Advanced Microeconomics Theory Chapter 8: Imperfect Competition
Outline Basic Game Theory Bertrand Model of Price Competition Cournot Model of Quantity Competition Product Differentiation Dynamic Competition Capacity Constraints Endogenous Entry Repeated Interaction Advanced Microeconomic Theory 2
Introduction Monopoly: a single firm Oligopoly: a limited number of firms When allowing for NN firms, the equilibrium predictions embody the results in perfectly competitive and monopoly markets as special cases. Advanced Microeconomic Theory 3
Basic Game Theory Advanced Microeconomic Theory 4
Basic Game Theory Consider a setting with JJ firms, each with a strategy ss jj Example: an output level, a price, or an advertising expenditure. Let (ss jj, ss jj ) denote a strategy profile where ss jj represents the strategies selected by all firms kk jj, i.e., ss jj = (ss 1,, ss jj 1, ss jj+1,, ss JJ ). Strategy ss jj strictly dominates another strategy ss jj ss jj for firm jj if ππ jj (ss jj, ss jj ) > ππ jj (ss jj, ss jj ) for all ss jj That is, ss jj yields a higher profit than ss jj does, regardless of the strategy ss jj selected by all of firm jj s rivals. Advanced Microeconomic Theory 5
Basic Game Theory Payoff matrix (Normal Form Game) Firm A Firm B Low prices High prices Low prices 5, 5 9, 1 High prices 1, 9 7, 7 Low prices yields a higher payoff than high prices both when a firm s rival chooses low prices and when it selects high prices. Low prices is a strictly dominant strategy for both firms (i.e., ss jj ). High prices is referred to as a strictly dominated strategy (i.e., ss jj ). Advanced Microeconomic Theory 6
Basic Game Theory A strictly dominated strategy can be deleted from the set of strategies a rational player would use. This helps to reduce the number of strategies to consider as optimal for each player. In the above payoff matrix, both firms will select low prices in the unique equilibrium of the game. However, games do not always have a strictly dominated strategy. Advanced Microeconomic Theory 7
Basic Game Theory Firm A Firm B L R U 3, 1 0, 0 D 0, 0 1, 3 UU is better than DD for firm AA if its opponent selects LL, but becomes worse otherwise. Similarly, LL is better than RR for firm BB if its opponent chooses UU, but becomes worse otherwise. Hence, no strictly dominated strategies for either player. Advanced Microeconomic Theory 8
Basic Game Theory What is the equilibrium of the game then? The Nash equilibrium solution concept A strategy profile (ss jj, ss jj ) is a Nash equilibrium (NE) if, for every player jj, ππ jj ss jj, ss jj ππ jj ss jj, ss jj for every ss jj ss jj That is, ss jj is player jj s best response to his opponents choosing ss jj as ss jj yields a better payoff than any of his available strategies ss jj ss jj. Advanced Microeconomic Theory 9
Basic Game Theory In the above game: Firm AA s best response to firm BB s playing LL is BBRR AA LL = UU, while to firm BB playing RR is BBRR AA RR = DD. Similarly, firm BB s best response to firm AA choosing UU is BBRR BB UU = LL, whereas to firm AA selecting DD is BBRR BB DD = RR. Hence, strategy profiles (UU, LL) and (DD, RR) are mutual best responses (i.e., the two Nash equilibria). Advanced Microeconomic Theory 10
Bertrand Model of Price Competition Advanced Microeconomic Theory 11
Bertrand Model of Price Competition Consider: An industry with two firms, 1 and 2, selling a homogeneous product Firms face market demand xx(pp), where xx(pp) is continuous and strictly decreasing in pp There exists a high enough price (choke price) pp < such that xx(pp) = 0 for all pp > pp Both firms are symmetric in their constant marginal cost cc > 0, where xx cc (0, ) Every firm jj simultaneously sets a price pp jj Advanced Microeconomic Theory 12
Bertrand Model of Price Competition Firm jj s demand is xx jj (pp jj, pp kk ) = xx(pp jj ) if pp jj < pp kk 1 2 xx(pp jj) if pp jj = pp kk 0 if pp jj > pp kk Intuition: Firm jj captures all market if its price is the lowest, pp jj < pp kk no market if its price is the highest, pp jj > pp kk shares the market with firm kk if the price of both firms coincide, pp jj = pp kk Advanced Microeconomic Theory 13
Bertrand Model of Price Competition Given prices pp jj and pp kk, firm jj s profits are therefore (pp jj cc) xx jj (pp jj, pp kk ) We are now ready to find equilibrium prices in the Bertrand duopoly model. There is a unique NE (pp jj, pp kk ) in the Bertrand duopoly model. In this equilibrium, both firms set prices equal to marginal cost, pp jj = pp kk = cc. Advanced Microeconomic Theory 14
