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.

Advanced Microeconomic Theory 24 Cournot Model of Quantity Competition

• Equilibrium output under Cournot does not coincide with the monopoly output either. – That’s because, every firm 푖, individually increasing its output level 푞푖, takes into account how the reduction in market price affects its own profits, but ignores the profit loss (i.e., a negative external effect) that its rival suffers from such a lower price. – Since every firm does not take into account this external effect, aggregate output is too large, relative to the output that would maximize firms’ joint profits.

Advanced Microeconomic Theory 25 Cournot Model of Quantity Competition

• Example (Cournot vs. Cartel): – Let us demonstrate that firms’ Cournot output is larger than that under the cartel. – PMP of the cartel is max (푎 − 푏(푞1+푞2))푞1 − 푐푞1 푞1,푞2 + (푎 − 푏(푞1+푞2))푞2 − 푐푞2 – Since 푄 = 푞1 + 푞2, the PMP can be written as max 푎 − 푏(푞1+푞2) (푞1+푞2) − 푐(푞1+푞2) 푞1,푞2 = max 푎 − 푏푄 푄 − 푐푄 = 푎푄 − 푏푄2 − 푐푄 푄

Advanced Microeconomic Theory 26 Cournot Model of Quantity Competition

• Example (continued): – FOC wrt 푄 푎 − 2푏푄 − 푐 ≤ 0 – Solving for 푄, we obtain the aggregate output 푎−푐 푄∗ = 2푏 which is positive since 푎 > 푐, i.e., 푝(0) = 푎 > 푐. – Since firms are symmetric in costs, each produces 푄 푎−푐 푞 = = 푖 2 4푏

Advanced Microeconomic Theory 27 Cournot Model of Quantity Competition

• Example (continued): – The equilibrium price is 푎−푐 푎+푐 푝 = 푎 − 푏푄 = 푎 − 푏 = 2푏 2 – Finally, the equilibrium profits are

휋푖 = 푝 ⋅ 푞푖 − 푐푞푖 푎+푐 푎−푐 푎−푐 푎−푐 2 = ⋅ − 푐 = 2 4푏 4푏 8푏 which is larger than firms would obtain under 푎−푐 2 Cournot competition, . 9푏 Advanced Microeconomic Theory 28 Cournot Model of Quantity Competition: Cournot Pricing Rule • Firms’ market power can be expressed using a variation of the Lerner index. – Consider firm 푗’s profit maximization problem 휋푗 = 푝(푞)푞푗 − 푐푗(푞푗) – FOC for every firm 푗

푝′ 푞 푞푗 + 푝 푞 − 푐푗 = 0 or 푝(푞) − 푐푗 = −푝′ 푞 푞푗 – Multiplying both sides by 푞 and dividing them by 푝(푞) yield ′ 푝 푞 − 푐푗 −푝 푞 푞푗 푞 = 푞 푝(푞) 푝(푞) Advanced Microeconomic Theory 29 Cournot Model of Quantity Competition: Cournot Pricing Rule

1 푞 – Recalling = −푝′ 푞 ⋅ , we have 휀 푝 푞 푝 푞 −푐 1 푞 푗 = 푞 푝(푞) 휀 푗 푝 푞 −푐 1 푞 or 푗 = 푗 푝(푞) 휀 푞 푞 – Defining 훼 ≡ 푗 as firm 푗’s market share, we obtain 푗 푞 푝 푞 − 푐푗 훼푗 = 푝(푞) 휀 which is referred to as the Cournot pricing rule.

Advanced Microeconomic Theory 30 Cournot Model of Quantity Competition: Cournot Pricing Rule – Note:

. When 훼푗 = 1, implying that firm 푗 is a monopoly, the IEPR becomes a special case of the Cournot price rule.

. The larger the market share 훼푗 of a given firm, the larger the price markup of firm 푗. . The more inelastic demand 휀 is, the larger the price markup of firm 푗.

Advanced Microeconomic Theory 31 Cournot Model of Quantity Competition: Cournot Pricing Rule • Example (Merger effects on Cournot Prices): – Consider an industry with 푛 firms and a constant- elasticity demand function 푞(푝) = 푎푝−1, where 푎 > 0 and 휀 = 1. – Before merger, we have 푝퐵 − 푐 1 푛푐 = ⟹ 푝퐵 = 푝퐵 푛 푛 − 1 – After the merger of 푘 < 푛 firms 푛 − 푘 + 1 firms remain in the industry, and thus 푝퐴 − 푐 1 푛 − 푘 + 1 푐 = ⟹ 푝퐴 = 푝퐴 푛 − 푘 + 1 푛 − 푘 Advanced Microeconomic Theory 32 Cournot Model of Quantity Competition: Cournot Pricing Rule • Example (continued): – The percentage change in prices is 푛 − 푘 + 1 푐 푛푐 푝퐴 − 푝퐵 − %Δ푝 = = 푛 − 푘 푛 − 1 푝퐵 푛푐 푛 − 1 푘 − 1 = > 0 푛(푛 − 푘) – Hence, prices increase after the merger. – Also, %Δ푝 increases as the number of merging firms 푘 increases 휕%Δ푝 푛 − 1 = > 0 휕푘 푛 푛 − 푘 2 Advanced Microeconomic Theory 33 Cournot Model of Quantity Competition: Cournot Pricing Rule • Example (continued): – The percentage %Δp increase in price after the merger, %Δ푝, as a 0.20 function of the number %Δp(k) of merging firms, 푘. – For simplicity, 푛 = 0.10 100. k 20 40 60 80 100

