’s demand is 푥(푝푗) if 푝푗 < 푝푘 1 푥 (푝 , 푝 ) = 푥(푝 ) if 푝 = 푝 푗 푗 푘 2 푗 푗 푘 0 if 푝푗 > 푝푘 • Intuition: Firm 푗 captures – all market if its price is the lowest, 푝푗 < 푝푘 – no market if its price is the highest, 푝푗 > 푝푘 – shares the market with firm 푘 if the price of both firms coincide, 푝푗 = 푝푘 Advanced Microeconomic Theory 3 Bertrand Model of Price Competition • Given prices 푝푗 and 푝푘, firm 푗’s profits are therefore (푝푗 − 푐) ∙ 푥푗 (푝푗, 푝푘) • We are now ready to find equilibrium prices in the Bertrand duopoly model. ∗ ∗ – There is a unique NE (푝푗 , 푝푘) in the Bertrand duopoly model. In this equilibrium, both firms ∗ ∗ set prices equal to marginal cost, 푝푗 = 푝푘 = 푐. Advanced Microeconomic Theory 4 Bertrand Model of Price Competition • Let’s us describe the best response function of firm 푗. • If 푝푘 < 푐, firm 푗 sets its price at 푝푗 = 푐. – Firm 푗 does not undercut firm 푘 since that would entail negative profits. • If 푐 < 푝푘 < 푝푗, firm 푗 slightly undercuts firm 푘, i.e., 푝푗 = 푝푘 − 휀. – This allows firm 푗 to capture all sales and still make a positive margin on each unit. • If 푝푘 > 푝푚, where 푝푚 is a monopoly price, firm 푗 does not need to charge more than 푝푚, i.e., 푝푗 = 푝푚. – 푝푚 allows firm 푗 to capture all sales and maximize profits as the only firm selling a positive output. Advanced Microeconomic Theory 5 Bertrand Model of Price Competition • Firm 푗’s best response has: pj – a flat segment for all 45°-line (pj = pk) 푝푘 < 푐, where p p (p ) 푝푗(푝푘) = 푐 m j k – a positive slope for all 푐 < 푝푘 < 푝푗, where c firm 푗 charges a price slightly below firm 푘 pk – a flat segment for all c pm 푝푘 > 푝푚, where 푝푗(푝푘) = 푝푚 Advanced Microeconomic Theory 6 Bertrand Model of Price Competition • A symmetric argument applies to the construction pj pk (pj) of the best response function of firm 푘. 45°-line (pj = pk) • A mutual best response for both firms is pm pj (pk) ∗ ∗ (푝1, 푝2) = (푐, 푐) where the two best response functions cross c each other. • This is the NE of the Bertrand model pk – Firms make no economic c pm profits. Advanced Microeconomic Theory 7 Bertrand Model of Price Competition • With only two firms competing in prices we obtain the perfectly competitive outcome, where firms set prices equal to marginal cost. • Price competition makes each firm 푗 face an infinitely elastic demand curve at its rival’s price, 푝푘. – Any increase (decrease) from 푝푘 infinitely reduces (increases, respectively) firm 푗’s demand. Advanced Microeconomic Theory 8 Bertrand Model of Price Competition • How much does Bertrand equilibrium hinge into our assumptions? – Quite a lot • The competitive pressure in the Bertrand model with homogenous products is ameliorated if we instead consider: – Price competition (but allowing for heterogeneous products) – Quantity competition (still with homogenous products) – Capacity constraints Advanced Microeconomic Theory 9 Bertrand Model of Price Competition • Remark: – How our results would be affected if firms face different production costs, i.e., 0 < 푐1 < 푐2? – The most efficient firm sets a price equal to the marginal cost of the least efficient firm, 푝1 = 푐2. – Other firms will set a random price in the uniform interval [푐1, 푐1 + 휂] where 휂 > 0 is some small random increment with probability distribution 푓 푝, 휂 > 0 for all 푝. Advanced Microeconomic Theory 10 Cournot Model of Quantity Competition Advanced Microeconomic Theory 11 Cournot Model of Quantity Competition • Let us now consider that firms compete in quantities. • Assume that: – Firms bring their output 푞1 and 푞2 to a market, the market clears, and the price is determined from the inverse demand function 푝(푞), where 푞 = 푞1 + 푞2. – 푝(푞) satisfies 푝’(푞) < 0 at all output levels 푞 ≥ 0, – Both firms face a common marginal cost 푐 > 0 – 푝(0) > 푐 in order to guarantee that the inverse demand curve crosses the constant marginal cost curve at an interior point. Advanced Microeconomic Theory 12 Cournot Model of Quantity Competition • Let us first identify every firm’s best response function • Firm 1’s PMP, for a given output level of its rival, 푞ത2, max 푝 푞1 + 푞ത2 푞1 − 푐푞1 푞 ≥0 1 Price • When solving this PMP, firm 1 treats firm 2’s production, 푞ത2, as a parameter, since firm 1 cannot vary its level. Advanced Microeconomic