Bertrand Model of Price Competition Let s us describe the best response function of firm jj. If pp kk < cc, firm jj sets its price at pp jj = cc. Firm jj does not undercut firm kk since that would entail negative profits. If cc < pp kk < pp jj, firm jj slightly undercuts firm kk, i.e., pp jj = pp kk εε. This allows firm jj to capture all sales and still make a positive margin on each unit. If pp kk > pp mm, where pp mm is a monopoly price, firm jj does not need to charge more than pp mm, i.e., pp jj = pp mm. pp mm allows firm jj to capture all sales and maximize profits as the only firm selling a positive output. Advanced Microeconomic Theory 15
Bertrand Model of Price Competition Firm jj s best response has: a flat segment for all pp kk < cc, where pp jj (pp kk ) = cc a positive slope for all cc < pp kk < pp jj, where firm jj charges a price slightly below firm kk a flat segment for all pp kk > pp mm, where pp jj (pp kk ) = pp mm p m c p j c p m 45 -line (p j = p k ) p j (p k ) p k Advanced Microeconomic Theory 16
Bertrand Model of Price Competition A symmetric argument applies to the construction of the best response function of firm kk. A mutual best response for both firms is (pp 1, pp 2 ) = (cc, cc) where the two best response functions cross each other. This is the NE of the Bertrand model Firms make no economic profits. p m c p j c p k (p j ) p m 45 -line (p j = p k ) p j (p k ) p k Advanced Microeconomic Theory 17
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 jj face an infinitely elastic demand curve at its rival s price, pp kk. Any increase (decrease) from pp kk infinitely reduces (increases, respectively) firm jj s demand. Advanced Microeconomic Theory 18
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 19
Bertrand Model of Price Competition Remark: How our results would be affected if firms face different production costs, i.e., 0 < cc 1 < cc 2? The most efficient firm sets a price equal to the marginal cost of the least efficient firm, pp 1 = cc 2. Other firms will set a random price in the uniform interval [cc 1, cc 1 + ηη] where ηη > 0 is some small random increment with probability distribution ff pp, ηη > 0 for all pp. Advanced Microeconomic Theory 20
Cournot Model of Quantity Competition Advanced Microeconomic Theory 21
Cournot Model of Quantity Competition Let us now consider that firms compete in quantities. Assume that: Firms bring their output qq 1 and qq 2 to a market, the market clears, and the price is determined from the inverse demand function pp(qq), where qq = qq 1 + qq 2. pp(qq) satisfies ppp(qq) < 0 at all output levels qq 0, Both firms face a common marginal cost cc > 0 pp(0) > cc in order to guarantee that the inverse demand curve crosses the constant marginal cost curve at an interior point. Advanced Microeconomic Theory 22
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, qq 2, max qq 1 0 pp qq 1 + qq 2 Price qq 1 ccqq 1 When solving this PMP, firm 1 treats firm 2 s production, qq 2, as a parameter, since firm 1 cannot vary its level. Advanced Microeconomic Theory 23
Cournot Model of Quantity Competition FOCs: pp (qq 1 + qq 2 )qq 1 + pp(qq 1 + qq 2 ) cc 0 with equality if qq 1 > 0 Solving this expression for qq 1, we obtain firm 1 s best response function (BRF), qq 1 (qq 2 ). A similar argument applies to firm 2 s PMP and its best response function qq 2 (qq 1 ). Therefore, a pair of output levels (qq 1, qq 2 ) is NE of the Cournot model if and only if qq 1 qq 1 (qq 2 ) for firm 1 s output qq 2 qq 2 (qq 1 ) for firm 2 s output Advanced Microeconomic Theory 24
Cournot Model of Quantity Competition To show that qq 1, qq 2 > 0, let us work by contradiction, assuming qq 1 = 0. Firm 2 becomes a monopolist since it is the only firm producing a positive output. Using the FOC for firm 1, we obtain pp (0 + qq 2 )0 + pp(0 + qq 2 ) cc or pp(qq 2 ) cc And using the FOC for firm 2, we have pp (qq 2 + 0)qq 2 + pp(qq 2 + 0) cc or pp (qq 2 )qq 2 + pp(qq 2 ) cc This implies firm 2 s MR under monopoly is lower than its MC. Thus, qq 2 = 0. Advanced Microeconomic Theory 25
Cournot Model of Quantity Competition Hence, if qq 1 = 0, firm 2 s output would also be zero, qq 2 = 0. But this implies that pp(0) < cc since no firm produces a positive output, thus violating our initial assumption pp(0) > cc. Contradiction! As a result, we must have that both qq 1 > 0 and qq 2 > 0. Note: This result does not necessarily hold when both firms are asymmetric in their production cost. Advanced Microeconomic Theory 26
Cournot Model of Quantity Competition Example (symmetric costs): Consider an inverse demand curve pp(qq) = aa bbbb, and two firms competing à la Cournot both facing a marginal cost cc > 0. Firm 1 s PMP is aa bb(qq 1 + qq 2 ) qq 1 ccqq 1 FOC wrt qq 1 : aa 2bbqq 1 bbqq 2 cc 0 with equality if qq 1 > 0 Advanced Microeconomic Theory 27
Cournot Model of Quantity Competition Example (continue): Solving for qq 1, we obtain firm 1 s BRF qq 1 (qq 2 ) = aa cc 2bb qq 2 2 Analogously, firm 2 s BRF qq 2 (qq 1 ) = aa cc 2bb qq 1 2 Advanced Microeconomic Theory 28