Advanced Microeconomic Theory 34 Cournot Model of Quantity Competition: Asymmetric Costs • Assume that firm 1 and 2’s constant marginal costs of production differ, i.e., 푐1 > 푐2, so firm 2 is more efficient than firm 1. Assume also that the inverse demand function is 푝 푄 = 푎 − 푏푄, and 푄 = 푞1 + 푞2. • Firm 푖’s PMP is max 푎 − 푏(푞푖 + 푞푗) 푞푖 − 푐푖푞푖 푞푖 • FOC: 푎 − 2푏푞푖 − 푏푞푗 − 푐푖 = 0

Advanced Microeconomic Theory 35 Cournot Model of Quantity Competition: Asymmetric Costs

• Solving for 푞푖 (assuming an interior solution) yields firm 푖’s BRF 푎 − 푐 푞푗 푞 (푞 ) = 푖 − 푖 푗 2푏 2 • Firm 1’s optimal output level can be found by plugging firm 2’s BRF into firm 1’s 푎 − 푐 1 푎 − 푐 푞∗ 푎 − 2푐 + 푐 푞∗ = 1 − 2 − 1 ⟺ 푞∗ = 1 2 1 2푏 2 2푏 2 1 3푏 • Similarly, firm 2’s optimal output level is 푎 − 푐 푞∗ 푎 + 푐 − 2푐 푞∗ = 2 − 1 = 1 2 2 2푏 2 3푏

Advanced Microeconomic Theory 36 Cournot Model of Quantity Competition: Asymmetric Costs ∗ ∗ • The output levels (푞1, 푞2) also vary when (푐1, 푐2) changes 휕푞∗ 2 휕푞∗ 1 1 = − < 0 and 1 = > 0 휕푐1 3푏 휕푐2 3푏 휕푞∗ 1 휕푞∗ 2 2 = > 0 and 2 = − < 0 휕푐1 3푏 휕푐2 3푏 • Intuition: Each firm’s output decreases in its own costs, but increases in its rival’s costs.

Advanced Microeconomic Theory 37 Cournot Model of Quantity Competition: Asymmetric Costs • BRFs for firms 1 and 2 q1 푎+푐2 when 푐 > (i.e., a – c2 1 2 b only firm 2 produces).

• BRFs cross at the vertical q2(q1) ∗ axis where 푞1 = 0 and ∗ 푞2 > 0 (i.e., a corner

solution) a – c1 2b q1(q2) * * (q 1,q2 ) q2 a – c1 a – c2 b 2b

Advanced Microeconomic Theory 38 Cournot Model of Quantity Competition: 퐽 > 2 firms • Consider 퐽 > 2 firms, all facing the same constant marginal cost 푐 > 0. The linear inverse demand curve is 푝 푄 = 푎 − 푏푄, where 푄 = σ퐽 푞푘. • Firm 푖’s PMP is

max 푎 − 푏 푞푖 + ෍ 푞푘 푞푖 − 푐푞푖 푞푖 푘≠푖 • FOC: ∗ ∗ 푎 − 2푏푞푖 − 푏 ෍ 푞푘 − 푐 ≤ 0 푘≠푖

Advanced Microeconomic Theory 39 Cournot Model of Quantity Competition: 퐽 > 2 firms ∗ • Solving for 푞푖 , we obtain firm 푖’s BRF 푎 − 푐 1 푞∗ = − ෍ 푞∗ 푖 2푏 2 푘 푘≠푖 • Since all firms are symmetric, their BRFs are also ∗ ∗ ∗ symmetric, implying 푞1 = 푞2 = ⋯ = 푞퐽. This ∗ ∗ ∗ ∗ implies σ푘≠푖 푞푘 = 퐽푞푖 − 푞푖 = 퐽 − 1 푞푖 . • Hence, the BRF becomes 푎 − 푐 1 푞∗ = − 퐽 − 1 푞∗ 푖 2푏 2 푖

Advanced Microeconomic Theory 40 Cournot Model of Quantity Competition: 퐽 > 2 firms ∗ • Solving for 푞푖 푎 − 푐 푞∗ = 푖 퐽 + 1 푏 which is also the equilibrium output for other 퐽 − 1 firms. • Therefore, aggregate output is 퐽 푎 − 푐 푄∗ = 퐽푞∗ = 푖 퐽 + 1 푏 and the corresponding equilibrium price is 푎 + 퐽푐 푝∗ = 푎 − 푏푄∗ = 퐽 + 1 Advanced Microeconomic Theory 41 Cournot Model of Quantity Competition: 퐽 > 2 firms • Firm 푖’s equilibrium profits are ∗ ∗ ∗ ∗ 휋푖 = 푎 − 푏푄 푞푖 − 푐푞푖 퐽 푎 − 푐 푎 − 푐 푎 − 푐 = 푎 − 푏 − 푐 퐽 + 1 푏 퐽 + 1 푏 퐽 + 1 푏 푎 − 푐 2 = = 푞∗ 2 퐽 + 1 푏 푖

Advanced Microeconomic Theory 42 Cournot Model of Quantity Competition: 퐽 > 2 firms • We can show that 푎 − 푐 푎 − 푐 lim 푞∗ = = 퐽→2 푖 2 + 1 푏 3푏 2(푎 − 푐) 2(푎 − 푐) lim 푄∗ = = 퐽→2 2 + 1 푏 3푏 푎 + 2푐 푎 + 2푐 lim 푝∗ = = 퐽→2 2 + 1 3 which exactly coincide with our results in the Cournot duopoly model.