Theory 13 Cournot Model of Quantity Competition • FOCs: ′ 푝 (푞1 + 푞ത2)푞1 + 푝(푞1 + 푞ത2) − 푐 ≤ 0 with equality if 푞1 > 0 • Solving this expression for 푞1, we obtain firm 1’s best response function (BRF), 푞1(푞ത2). • A similar argument applies to firm 2’s PMP and its best response function 푞2(푞ത1). ∗ ∗ • Therefore, a pair of output levels (푞1, 푞2) is NE of the Cournot model if and only if ∗ 푞1 ∈ 푞1(푞ത2) for firm 1’s output ∗ 푞2 ∈ 푞2(푞ത1) for firm 2’s output Advanced Microeconomic Theory 14 Cournot Model of Quantity Competition ∗ ∗ • To show that 푞1, 푞2 > 0, let us work by contradiction, ∗ assuming 푞1 = 0. – Firm 2 becomes a monopolist since it is the only firm producing a positive output. • Using the FOC for firm 1, we obtain ′ ∗ ∗ 푝 (0 + 푞2)0 + 푝(0 + 푞2) ≤ 푐 ∗ or 푝(푞2) ≤ 푐 • And using the FOC for firm 2, we have ′ ∗ ∗ ∗ 푝 (푞2 + 0)푞2 + 푝(푞2 + 0) ≤ 푐 ′ ∗ ∗ ∗ or 푝 (푞2)푞2 + 푝(푞2) ≤ 푐 • This implies firm 2’s MR under monopoly is lower than ∗ its MC. Thus, 푞2 = Advanced0. Microeconomic Theory 15 Cournot Model of Quantity Competition ∗ • Hence, if 푞1 = 0, firm 2’s output would also be ∗ zero, 푞2 = 0. • But this implies that 푝(0) < 푐 since no firm produces a positive output, thus violating our initial assumption 푝(0) > 푐. – Contradiction! ∗ • As a result, we must have that both 푞1 > 0 and ∗ 푞2 > 0. • Note: This result does not necessarily hold when both firms are asymmetric in their production cost. Advanced Microeconomic Theory 16 Cournot Model of Quantity Competition • Example (symmetric costs): – Consider an inverse demand curve 푝(푞) = 푎 − 푏푞, and two firms competing à la Cournot both facing a marginal cost 푐 > 0. – Firm 1’s PMP is 푎 − 푏(푞1 + 푞ത2) 푞1 − 푐푞1 – FOC wrt 푞1: 푎 − 2푏푞1 − 푏푞ത2 − 푐 ≤ 0 with equality if 푞1 > 0 Advanced Microeconomic Theory 17 Cournot Model of Quantity Competition • Example (continue): – Solving for 푞1, we obtain firm 1’s BRF 푎−푐 푞ത 푞 (푞ത ) = − 2 1 2 2푏 2 – Analogously, firm 2’s BRF 푎−푐 푞ത 푞 (푞ത ) = − 1 2 1 2푏 2 Advanced Microeconomic Theory 18 Cournot Model of Quantity Competition • Firm 1’s BRF: – When 푞2 = 0, then 푎−푐 푞 = , which 1 2푏 coincides with its output under monopoly. – As 푞2 increases, 푞1 decreases (i.e., firm 1’s and 2’s output are strategic substitutes) 푎−푐 – When 푞 = , then 2 푏 푞1 = 0. Advanced Microeconomic Theory 19 Cournot Model of Quantity Competition • A similar argument applies for firm 2’s BRF. • Superimposing both firms’ BRFs, we obtain the Cournot equilibrium output ∗ ∗ pair (푞1, 푞2). Advanced Microeconomic Theory 20 Cournot Model of Quantity Competition q1 Perfect competition a – c q1 + q2 = qc = a – c b 45°-line (q1 = q2) b q (q ) a – c 2 1 2b * * a – c (q 1,q2 ) Monopoly a – c 3b q1 + q2 = qm = q1(q2) 2b 45° q a – c a – c a – c 2 3b 2b b Advanced Microeconomic Theory 21 Cournot Model of Quantity Competition ∗ ∗ • Cournot equilibrium output pair (푞1, 푞2) occurs at the intersection of the two BRFs, i.e., 푎−푐 푎−푐 (푞∗, 푞∗) = , 1 2 3푏 3푏 • Aggregate output becomes 푎−푐 푎−푐 2(푎−푐) 푞∗ = 푞∗ + 푞∗ = + = 1 2 3푏 3푏 3푏 푎−푐 which is larger than under monopoly, 푞 = , 푚 2푏 but smaller than under perfect competition, 푞푐 = 푎−푐 . 푏 Advanced Microeconomic Theory 22 Cournot Model of Quantity Competition • The equilibrium price becomes 2 푎−푐 푎+2푐 푝 푞∗ = 푎 − 푏푞∗ = 푎 − 푏 = 3푏 3 푎+푐 which is lower than under monopoly, 푝 = , but 푚 2 higher than under perfect competition, 푝푐 = 푐. • Finally, the equilibrium profits of every firm 푗 푎+2푐 푎−푐 푎−푐 푎−푐 2 휋∗ = 푝 푞∗ 푞∗ − 푐푞∗ = − 푐 = 푗 푗 푗 3 3푏 3푏 4푏 푎−푐 2 which are lower than under monopoly, 휋 = , 푚 4푏 but higher than under perfect competition, 휋푐 = 0. Advanced Microeconomic Theory 23 Cournot Model of Quantity Competition • Quantity competition (Cournot model) yields less competitive outcomes than price competition (Bertrand model), whereby firms’ behavior mimics that in perfectly competitive markets – That’s because, the demand that every firm faces in the Cournot game is not infinitely elastic. – A reduction in output does not produce an infinite increase in market price, but instead an increase of − 푝′(푞1 + 푞2). – Hence, if firms produce the same output as under 푎−푐 marginal cost pricing, i.e., half of , each firm would 2 have incentives to deviate from such a high output level by marginally reducing its output.