Cournot Model of Quantity Competition Firm 1 s BRF: When qq 2 = 0, then qq 1 = aa cc 2bb, which coincides with its output under monopoly. As qq 2 increases, qq 1 decreases (i.e., firm 1 s and 2 s output are strategic substitutes) When qq 2 = aa cc, then bb qq 1 = 0. Advanced Microeconomic Theory 29
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 (qq 1, qq 2 ). Advanced Microeconomic Theory 30
Cournot Model of Quantity Competition q 1 a c b Perfect competition q 1 + q 2 = q c = a c b 45 -line (q 1 = q 2 ) a c 2b a c 3b q 2 (q 1 ) (q* 1,q * 2 ) q 1 (q 2 ) Monopoly q 1 + q 2 = q m = a c 2b 45 a c 3b a c 2b a c b Advanced Microeconomic Theory 31 q 2
Cournot Model of Quantity Competition Cournot equilibrium output pair (qq 1, qq 2 ) occurs at the intersection of the two BRFs, i.e., (qq 1, qq 2 ) = Aggregate output becomes qq = qq 1 + qq 2 = aa cc 3bb aa cc, aa cc 3bb 3bb + aa cc 3bb = 2(aa cc) 3bb which is larger than under monopoly, qq mm = aa cc 2bb, but smaller than under perfect competition, qq cc = aa cc bb. Advanced Microeconomic Theory 32
Cournot Model of Quantity Competition The equilibrium price becomes pp qq = aa bbqq = aa bb 2 aa cc 3bb = aa+2cc 3 which is lower than under monopoly, pp mm = aa+cc 2, but higher than under perfect competition, pp cc = cc. Finally, the equilibrium profits of every firm jj ππ jj = pp qq qq jj ccqq jj = aa+2cc 3 aa cc 3bb cc aa cc 3bb = aa cc 2 9bb which are lower than under monopoly, ππ mm = aa cc 2 4bb, but higher than under perfect competition, ππ cc = 0. Advanced Microeconomic Theory 33
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 pp (qq 1 + qq 2 ). Hence, if firms produce the same output as under marginal cost pricing, i.e., half of aa cc, each firm would 2 have incentives to deviate from such a high output level by marginally reducing its output. Advanced Microeconomic Theory 34
Cournot Model of Quantity Competition Equilibrium output under Cournot does not coincide with the monopoly output either. That s because, every firm ii, individually increasing its output level qq ii, 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 35
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 (aa bb(qq 1 +qq 2 ))qq 1 ccqq 1 qq 1,qq 2 + (aa bb(qq 1 +qq 2 ))qq 2 ccqq 2 Since QQ = qq 1 + qq 2, the PMP can be written as max aa bb(qq 1 +qq 2 ) (qq 1 +qq 2 ) cc(qq 1 +qq 2 ) qq 1,qq 2 = max aa bbbb QQ cccc = aaaa bbqq 2 cccc QQ Advanced Microeconomic Theory 36
Cournot Model of Quantity Competition Example (continued): FOC with respect to QQ aa 2bbbb cc 0 Solving for QQ, we obtain the aggregate output QQ = aa cc 2bb which is positive since aa > cc, i.e., pp(0) = aa > cc. Since firms are symmetric in costs, each produces qq ii = QQ 2 = aa cc 4bb Advanced Microeconomic Theory 37
Cournot Model of Quantity Competition Example (continued): The equilibrium price is pp = aa bbbb = aa bb aa cc 2bb Finally, the equilibrium profits are = aa+cc 2 = aa+cc aa cc 2 4bb ππ ii = pp qq ii ccqq ii aa cc cc = aa cc 2 4bb 8bb which is larger than firms would obtain under Cournot competition, aa cc 2 9bb. Advanced Microeconomic Theory 38
Cournot Model of Quantity Competition: Cournot Pricing Rule Firms market power can be expressed using a variation of the Lerner index. Consider firm jj s profit maximization problem ππ jj = pp(qq)qq jj cc jj (qq jj ) FOC for every firm jj pp qq qq jj + pp qq cc jj = 0 or pp(qq) cc jj = pp qq qq jj Multiplying both sides by qq and dividing them by pp(qq) yield qq pp qq cc jj pp(qq) = pp qq qq jj pp(qq) qq Advanced Microeconomic Theory 39
Cournot Model of Quantity Competition: Cournot Pricing Rule Recalling 1 εε = pp qq qq pp qq, we have qq pp qq cc jj pp(qq) = 1 εε qq jj Defining αα jj qq jj qq or pp qq cc jj pp(qq) = 1 εε qq jj qq as firm jj s market share, we obtain pp qq cc jj pp(qq) = αα jj εε which is referred to as the Cournot pricing rule. Advanced Microeconomic Theory 40
Cournot Model of Quantity Competition: Note: Cournot Pricing Rule When αα jj = 1, implying that firm jj is a monopoly, the IEPR becomes a special case of the Cournot price rule. The larger the market share αα jj of a given firm, the larger the price markup of firm jj. The more inelastic demand εε is, the larger the price markup of firm jj. Advanced Microeconomic Theory 41
Cournot Model of Quantity Competition: Cournot Pricing Rule Example (Merger effects on Cournot Prices): Consider an industry with nn firms and a constantelasticity demand function qq(pp) = aapp 1, where aa > 0 and εε = 1. Before merger, we have pp BB cc pp BB = 1 nn ppbb = nnnn nn 1 After the merger of kk < nn firms nn kk + 1 firms remain in the industry, and thus pp AA cc pp AA = 1 nn kk + 1 ppaa = nn kk + 1 cc nn kk Advanced Microeconomic Theory 42