Advanced Microeconomic Theory 43 Cournot Model of Quantity Competition: 퐽 > 2 firms • We can show that 푎 − 푐 푎 − 푐 lim 푞∗ = = 퐽→1 푖 1 + 1 푏 2푏 1(푎 − 푐) 푎 − 푐 lim 푄∗ = = 퐽→1 1 + 1 푏 2푏 푎 + 1푐 푎 + 푐 lim 푝∗ = = 퐽→1 1 + 1 2 which exactly coincide with our findings in the monopoly.

Advanced Microeconomic Theory 44 Cournot Model of Quantity Competition: 퐽 > 2 firms • We can show that lim 푞∗ = 0 퐽→∞ 푖 푎 − 푐 lim 푄∗ = 퐽→∞ 푏 lim 푝∗ = 푐 퐽→∞ which coincides with the solution in a perfectly competitive market.

Advanced Microeconomic Theory 45 Product Differentiation

Advanced Microeconomic Theory 46 Product Differentiation

• So far we assumed that firms sell homogenous (undifferentiated) products. • What if the goods firms sell are differentiated? – For simplicity, we will assume that product attributes are exogenous (not chosen by the firm)

Advanced Microeconomic Theory 47 Product Differentiation: Bertrand Model • Consider the case where every firm 푖, for 푖 = {1,2}, faces demand curve 푞푖(푝푖, 푝푗) = 푎 − 푏푝푖 + 푐푝푗 where 푎, 푏, 푐 > 0 and 푗 ≠ 푖.

• Hence, an increase in 푝푗 increases firm 푖’s sales. • Firm 푖’s PMP: max (푎 − 푏푝푖 + 푐푝푗)푝푖 푝푖≥0 • FOC: −2푏푝푖 + 푐푝푗 = 0 Advanced Microeconomic Theory 48 Product Differentiation: Bertrand Model

• Solving for 푝푖, we find firm 푖’s BRF 푎 + 푐푝푗 푝 (푝 ) = 푖 푗 2푏 • Firm 푗 also has a symmetric BRF. • Note: – BRFs are now positively sloped – An increase in firm 푗’s price leads firm 푖 to increase his, and vice versa – In this case, firms’ choices (i.e., prices) are strategic complements Advanced Microeconomic Theory 49 Product Differentiation: Bertrand Model

p1

p2(p1)

p (p ) * 1 2 p 1

a 2b

p2 a * p2 2b Advanced Microeconomic Theory 50 Product Differentiation: Bertrand Model • Simultaneously solving the two BRS yields 푎 푝∗ = 푖 2푏 − 푐 with corresponding equilibrium sales of 푎푏 푞∗(푝∗, 푝∗) = 푎 − 푏푝∗ + 푐푝∗ = 푖 푖 푗 푖 푗 2푏 − 푐 and equilibrium profits of 푎 푎푏 휋∗ = 푝∗ ∙ 푞∗ 푝∗, 푝∗ = 푖 푖 푖 푖 푗 2푏 − 푐 2푏 − 푐 푎2푏 = 2푏 − 푐 2 Advanced Microeconomic Theory 51 Product Differentiation: Cournot Model • Consider two firms with the following linear inverse demand curves 푝1(푞1, 푞2) = 훼 − 훽푞1 − 훾푞2 for firm 1 푝2(푞1, 푞2) = 훼 − 훾푞1 − 훽푞2 for firm 2 • We assume that 훽 > 0 and 훽 > 훾 – That is, the effect of increasing 푞1 on 푝1 is larger than the effect of increasing 푞1 on 푝2 – Intuitively, the price of a particular brand is more sensitive to changes in its own output than to changes in its rival’s output – In other words, own-price effects dominate the cross- price effects.

Advanced Microeconomic Theory 52 Product Differentiation: Cournot Model • Firm 푖’s PMP is (assuming no costs) max (훼 − 훽푞푖 − 훾푞푗)푞푖 푞푖≥0 • FOC: 훼 − 2훽푞푖 − 훾푞푗 = 0

• Solving for 푞푖 we find firm 푖’s BRF 훼 훾 푞 (푞 ) = − 푞 푖 푗 2훽 2훽 푗 • Firm 푗 also has a symmetric BRF

Advanced Microeconomic Theory 53 Product Differentiation: Cournot Model

q1

α γ

α q2(q1) 2β

* * (q 1,q2 )

q1(q2)

q α α 2 2β γ Advanced Microeconomic Theory 54 Product Differentiation: Cournot Model • Comparative statics of firm 푖’s BRF – As 훽 → 훾 (products become more homogeneous), BRF becomes steeper. That is, the profit- maximizing choice of 푞푖 is more sensitive to changes in 푞푗 (tougher competition) – As 훾 → 0 (products become very differentiated), firm 푖’s BRF no longer depends on 푞푗 and becomes flat (milder competition)

Advanced Microeconomic Theory 55 Product Differentiation: Cournot Model • Simultaneously solving the two BRF yields 훼 푞∗ = for all 푖 = {1,2} 푖 2훽 + 훾 with a corresponding equilibrium price of 훼훽 푝∗ = 훼 − 훽푞∗ − 훾푞∗ = 푖 푖 푗 2훽 + 훾 and equilibrium profits of 훼훽 훼 훼2훽 휋∗ = 푝∗푞∗ = = 푖 푖 푖 2훽 + 훾 2훽 + 훾 2훽 + 훾 2

Advanced Microeconomic Theory 56 Product Differentiation: Cournot Model • Note: – As 훾 increases (products become more homogeneous), individual and aggregate output decrease, and individual profits decrease as well. – If 훾 → 훽 (indicating undifferentiated products), then 훼 훼 푞∗ = = as in standard Cournot models of 푖 2훽+훽 3훽 homogeneous products. – If 훾 → 0 (extremely differentiated products), then 훼 훼 푞∗ = = as in monopoly. 푖 2훽+0 2훽

Advanced Microeconomic Theory 57 Dynamic Competition

Advanced Microeconomic Theory 58 Dynamic Competition: Sequential Bertrand Model with Homogeneous Products

• Assume that firm 1 chooses its price 푝1 first, whereas firm 2 observes that price and responds with its own price 푝2. • Since the game is a sequential-move game (rather than a simultaneous-move game), we should use backward induction.

Advanced Microeconomic Theory 59 Dynamic Competition: Sequential Bertrand Model with Homogeneous Products • Firm 2 (the follower) has a BRF given by 푝1 − 휀 if 푝1 > 푐 푝2(푝1) = ቊ 푐 if 푝1 ≤ 푐 while firm 1’s (the leader’s) BRF is 푝1 = 푐 • Intuition: the follower undercuts the leader’s price 푝1 by a small 휀 > 0 if 푝1 > 푐, or keeps it at 푝2 = 푐 if the leader sets 푝1 = 푐.

Advanced Microeconomic Theory 60 Dynamic Competition: Sequential Bertrand Model with Homogeneous Products • The leader expects that its price will be: – undercut by the follower when 푝1 > 푐 (thus yielding no sales) – mimicked by the follower when 푝1 = 푐 (thus entailing half of the market share) • Hence, the leader has (weak) incentives to set a price 푝1 = 푐. • As a consequence, the equilibrium price pair ∗ ∗ remains at (푝1, 푝2) = (푐, 푐), as in the simultaneous-move version of the Bertrand model.

Advanced Microeconomic Theory 61 Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products • Assume that firms sell differentiated products, where firm 푗’s demand is 푞푗 = 퐷푗(푝푗, 푝푘) – Example: 푞푗(푝푗, 푝푘) = 푎 − 푏푝푗 + 푐푝푘, where 푎, 푏, 푐 > 0 and 푏 > 푐 • In the second stage, firm 2 (the follower) solves following PMP max 휋2 = 푝2푞2 − 푇퐶(푞2) 푝2≥0 = 푝2퐷2(푝2, 푝1) − 푇퐶(퐷2(푝2, 푝1)) 푞2 Advanced Microeconomic Theory 62 Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products

• FOCs wrt 푝2 yield 휕퐷2(푝2, 푝1) 퐷2(푝2, 푝1) + 푝2 휕푝2 휕푇퐶 퐷 (푝 , 푝 ) 휕퐷 (푝 , 푝 ) − 2 2 1 2 2 1 = 0 휕퐷2(푝2, 푝1) 휕푝2 Using the chain rule

• Solving for 푝2 produces the follower’s BRF for every price set by the leader, 푝1, i.e., 푝2(푝1).

Advanced Microeconomic Theory 63 Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products • In the first stage, firm 1 (leader) anticipates that the follower will use BRF 푝2(푝1) to respond to each possible price 푝1, hence solves following PMP

max 휋1 = 푝1푞1 − 푇퐶 푞1 푝1≥0 푞1

= 푝1퐷1 푝1, 푝2 푝1 − 푇퐶 퐷1 푝1, 푝2(푝1)

퐵푅퐹2

Advanced Microeconomic Theory 64 Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products

• FOCs wrt 푝1 yield 휕퐷1(푝1, 푝2) 휕퐷1(푝1, 푝2) 휕푝2(푝1) 퐷1(푝1, 푝2) + 푝1 + 휕푝1 휕푝2(푝1) 휕푝1 New Strategic Effect 휕푇퐶 퐷 (푝 , 푝 ) 휕퐷 (푝 , 푝 ) 휕퐷 (푝 , 푝 ) 휕푝 (푝 ) − 1 1 2 1 1 2 + 1 1 2 2 1 = 0 휕퐷1(푝1, 푝2) 휕푝1 휕푝2(푝1) 휕푝1 New Strategic Effect • Or more compactly as 휕푇퐶 퐷1(푝1, 푝2) 휕퐷1(푝1, 푝2) 휕푝2(푝1) 퐷1(푝1, 푝2) + 푝1 − 1 + = 0 휕퐷1(푝1, 푝2) 휕푝1 휕푝1 푁푒푤

Advanced Microeconomic Theory 65 Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products • In contrast to the Bertrand model with simultaneous price competition, an increase in firm 1’s price now produces an increase in firm 2’s price in the second stage. • Hence, the leader has more incentives to raise its price, ultimately softening the price competition. • While a softened competition benefits both the leader and the follower, the real beneficiary is the follower, as its profits increase more than the leader’s.

Advanced Microeconomic Theory 66 Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products • Example:

– Consider a linear demand 푞푖 = 1 − 2푝푖 + 푝푗, with no marginal costs, i.e., 푐 = 0. – Simultaneous Bertrand model: the PMP is max 휋푗 = 푝푗 ∙ (1 − 2푝푗 + 푝푘) for any 푘 ≠ 푗 푝푗≥0

where FOC wrt 푝푗 produces firm 푗’s BRF 1 1 푝 (푝 ) = + 푝 푗 푘 4 4 푘 1 – Simultaneously solving the two BRFs yields 푝∗ = ≃ 푗 3 2 0.33, entailing equilibrium profits of 휋∗ = ≃ 0.222. 푗 9 Advanced Microeconomic Theory 67 Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products • Example (continued): – Sequential Bertrand model: in the second stage, firm 2’s (the follower’s) PMP is max 휋2 = 푝2 ∙ 1 − 2푝2 − 푝1 푝2≥0 where FOC wrt 푝2 produces firm 2’s BRF 1 1 푝 (푝 ) = + 푝 2 1 4 4 1 – In the first stage, firm 1’s (the leader’s) PMP is 1 1 1 max 휋1 = 푝1 ∙ 1 − 2푝1 + + 푝1 = 푝1 ∙ (5 − 7푝1) 푝1≥0 4 4 4