Cournot Model of Quantity Competition: Example (continued): Cournot Pricing Rule The percentage change in prices is %Δpp = ppaa pp BB pp BB = nn kk + 1 cc nn kk nnnn nn 1 nnnn nn 1 = kk 1 nn(nn kk) > 0 Hence, prices increase after the merger. Also, %Δpp increases as the number of merging firms kk increases Δpp = nn 1 nn nn kk 2 > 0 Advanced Microeconomic Theory 43
Cournot Model of Quantity Competition: Cournot Pricing Rule Example (continued): The percentage increase in price after the merger, %Δpp, as a function of the number of merging firms, kk. For simplicity, nn = 100. %Δp 0.20 0.10 %Δp(k) 20 40 60 80 100 k Advanced Microeconomic Theory 44
Cournot Model of Quantity Competition: SOC Let us check if the first order (necessary) conditions are also sufficient. Recall that FOCs are pp qq qq jj + pp qq cc jj (qq jj ) 0 Differentiating FOCs wrt qq jj yields pp qq qq jj + pp qq + pp qq cc jj (qq jj ) 0 pp qq < 0: by definition (negatively sloped inverse demand curve) cc jj (qq jj ) 0: by assumption (constant or increasing marginal costs) pp qq qq jj 0: as long as the demand curve decreases at a constant or decreasing rate Advanced Microeconomic Theory 45
Cournot Model of Quantity Competition: SOC Example (linear demand): The linear inverse demand curve is pp(qq) = aa bbbb and constant marginal cost is cc > 0. Since pp qq = bb < 0, pp qq = 0, cc qq = cc and cc (qq) = 0, the SOC reduces to 0 2bb 0 = 2bb < 0 where bb > 0 by definition. Hence the equilibrium output is indeed profit maximizing. Advanced Microeconomic Theory 46
Cournot Model of Quantity Competition: SOC Note that SOCs coincides with the crossderivative 2 ππ jj = pp qq qq qq jj qq kk qq jj + pp qq cc (qq jj ) kk = pp qq qq jj pp qq for all kk jj. Hence, the firm jj s BRF decreases in qq kk as long as pp qq qq jj pp qq < 0 That is, firm jj s BRF is negatively sloped. Advanced Microeconomic Theory 47
Cournot Model of Quantity Competition: Asymmetric Costs Assume that firm 1 and 2 s constant marginal costs of production differ, i.e., cc 1 > cc 2, so firm 2 is more efficient than firm 1. Assume also that the inverse demand function is pp QQ = aa bbbb, and QQ = qq 1 + qq 2. Firm ii s PMP is max aa bb(qq ii + qq jj ) qq ii cc ii qq ii qq ii FOC: aa 2bbqq ii bbqq jj cc ii = 0 Advanced Microeconomic Theory 48
Cournot Model of Quantity Competition: Asymmetric Costs Solving for qq ii (assuming an interior solution) yields firm ii s BRF qq ii (qq jj ) = aa cc ii 2bb qq jj 2 Firm 1 s optimal output level can be found by plugging firm 2 s BRF into firm 1 s qq 1 = aa cc 1 2bb 1 aa cc 2 2 2bb qq 1 2 qq 1 = aa 2cc 1 + cc 2 3bb Similarly, firm 2 s optimal output level is qq 2 = aa cc 2 2bb qq 1 2 = aa + cc 1 2cc 2 3bb Advanced Microeconomic Theory 49
Cournot Model of Quantity Competition: Asymmetric Costs If firm ii s costs are sufficiently high it will not produce at all. Firm 1: qq 1 0 if aa+cc 2 2 Firm 2: qq 2 0 if aa+cc 1 2 cc 1 cc 2 Thus, we can identify three different cases: If cc ii aa+cc jj for all firms ii = {1,2}, no firm produces a 2 positive output If cc ii aa+cc jj but cc 2 jj < aa+cc ii, then only firm jj produces 2 positive output If cc ii < aa+cc jj for all firms ii = {1,2}, both firms produce 2 positive output Advanced Microeconomic Theory 50
Cournot Model of Quantity Competition: Asymmetric Costs c 2 Only firm 1 produces c 1 = a + c 2 2 No firms produce 45 (c 1 = c 2 ) a c 2 = a + c 1 2 a 2 Both firms produce Only firm 2 produces a 2 a Advanced Microeconomic Theory 51 c 1
Cournot Model of Quantity Competition: Asymmetric Costs The output levels (qq 1, qq 2 ) also vary when (cc 1, cc 2 ) changes qq 1 = 2 = 1 cc 1 cc 2 qq 2 cc 1 = 1 3bb < 0 and qq 1 3bb > 0 3bb > 0 and qq 2 = 2 cc 2 3bb < 0 Intuition: Each firm s output decreases in its own costs, but increases in its rival s costs. Advanced Microeconomic Theory 52
Cournot Model of Quantity Competition: Asymmetric Costs BRFs for firms 1 and 2 when cc 1 > aa+cc 2 2 (i.e., only firm 2 produces). a c 2 b q 1 BRFs cross at the vertical axis where qq 1 = 0 and qq 2 > 0 (i.e., a corner solution) a c 1 2b q 1 (q 2 ) q 2 (q 1 ) (q* 1,q * 2 ) a c 1 b a c2 2b q 2 Advanced Microeconomic Theory 53
Cournot Model of Quantity Competition: JJ > 2 firms Consider JJ > 2 firms, all facing the same constant marginal cost cc > 0. The linear inverse demand curve is pp QQ = aa bbbb, where QQ = JJ qq kk. Firm ii s PMP is FOC: max qq ii aa bb qq ii + qq kk kk ii aa 2bbqq ii bb qq kk kk ii qq ii ccqq ii cc 0 Advanced Microeconomic Theory 54
Cournot Model of Quantity Competition: JJ > 2 firms Solving for qq ii, we obtain firm ii s BRF qq aa cc ii = 2bb 1 2 qq kk kk ii Since all firms are symmetric, their BRFs are also symmetric, implying qq 1 = qq 2 = = qq JJ. This implies kk ii qq kk = JJqq ii qq ii = JJ 1 qq ii. Hence, the BRF becomes qq aa cc ii = 2bb 1 2 JJ 1 qq ii Advanced Microeconomic Theory 55