퐵푅퐹2 Advanced Microeconomic Theory 68 Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products • Example (continued):

– FOC wrt 푝1, and solving for 푝1, produces firm 1’s 5 equilibrium price 푝∗ = = 0.36. 1 14 ∗ – Substituting 푝1 into the BRF of firm 2 yields 1 1 푝∗ 0.36 = + 0.36 = 0.34. 2 4 4 – Equilibrium profits are hence 1 휋∗ = 0.36 5 − 7 0.36 = 0.223 for firm 1 1 4 ∗ 휋2 = 0.34 1 − 2 0.34 + 0.36 = 0.230 for firm 2

Advanced Microeconomic Theory 69 Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products • Example (continued): p1 – Both firms’ prices and

profits are higher in p2(p1)

the sequential than in Prices with sequential price competition the simultaneous 0.36 p1(p2) game. ⅓ Prices with simultaneous – price competition However, the follower ¼ earns more than the leader in the sequential game! p2 (second mover’s ¼ ⅓ 0.34 advantage) Advanced Microeconomic Theory 70 Dynamic Competition: Sequential Cournot Model with Homogenous Products • Stackelberg model: firm 1 (the leader) chooses output level 푞1, and firm 2 (the follower) observing the output decision of the leader, responds with its own output 푞2(푞1). • By backward induction, the follower’s BRF is 푞2(푞1) for any 푞1. • Since the leader anticipates 푞2(푞1) from the follower, the leader’s PMP is max 푝 푞1 + 푞2(푞1) 푞1 − 푇퐶1(푞1) 푞1≥0 퐵푅퐹2

Advanced Microeconomic Theory 71 Dynamic Competition: Sequential Cournot Model with Homogenous Products

• FOCs wrt 푞1 yields ′ 휕푞2(푞1) 푝 푞1 + 푞2(푞1) + 푝 푞1 + 푞2(푞1) 푞1 + 푞1 휕푞1 휕푇퐶 (푞 ) − 1 1 = 0 휕푞1 or more compactly

′ ′ 휕푞2(푞1) 휕푇퐶1 푞1 푝 푄 + 푝 푄 푞1 + 푝 푄 푞1 − = 0 휕푞1 휕푞1 Strategic Effect • This FOC coincides with that for standard Cournot model with simultaneous output decisions, except for the strategic effect. Advanced Microeconomic Theory 72 Dynamic Competition: Sequential Cournot Model with Homogenous Products • The strategic effect is positive since 푝′(푄) < 0 휕푞 (푞 ) and 2 1 < 0. 휕푞1 • Firm 1 (the leader) has more incentive to raise 푞1 relative to the Cournot model with simultaneous output decision. • Intuition (first-mover advantage): – By overproducing, the leader forces the follower to 휕푞2(푞1) reduce its output 푞2 by the amount . 휕푞1 – This helps the leader sell its production at a higher price, as reflected by 푝′(푄); ultimately earning a larger profit than in the standard Cournot model.

Advanced Microeconomic Theory 73 Dynamic Competition: Sequential Cournot Model with Homogenous Products • Example: – Consider linear inverse demand 푝 = 푎 − 푄, where 푄 = 푞1 + 푞2, and a constant marginal cost of 푐. – Firm 2’s (the follower’s) PMP is

max (푎 − 푞1 − 푞2)푞2 − 푐푞2 푞2 – FOC: 푎 − 푞1 − 2푞2 − 푐 = 0 – Solving for 푞2 yields the follower’s BRF 푎−푞 −푐 푞 푞 = 1 2 1 2

Advanced Microeconomic Theory 74 Dynamic Competition: Sequential Cournot Model with Homogenous Products • Example (continued): – Plugging 푞2 푞1 into the leader’s PMP, we get 푎−푞1−푐 1 max 푎 − 푞1 − 푞1 − 푐푞1 = (푎 − 푞1 − 푐) 푞1 2 2 – FOC: 1 푎 − 2푞 − 푐 = 0 2 1 – Solving for 푞1, we obtain the leader’s equilibrium 푎−푐 output level 푞∗ = . 1 2 ∗ – Substituting 푞1 into the follower’s BRF yields the 푎−푐 follower’s equilibrium output 푞∗ = . 2 4 Advanced Microeconomic Theory 75 Dynamic Competition: Sequential Cournot Model with Homogenous Products

q1

q2(q1)

Stackelberg Quantities a – c 2 Cournot Quantities

q1(q2) q a – c 2 2 Advanced Microeconomic Theory 76 Dynamic Competition: Sequential Cournot Model with Homogenous Products • Example (continued): – The equilibrium price is 푎 + 3푐 푝 = 푎 − 푞∗ − 푞∗ = 1 2 4 – And the resulting equilibrium profits are 푎+3푐 푎−푐 푎−푐 푎−푐 2 휋∗ = − 푐 = 1 4 2 2 8 푎+3푐 푎−푐 푎−푐 푎−푐 2 휋∗ = − 푐 = 2 4 4 4 16

Advanced Microeconomic Theory 77 Dynamic Competition: Sequential Cournot Model with Homogenous Products Price • Linear inverse demand 푝 푄 = 푎 − 푄 a • Symmetric marginal costs 푐 > 0 pm = a + c Monopoly 2

pCournot = a + 2c Cournot 3

pStackelberg = a + 3c Stackelberg 4

Bertrand and Perfect Competition pP.C. =pBertrand = c

Units a – c 2(a – c) 3(a – c) a – c a 2b 3b 4b b b Advanced Microeconomic Theory 78 Dynamic Competition: Sequential Cournot Model with Heterogeneous Products • Assume that firms sell differentiated products, with inverse demand curves for firms 1 and 2

푝1(푞1, 푞2) = 훼 − 훽푞1 − 훾푞2 for firm 1 푝2(푞1, 푞2) = 훼 − 훾푞1 − 훽푞2 for firm 2 • Firm 2’s (the follower’s) PMP is

max (훼 − 훾푞1 − 훽푞2) ∙ 푞2 푞2 where, for simplicity, we assume no marginal costs. • FOC: 훼 − 훾푞1 − 2훽푞2 = 0

Advanced Microeconomic Theory 79 Dynamic Competition: Sequential Cournot Model with Heterogeneous Products

• Solving for 푞2 yields firm 2’s BRF 훼−훾푞 푞 (푞 ) = 1 2 1 2훽 • Plugging 푞2 푞1 into the leader’s firm 1’s (the leader’s) PMP, we get

훼−훾푞1 max 훼 − 훽푞1 − 훾 푞1 = 푞1 2훽 2훽−훾 2훽2−훾2 max 훼 − 푞1 푞1 푞1 2훽 2훽 • FOC: 2훽−훾 2훽2−훾2 훼 − 푞 = 0 2훽 훽 1 Advanced Microeconomic Theory 80 Dynamic Competition: Sequential Cournot Model with Heterogeneous Products

• Solving for 푞1, we obtain the leader’s equilibrium 훼(2훽−훾) output level 푞∗ = 1 2(2훽2−훾2) ∗ • Substituting 푞1 into the follower’s BRF yields the follower’s equilibrium output 훼−훾푞∗ 훼(4훽2−2훽훾−훾2) 푞∗ = 1 = 2 2훽 4훽(2훽2−훾2) • Note: ∗ ∗ – 푞1 > 푞2 – If 훾 → 훽 (i.e., the products become more homogeneous), ∗ ∗ (푞1, 푞2) convege to the standard Stackelberg values. – If 훾 → 0 (i.e., the products become very differentiated), 훼 (푞∗, 푞∗) converge to the monopoly output 푞푚 = . 1 2 2훽

Advanced Microeconomic Theory 81 Capacity Constraints

Advanced Microeconomic Theory 82 Capacity Constraints

• How come are equilibrium outcomes in the standard Bertrand and Cournot models so different? • Do firms really compete in prices without facing capacity constraints? – Bertrand model assumes a firm can supply infinitely large amount if its price is lower than its rivals. • Extension of the Bertrand model: – First stage: firms set capacities, 푞ത1 and 푞ത2, with a cost of capacity 푐 > 0 – Second stage: firms observe each other’s capacities and compete in prices, simultaneously setting 푝1 and 푝2 Advanced Microeconomic Theory 83 Capacity Constraints

• What is the role of capacity constraint? – When a firm’s price is lower than its capacity, not all consumers can be served. – Hence, sales must be rationed through efficient rationing: the customers with the highest willingness to pay get the product first. • Intuitively, if 푝1 < 푝2 and the quantity demanded at 푝1 is so large that 푄(푝1) > 푞ത1, then the first 푞ത1 units are served to the customers with the highest willingness to pay (i.e., the upper segment of the demand curve), while some customers are left in the form of residual demand to firm 2. Advanced Microeconomic Theory 84 Capacity Constraints

p q , firm 1's capacity • At 푝1 the quantity 1 demanded is 푄(푝1), but only 푞ത1 units can be served. • Hence, the residual demand is 푄(푝1) − 푞ത1. 4th • Since firm 2 sets a p2 price of 푝2, its 1st demand will be p1 5th 푄(푝 ). 2nd Q(p) 2 3rd • Thus, a portion of the q q 1 Q(p2) Q(p1) Unserved customers by firm 1 residual demand , 6th 푄(푝 ) − 푞ത These units become residual i.e., 2 1, is demand for firm 2. captured. Q2(p2) – q1 Advanced Microeconomic Theory 85 Capacity Constraints

• Hence, firm 2’s residual demand can be expressed as 푄 푝 − 푞ത 푄 푝 − 푞ത ≥ 0 ቊ 2 1 if 2 1 0 otherwise

• Should we restrict 푞ത1 and 푞ത2 somewhat? – Yes. A firm will never set a huge capacity if such capacity entails negative profits, independently of the decision of its competitor.

Advanced Microeconomic Theory 86 Capacity Constraints

• How to express this rather obvious statement with a simple mathematical condition? – The maximal revenue of a firm under monopoly is 푎 max (푎 − 푞)푞, which is maximized at 푞 = , yielding 푞 2 푎2 profits of . 4 푎2 – Maximal revenues are larger than costs if ≥ 푐푞ത , or 4 푗 푎2 solving for 푞ത , ≥ 푞ത . 푗 4푐 푗 – Intuitively, the capacity cannot be too high, as otherwise the firm would not obtain positive profits regardless of the opponent’s decision. Advanced Microeconomic Theory 87 Capacity Constraints: Second Stage

• By backward induction, we start with the second stage (pricing game), where firms simultaneously choose prices 푝1 and 푝2 as a function of the capacity choices 푞ത1 and 푞ത2. • We want to show that in this second stage, both firms set a common price ∗ 푝1 = 푝2 = 푝 = 푎 − 푞ത1 − 푞ത2 where demand equals supply, i.e., total capacity, ∗ 푝 = 푎 − 푄ത, where 푄ത ≡ 푞ത1 + 푞ത2

Advanced Microeconomic Theory 88 Capacity Constraints: Second Stage

• In order to prove this result, we start by assuming ∗ that firm 1 sets 푝1 = 푝 . We now need to show ∗ that firm 2 also sets 푝2 = 푝 , i.e., it does not have incentives to deviate from 푝∗. ∗ • If firm 2 does not deviate, 푝1 = 푝2 = 푝 , then it sells up to its capacity 푞ത2. • If firm 2 reduces its price below 푝∗, demand would exceed its capacity 푞ത2. As a result, firm 2 would sell the same units as before, 푞ത2, but at a lower price.