Cournot Model of Quantity Competition: JJ > 2 firms Solving for qq ii qq aa cc ii = JJ + 1 bb which is also the equilibrium output for other JJ 1 firms. Therefore, aggregate output is QQ = JJqq ii = JJ aa cc JJ + 1 bb and the corresponding equilibrium price is pp = aa bbqq aa + JJJJ = JJ + 1 Advanced Microeconomic Theory 56
Cournot Model of Quantity Competition: JJ > 2 firms Firm ii s equilibrium profits are ππ ii = aa bbqq qq ii ccqq ii JJ aa cc aa cc = aa bb JJ + 1 bb JJ + 1 bb aa cc aa cc 2 cc = = qq 2 JJ + 1 bb JJ + 1 bb ii Advanced Microeconomic Theory 57
Cournot Model of Quantity Competition: We can show that lim qq JJ 2 ii = lim JJ 2 QQ = lim JJ 2 pp = JJ > 2 firms aa cc 2 + 1 bb 2(aa cc) 2 + 1 bb aa + 2cc = aa cc 3bb = 2(aa cc) 3bb = aa + 2cc 3 2 + 1 which exactly coincide with our results in the Cournot duopoly model. Advanced Microeconomic Theory 58
Cournot Model of Quantity Competition: We can show that lim qq JJ 1 ii = lim JJ 1 QQ = lim JJ 1 pp = JJ > 2 firms aa cc 1 + 1 bb 1(aa cc) 1 + 1 bb aa + 1cc = aa cc 2bb = aa cc 2bb = aa + cc 2 1 + 1 which exactly coincide with our findings in the monopoly. Advanced Microeconomic Theory 59
Cournot Model of Quantity Competition: JJ > 2 firms We can show that lim qq JJ ii = 0 aa cc lim JJ QQ = bb lim JJ pp = cc which coincides with the solution in a perfectly competitive market. Advanced Microeconomic Theory 60
Product Differentiation Advanced Microeconomic Theory 61
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 62
Product Differentiation: Bertrand Model Consider the case where every firm ii, for ii = {1,2}, faces demand curve qq ii (pp ii, pp jj ) = aa bbpp ii + ccpp jj where aa, bb, cc > 0 and jj ii. Hence, an increase in pp jj increases firm ii s sales. Firm ii s PMP: max pp ii 0 FOC: (aa bbpp ii + ccpp jj )pp ii aa 2bbpp ii + ccpp jj = 0 Advanced Microeconomic Theory 63
Product Differentiation: Bertrand Model Solving for pp ii, we find firm ii s BRF pp ii (pp jj ) = aa + ccpp jj 2bb Firm jj also has a symmetric BRF. Note: BRFs are now positively sloped An increase in firm jj s price leads firm ii to increase his, and vice versa In this case, firms choices (i.e., prices) are strategic complements Advanced Microeconomic Theory 64
Product Differentiation: Bertrand Model p 1 p 2 (p 1 ) p 1 * p 1 (p 2 ) a 2b a 2b p 2 * Advanced Microeconomic Theory 65 p 2
Product Differentiation: Bertrand Model Simultaneously solving the two BRS yields pp ii = aa 2bb cc with corresponding equilibrium sales of qq ii (pp ii, pp jj ) = aa bbpp ii + ccpp jj = and equilibrium profits of ππ ii = pp ii qq ii pp aa ii, pp jj = 2bb cc aa 2 bb = 2bb cc 2 aaaa 2bb cc aaaa 2bb cc Advanced Microeconomic Theory 66
Product Differentiation: Cournot Model Consider two firms with the following linear inverse demand curves pp 1 (qq 1, qq 2 ) = αα ββqq 1 γγqq 2 for firm 1 pp 2 (qq 1, qq 2 ) = αα γγqq 1 ββqq 2 for firm 2 We assume that ββ > 0 and ββ > γγ That is, the effect of increasing qq 1 on pp 1 is larger than the effect of increasing qq 1 on pp 2 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 crossprice effects. Advanced Microeconomic Theory 67
Product Differentiation: Cournot Model Firm ii s PMP is (assuming no costs) max (αα ββqq ii γγqq jj )qq ii qq ii 0 FOC: αα 2ββqq ii γγqq jj = 0 Solving for qq ii we find firm ii s BRF qq ii (qq jj ) = αα 2ββ γγ 2ββ qq jj Firm jj also has a symmetric BRF Advanced Microeconomic Theory 68
Product Differentiation: q 1 Cournot Model α γ α 2β q 2 (q 1 ) (q* 1,q * 2 ) q 1 (q 2 ) α 2β Advanced Microeconomic Theory 69 α γ q 2
Product Differentiation: Cournot Model Comparative statics of firm ii s BRF As ββ γγ (products become more homogeneous), BRF becomes steeper. That is, the profitmaximizing choice of qq ii is more sensitive to changes in qq jj (tougher competition) As γγ 0 (products become very differentiated), firm ii s BRF no longer depends on qq jj and becomes flat (milder competition) Advanced Microeconomic Theory 70
Product Differentiation: Cournot Model Simultaneously solving the two BRF yields qq ii = αα for all ii = {1,2} 2ββ + γγ with a corresponding equilibrium price of pp ii = αα ββqq ii γγqq jj = αααα 2ββ + γγ and equilibrium profits of ππ ii = pp ii qq αααα αα ii = 2ββ + γγ 2ββ + γγ = αα 2 ββ 2ββ + γγ 2 Advanced Microeconomic Theory 71
Note: Product Differentiation: Cournot Model As γγ increases (products become more homogeneous), individual and aggregate output decrease, and individual profits decrease as well. If γγ ββ (indicating undifferentiated products), then qq ii = as in standard Cournot models of αα = αα 2ββ+ββ 3ββ homogeneous products. If γγ 0 (extremely differentiated products), then qq ii = as in monopoly. αα = αα 2ββ+0 2ββ Advanced Microeconomic Theory 72
Dynamic Competition Advanced Microeconomic Theory 73