Advanced Microeconomic Theory 89 Capacity Constraints: Second Stage

• If, instead, firm 2 charges a price above 푝∗, then ∗ 푝1 = 푝 < 푝2 and its revenues become 푝 (푎 − 푝 − 푞ത ) if 푎 − 푝 − 푞ത ≥ 0 푝 푄෠(푝 ) = ቊ 2 2 1 2 1 2 2 0 otherwise • Note: – This is fundamentally different from the standard Bertrand model without capacity constraints, where an increase in price by a firm reduces its sales to zero. – When capacity constraints are present, the firm can still capture a residual demand, ultimately raising its revenues after increasing its price.

Advanced Microeconomic Theory 90 Capacity Constraints: Second Stage

• We now find the maximum of this revenue function. FOC wrt 푝2 yields: 푎 − 푞ത 푎 − 2푝 − 푞ത = 0 ⟺ 푝 = 1 2 1 2 2 ∗ • The non-deviating price 푝 = 푎 − 푞ത1 − 푞ത2 lies above 푎−푞ത the maximum-revenue price 푝 = 1 when 2 2 푎 − 푞ത 푎 − 푞ത − 푞ത > 1 ⟺ 푎 > 푞ത + 2푞ത 1 2 2 1 2 푎2 • Since ≥ 푞ത (capacity constraint), we can obtain 4푐 푗 푎2 푎2 3푎2 + 2 > 푞ത + 2푞ത ⇔ > 푞ത + 2푞ത 4푐 4푐 1 2 4푐 1 2

Advanced Microeconomic Theory 91 Capacity Constraints: Second Stage

3푎2 • Therefore, 푎 > 푞ത1 + 2푞ത2 holds if 푎 > which, 4푐 4푐 solving for 푎, is equivalent to > 푎. 3

Advanced Microeconomic Theory 92 Capacity Constraints: Second Stage

4푐 • When > 푎 holds, 3 푎2 capacity constraint ≥ 4푐 3푎2 푞ത transforms into > 푗 4푐 ∗ 푞ത1 + 2푞ത2, implying 푝 > 푞ത 푝 = 푎 − 1. 2 2 • Thus, firm 2 does not have incentives to increase its price 푝2 from 푝∗, since that would lower its revenues.

Advanced Microeconomic Theory 93 Capacity Constraints: Second Stage

• In short, firm 2 does not have incentives to deviate from the common price ∗ 푝 = 푎 − 푞ത1 − 푞ത2 • A similar argument applies to firm 1 (by symmetry). • Hence, we have found an equilibrium in the pricing stage.

Advanced Microeconomic Theory 94 Capacity Constraints: First Stage

• In the first stage (capacity setting), firms simultaneously select their capacities 푞ത1 and 푞ത2. • Inserting stage 2 equilibrium prices, i.e., ∗ 푝1 = 푝2 = 푝 = 푎 − 푞ത1 − 푞ത2, into firm 푗’s profit function yields 휋푗(푞ത1, 푞ത2) = (푎 − 푞ത1 − 푞ത2)푞ത푖 − 푐푞ത푖 푝∗

• FOC wrt capacity 푞ത푗 yields firm 푗’s BRF 푎 − 푐 1 푞ത (푞ത ) = − 푞ത 푗 푘 2 2 푘 Advanced Microeconomic Theory 95 Capacity Constraints: First Stage

• Solving the two BRFs simultaneously, we obtain a symmetric solution 푎 − 푐 푞ത = 푞ത = 푗 푘 3 • These are the same equilibrium predictions as those in the standard Cournot model. • Hence, capacities in this two-stage game coincide with output decisions in the standard Cournot model, while prices are set equal to total capacity.

Advanced Microeconomic Theory 96 Endogenous Entry

Advanced Microeconomic Theory 97 Endogenous Entry

• So far the number of firms was exogenous • What if the number of firms operating in a market is endogenously determined? • That is, how many firms would enter an industry where – They know that competition will be a la Cournot – They must incur a fixed entry cost 퐹 > 0.

Advanced Microeconomic Theory 98 Endogenous Entry

• Consider inverse demand function 푝(푞), where 푞 denotes aggregate output

• Every firm 푗 faces the same total cost function, 푐(푞푗), of producing 푞푗 units • Hence, the Cournot equilibrium must be symmetric – Every firm produces the same output level 푞(푛), which is a function of the number of entrants. • Entry profits for firm 푗 are

휋푗 푛 = 푝 푛 ∙ 푞 푛 푞 푛 − 푐 푞 푛 − 퐹ณ 푄 Production Costs Fixed Entry Cost 푝(푄)

Advanced Microeconomic Theory 99 Endogenous Entry

• Three assumptions (valid under most demand and cost functions): – individual equilibrium output 푞(푛) is decreasing in 푛; – aggregate output 푞 ≡ 푛 ∙ 푞(푛) increases in 푛; – equilibrium price 푝(푛 ∙ 푞(푛)) remains above marginal costs regardless of the number of entrants 푛.