Dynamic Competition: Sequential Bertrand Model with Homogeneous Products Assume that firm 1 chooses its price pp 1 first, whereas firm 2 observes that price and responds with its own price pp 2. Since the game is a sequential-move game (rather than a simultaneous-move game), we should use backward induction. Advanced Microeconomic Theory 74
Dynamic Competition: Sequential Bertrand Model with Homogeneous Products Firm 2 (the follower) has a BRF given by pp 2 (pp 1 ) = pp 1 εε if pp 1 > cc cc if pp 1 cc while firm 1 s (the leader s) BRF is pp 1 = cc Intuition: the follower undercuts the leader s price pp 1 by a small εε > 0 if pp 1 > cc, or keeps it at pp 2 = cc if the leader sets pp 1 = cc. Advanced Microeconomic Theory 75
Dynamic Competition: Sequential Bertrand Model with Homogeneous Products The leader expects that its price will be: undercut by the follower when pp 1 > cc (thus yielding no sales) mimicked by the follower when pp 1 = cc (thus entailing half of the market share) Hence, the leader has (weak) incentives to set a price pp 1 = cc. As a consequence, the equilibrium price pair remains at (pp 1, pp 2 ) = (cc, cc), as in the simultaneous-move version of the Bertrand model. Advanced Microeconomic Theory 76
Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products Assume that firms sell differentiated products, where firm jj s demand is qq jj = DD jj (pp jj, pp kk ) Example: qq jj (pp jj, pp kk ) = aa bbpp jj + ccpp kk, where aa, bb, cc > 0 and bb > cc In the second stage, firm 2 (the follower) solves following PMP max pp 2 0 ππ 2 = pp 2 qq 2 TTTT(qq 2 ) = pp 2 DD 2 (pp 2, pp 1 ) TTTT(DD 2 (pp 2, pp 1 ) qq 2 ) Advanced Microeconomic Theory 77
Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products FOCs wrt pp 2 yield DD 2 (pp 2, pp 1 ) DD 2 (pp 2, pp 1 ) + pp 2 pp 2 DD 2(pp 2, pp 1 ) DD 2 (pp 2, pp 1 ) DD 2 (pp 2, pp 1 ) pp 2 Using the chain rule = 0 Solving for pp 2 produces the follower s BRF for every price set by the leader, pp 1, i.e., pp 2 (pp 1 ). Advanced Microeconomic Theory 78
Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products In the first stage, firm 1 (leader) anticipates that the follower will use BRF pp 2 (pp 1 ) to respond to each possible price pp 1, hence solves following PMP max pp 1 0 ππ 1 = pp 1 qq 1 TTTT qq 1 = pp 1 DD 1 pp 1, pp 2 pp 1 BBBBFF 2 TTTT DD 1 pp 1, pp 2 (pp 1 ) qq 1 Advanced Microeconomic Theory 79
Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products FOCs wrt pp 1 yield DD 1 (pp 1, pp 2 ) DD 1 (pp 1, pp 2 ) + pp 1 + DD 1(pp 1, pp 2 ) pp 2 (pp 1 ) pp 1 pp 2 (pp 1 ) pp 1 New Strategic Effect DD 1(pp 1, pp 2 ) DD 1 (pp 1, pp 2 ) + DD 1(pp 1, pp 2 ) pp 2 (pp 1 ) DD 1 (pp 1, pp 2 ) pp 1 pp 2 (pp 1 ) pp 1 New Strategic Effect Or more compactly as DD 1 (pp 1, pp 2 ) + pp 1 DD 1(pp 1, pp 2 ) DD 1 (pp 1, pp 2 ) = 0 DD 1 (pp 1, pp 2 ) pp 1 1 + pp 2(pp 1 ) pp 1 NNNNNN = 0 Advanced Microeconomic Theory 80
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 81
Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products Example: Consider a linear demand qq ii = 1 2pp ii + pp jj, with no marginal costs, i.e., cc = 0. Simultaneous Bertrand model: the PMP is max ππ jj = pp jj (1 2pp jj + pp kk ) for any kk jj pp jj 0 where FOC wrt pp jj produces firm jj s BRF pp jj (pp kk ) = 1 4 + 1 4 pp kk Simultaneously solving the two BRFs yields pp jj = 1 3 0.33, entailing equilibrium profits of ππ jj = 2 0.222. 9 Advanced Microeconomic Theory 82
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 pp 2 0 ππ 2 = pp 2 1 2pp 2 + pp 1 where FOC wrt pp 2 produces firm 2 s BRF pp 2 (pp 1 ) = 1 4 + 1 4 pp 1 In the first stage, firm 1 s (the leader s) PMP is max pp 1 0 ππ 1 = pp 1 1 2pp 1 + 1 4 + 1 4 pp 1 BBBBFF 2 = pp 1 1 4 (5 7pp 1) Advanced Microeconomic Theory 83
Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products Example (continued): FOC wrt pp 1, and solving for pp 1, produces firm 1 s equilibrium price pp 1 = 5 = 0.36. 14 Substituting pp 1 into the BRF of firm 2 yields pp 2 0.36 = 1 + 1 0.36 = 0.34. 4 4 Equilibrium profits are hence ππ 1 = 0.36 1 5 7 0.36 = 0.223 for firm 1 4 ππ 2 = 0.34 1 2 0.34 + 0.36 = 0.230 for firm 2 Advanced Microeconomic Theory 84
Dynamic Competition: Sequential Bertrand Model with Heterogeneous Products Example (continued): Both firms prices and profits are higher in the sequential than in the simultaneous game. However, the follower earns more than the leader in the sequential game! (second mover s advantage) 0.36 ⅓ ¼ p 1 Prices with sequential price competition ¼ p 2 (p 1 ) ⅓ 0.34 p 1 (p 2 ) Prices with simultaneous price competition Advanced Microeconomic Theory 85 p 2
Dynamic Competition: Sequential Cournot Model with Homogenous Products Stackelberg model: firm 1 (the leader) chooses output level qq 1, and firm 2 (the follower) observing the output decision of the leader, responds with its own output qq 2 (qq 1 ). By backward induction, the follower s BRF is qq 2 (qq 1 ) for any qq 1. Since the leader anticipates qq 2 (qq 1 ) from the follower, the leader s PMP is max pp qq 1 + qq 2 (qq 1 ) qq 1 TTCC 1 (qq 1 ) qq 1 0 BBBBFF 2 Advanced Microeconomic Theory 86
Dynamic Competition: Sequential Cournot Model with Homogenous Products FOCs wrt qq 1 yields pp qq 1 + qq 2 (qq 1 ) + pp qq 1 + qq 2 (qq 1 ) qq 1 + qq 2(qq 1 ) qq 1 qq 1 CC 1(qq 1 ) qq 1 = 0 or more compactly pp QQ + pp QQ qq 1 + pp QQ qq 2(qq 1 ) qq 1 qq 1 Strategic Effect CC 1 qq 1 qq 1 = 0 This FOC coincides with that for standard Cournot model with simultaneous output decisions, except for the strategic effect. Advanced Microeconomic Theory 87
Dynamic Competition: Sequential Cournot Model with Homogenous Products The strategic effect is positive since pp (QQ) < 0 and qq 2(qq 1 ) qq 1 < 0. Firm 1 (the leader) has more incentive to raise qq 1 relative to the Cournot model with simultaneous output decision. Intuition (first-mover advantage): By overproducing, the leader forces the follower to reduce its output qq 2 by the amount qq 2(qq 1 ) qq 1. This helps the leader sell its production at a higher price, as reflected by ppp(qq); ultimately earning a larger profit than in the standard Cournot model. Advanced Microeconomic Theory 88
Dynamic Competition: Sequential Cournot Model with Homogenous Products Example: Consider linear inverse demand pp = aa QQ, where QQ = qq 1 + qq 2, and a constant marginal cost of cc. Firm 2 s (the follower s) PMP is max qq 2 (aa qq 1 qq 2 )qq 2 ccqq 2 FOC: aa qq 1 2qq 2 cc = 0 Solving for qq 2 yields the follower s BRF qq 2 qq 1 = aa qq 1 cc 2 Advanced Microeconomic Theory 89
Dynamic Competition: Sequential Cournot Model with Homogenous Products Example (continued): Plugging qq 2 qq 1 into the leader s PMP, we get max qq 1 FOC: aa qq 1 aa qq 1 cc 2 1 qq 1 ccqq 1 = 1 2 (aa qq 1 cc) 2 aa 2qq 1 cc = 0 Solving for qq 1, we obtain the leader s equilibrium output level qq 1 = aa cc 2. Substituting qq 1 into the follower s BRF yields the follower s equilibrium output qq 2 = aa cc 4. Advanced Microeconomic Theory 90
Dynamic Competition: Sequential Cournot Model with Homogenous Products q 1 q 2 (q 1 ) a c 2 Stackelberg Quantities Cournot Quantities q 1 (q 2 ) a c 2 Advanced Microeconomic Theory 91 q 2
Dynamic Competition: Sequential Cournot Model with Homogenous Products Example (continued): The equilibrium price is pp = aa qq 1 qq aa + 3cc 2 = 4 And the resulting equilibrium profits are ππ 1 = ππ 2 = aa+3cc 4 aa+3cc 4 aa cc 2 aa cc 4 cc cc aa cc 2 aa cc 4 = aa cc 2 8 = aa cc 2 16 Advanced Microeconomic Theory 92
Dynamic Competition: Sequential Cournot Model with Homogenous Products p m = Price a a + c 2 Monopoly Linear inverse demand pp QQ = aa QQ Symmetric marginal costs cc > 0 p Cournot = a + 2c 3 Cournot p Stackelberg = a + 3c 4 Stackelberg p P.C. =p Bertrand = c Bertrand and Perfect Competition Units a c 2b 2(a c) 3b 3(a c) 4b a c b a b Advanced Microeconomic Theory 93
Dynamic Competition: Sequential Cournot Model with Heterogeneous Products Assume that firms sell differentiated products, with inverse demand curves for firms 1 and 2 pp 1 (qq 1, qq 2 ) = αα ββqq 1 γγqq 2 for firm 1 pp 2 (qq 1, qq 2 ) = αα γγqq 1 ββqq 2 for firm 2 Firm 2 s (the follower s) PMP is max qq 2 (αα γγqq 1 ββqq 2 ) qq 2 where, for simplicity, we assume no marginal costs. FOC: αα γγqq 1 2ββqq 2 = 0 Advanced Microeconomic Theory 94
Dynamic Competition: Sequential Cournot Model with Heterogeneous Products Solving for qq 2 yields firm 2 s BRF qq 2 (qq 1 ) = αα γγqq 1 2ββ Plugging qq 2 qq 1 into the leader s firm 1 s (the leader s) PMP, we get FOC: max αα ββqq 1 γγ αα γγqq 1 qq 1 2ββ max αα 2ββ γγ qq 1 2ββ αα 2ββ γγ 2ββ 2ββ2 γγ 2 2ββ 2ββ2 γγ 2 ββ qq 1 = 0 qq 1 = qq 1 qq 1 Advanced Microeconomic Theory 95
Dynamic Competition: Sequential Cournot Model with Heterogeneous Products Solving for qq 1, we obtain the leader s equilibrium output level qq 1 = αα(2ββ γγ) 2(2ββ 2 γγ 2 ) Substituting qq 1 into the follower s BRF yields the follower s equilibrium output qq 2 = αα γγqq 1 2ββ = αα(4ββ2 2ββββ γγ 2 ) 4ββ(2ββ 2 γγ 2 ) Note: qq 1 > qq 2 If γγ ββ (i.e., the products become more homogeneous), (qq 1, qq 2 ) convege to the standard Stackelberg values. If γγ 0 (i.e., the products become very differentiated), (qq 1, qq 2 ) converge to the monopoly output qq mm = αα 2ββ. Advanced Microeconomic Theory 96
Capacity Constraints Advanced Microeconomic Theory 97
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, qq 1 and qq 2, with a cost of capacity cc > 0 Second stage: firms observe each other s capacities and compete in prices, simultaneously setting pp 1 and pp 2 Advanced Microeconomic Theory 98
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 pp 1 < pp 2 and the quantity demanded at pp 1 is so large that QQ(pp 1 ) > qq 1, then the first qq 1 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 99