Advanced Microeconomic Theory 100 Endogenous Entry

• Equilibrium number of firms: – The equilibrium occurs when no more firms have incentives to enter or exit the market, i.e., 푒 휋푗(푛 ) = 0. – Note that individual profits decrease in 푛, i.e., − + 휕푞(푛) 휋′ 푛 = 푝 푛푞 푛 − 푐′ 푞 푛 휕푛 휕[푛푞 푛 ] + 푞 푛 푝′ 푛푞 푛 < 0 휕푛 − +

Advanced Microeconomic Theory 101 Endogenous Entry

• Social optimum: – The social planner chooses the number of entrants 푛표 that maximizes social welfare 푛푞(푛) max 푊 푛 ≡ න 푝 푠 푑푠 − 푛 ∙ 푐 푞 푛 − 푛 ∙ 퐹 푛 0

Advanced Microeconomic Theory 102 Endogenous Entry

푛푞(푛) p = ׬0 푝 푠 푑푠 • 퐴 + 퐵 + 퐶 + 퐷

• 푛 ∙ 푐 푞 푛 = n c (q) A 퐶 + 퐷 p(n q(n)) • Social welfare is B

thus 퐴 + 퐵 minus n c (q(n)) C total entry costs D p(Q) Q 푛 ∙ 퐹 n q(n)

Advanced Microeconomic Theory 103 Endogenous Entry

– FOC wrt 푛 yields 휕푞 푛 휕푞 푛 푝 푛푞 푛 푛 + 푞 푛 − 푐 푞 푛 − 푛푐′ 푞 푛 − 퐹 = 0 휕푛 휕푛 or, re-arranging, 휕푞(푛) 휋 푛 + 푛 푝 푛푞 푛 − 푐′ 푞 푛 = 0 휕푛 – Hence, marginal increase in 푛 entails two opposite effects on social welfare: a) the profits of the new entrant increase social welfare (+, appropriability effect) b) the entrant reduces the profits of all previous incumbents in the industry as the individual sales of each firm decreases upon entry (-, business stealing effect) Advanced Microeconomic Theory 104 Endogenous Entry

• The “business stealing” effect is represented by: 휕푞(푛) 푛 푝 푛푞 푛 − 푐′ 푞 푛 < 0 휕푛 휕푞(푛) which is negative since < 0 and 휕푛 푛 푝 푛푞 푛 − 푐′ 푞 푛 > 0 by definition.

• Therefore, an additional entry induces a 휕푞(푛) reduction in aggregate output by 푛 , which in 휕푛 turn produces a negative effect on social welfare.

Advanced Microeconomic Theory 105 Endogenous Entry

• Given the negative sign of the business stealing effect, we can conclude that 휕푞 푛 푊′ 푛 = 휋 푛 + 푛 푝 푛푞 푛 − 푐′ 푞 푛 < 휋(푛) 휕푛 − and therefore more firms enter in equilibrium than in the social optimum, i.e., 푛푒 > 푛표.

Advanced Microeconomic Theory 106 Endogenous Entry

Advanced Microeconomic Theory 107 Endogenous Entry

• Example: – Consider a linear inverse demand 푝 푄 = 1 − 푄 and no marginal costs. – The equilibrium quantity in a market with 푛 firms that compete a la Cournot is 1 푞 푛 = 푛+1 – Let’s check if the three assumptions from above hold.

Advanced Microeconomic Theory 108 Endogenous Entry

• Example (continued): – First, individual output decreases with entry 휕푞 푛 1 = − < 0 휕푛 푛+1 2 – Second, aggregate output 푛푞(푛) increases with entry 휕 푛푞 푛 1 = > 0 휕푛 푛+1 2 – Third, price lies above marginal cost for any number of firms 1 1 푝 푛 − 푐 = 1 − 푛 ∙ = > 0 for all 푛 푛+1 푛+1 Advanced Microeconomic Theory 109 Endogenous Entry

• Example (continued): – Every firm earns equilibrium profits of 1 1 1 휋 푛 = − 퐹 = − 퐹 푛 + 1 푛 + 1 푛 + 1 2 푝(푛) 푞(푛) 1 – Since equilibrium profits after entry, , is 푛+1 2 smaller than 1 even if only one firm enters the industry, 푛 = 1, we assume that entry costs are lower than 1, i.e., 퐹 < 1.

Advanced Microeconomic Theory 110 Endogenous Entry

• Example (continued): – Social welfare is 푛 푛+1 푊 푛 = න (1 − 푠)푑푠 − 푛 ∙ 퐹 0 푛 푠 푛+1 = 푠 − ฬ − 푛 ∙ 퐹 2 0 2 푛 푛 + 2 1 = − 푛 ∙ 퐹 2 푛 + 1

Advanced Microeconomic Theory 111 Endogenous Entry

• Example (continued): – The number of firms entering the market in equilibrium, 푛푒, is that solving 휋 푛푒 = 0, 1 1 − 퐹 = 0 ⟺ 푛푒 = − 1 푛푒 + 1 2 퐹 whereas the number of firms maximizing social welfare, i.e., 푛표 solving 푊′ 푛표 = 0, 1 1 푊′ 푛표 = = 0 ⟺ 푛표 = − 1 푛표 + 1 3 3 퐹 where 푛푒 < 푛표 for all admissible values of 퐹, i.e., 퐹 ∈ 0,1 .

Advanced Microeconomic Theory 112 Endogenous Entry

• Example (continued): Number of firms

1 ne = – 1 (Equilibrium) F ½

1 no = – 1 (Soc. Optimal) F ⅓

0 Entry costs, F Advanced Microeconomic Theory 113