Capacity Constraints At pp 1 the quantity demanded is QQ(pp 1 ), but only qq 1 units can be served. Hence, the residual demand is QQ(pp 1 ) qq 1. Since firm 2 sets a price of pp 2, its demand will be QQ(pp 2 ). Thus, a portion of the residual demand, i.e., QQ(pp 2 ) qq 1, is captured. p 2 p 1 p 6 th 4 th 1 st q 1, firm 1's capacity 5 th 3 rd Q(p) q q 1 Q(p 2 ) Q(p 1 ) Unserved customers by firm 1 These units become residual demand for firm 2. Q 2 (p 2 ) q 1 Advanced Microeconomic Theory 100 2 nd
Capacity Constraints Hence, firm 2 s residual demand can be expressed as QQ pp 2 qq 1 if QQ pp 2 qq 1 0 0 otherwise Should we restrict qq 1 and qq 2 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 101
Capacity Constraints How to express this rather obvious statement with a simple mathematical condition? The maximal revenue of a firm under monopoly is max (aa qq)qq, which is maximized at qq = aa, yielding qq 2 profits of aa2 4. Maximal revenues are larger than costs if aa2 4 ccqq jj, or solving for qq jj, aa2 4cc qq jj. 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 102
Capacity Constraints: Second Stage By backward induction, we start with the second stage (pricing game), where firms simultaneously choose prices pp 1 and pp 2 as a function of the capacity choices qq 1 and qq 2. We want to show that in this second stage, both firms set a common price pp 1 = pp 2 = pp = aa qq 1 qq 2 where demand equals supply, i.e., total capacity, pp = aa QQ, where QQ qq 1 + qq 2 Advanced Microeconomic Theory 103
Capacity Constraints: Second Stage In order to prove this result, we start by assuming that firm 1 sets pp 1 = pp. We now need to show that firm 2 also sets pp 2 = pp, i.e., it does not have incentives to deviate from pp. If firm 2 does not deviate, pp 1 = pp 2 = pp, then it sells up to its capacity qq 2. If firm 2 reduces its price below pp, demand would exceed its capacity qq 2. As a result, firm 2 would sell the same units as before, qq 2, but at a lower price. Advanced Microeconomic Theory 104
Capacity Constraints: Second Stage If, instead, firm 2 charges a price above pp, then pp 1 = pp < pp 2 and its revenues become pp 2 QQ (pp 2 ) = pp 2(aa pp 2 qq 1 ) if aa pp 2 qq 1 0 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 105
Capacity Constraints: Second Stage We now find the maximum of this revenue function. FOC wrt pp 2 yields: aa 2pp 2 qq 1 = 0 pp 2 = aa qq 1 2 The non-deviating price pp = aa qq 1 qq 2 lies above the maximum-revenue price pp 2 = aa qq 1 when 2 aa qq 1 qq 2 > aa qq 1 2 aa > qq 1 + 2qq 2 Since aa2 4cc qq jj (capacity constraint), we can obtain aa 2 4cc + 2 aa2 4cc > qq 1 + 2qq 2 3aa2 4cc > qq 1 + 2qq 2 Advanced Microeconomic Theory 106
Capacity Constraints: Second Stage Therefore, aa > qq 1 + 2qq 2 holds if aa > 3aa2 which, 4cc solving for aa, is equivalent to 4cc > aa. 3 Advanced Microeconomic Theory 107
Capacity Constraints: Second Stage When 4cc > aa holds, 3 capacity constraint aa 2 qq 4cc jj transforms into 3aa 2 > qq 1 + 2qq 2, implying 4cc pp > pp 2 = aa qq 1 2. Thus, firm 2 does not have incentives to increase its price pp 2 from pp, since that would lower its revenues. Advanced Microeconomic Theory 108
Capacity Constraints: Second Stage In short, firm 2 does not have incentives to deviate from the common price pp = aa qq 1 qq 2 A similar argument applies to firm 1 (by symmetry). Hence, we have found an equilibrium in the pricing stage. Advanced Microeconomic Theory 109
Capacity Constraints: First Stage In the first stage (capacity setting), firms simultaneously select their capacities qq 1 and qq 2. Inserting stage 2 equilibrium prices, i.e., pp 1 = pp 2 = pp = aa qq 1 qq 2, into firm jj s profit function yields ππ jj (qq 1, qq 2 ) = (aa qq 1 qq 2 ) qq jj ccqq jj FOC wrt capacity qq jj yields firm jj s BRF pp aa cc qq jj (qq kk ) = 2 1 2 qq kk Advanced Microeconomic Theory 110
Capacity Constraints: First Stage Solving the two BRFs simultaneously, we obtain a symmetric solution aa cc qq jj = qq kk = 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 111
Endogenous Entry Advanced Microeconomic Theory 112
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 FF > 0. Advanced Microeconomic Theory 113
Endogenous Entry Consider inverse demand function pp(qq), where qq denotes aggregate output Every firm jj faces the same total cost function, cc(qq jj ), of producing qq jj units Hence, the Cournot equilibrium must be symmetric Every firm produces the same output level qq(nn), which is a function of the number of entrants. Entry profits for firm jj are ππ jj nn = pp nn qq nn QQ pp(qq) qq nn cc qq nn Production Costs FF Fixed Entry Cost Advanced Microeconomic Theory 114
Endogenous Entry Three assumptions (valid under most demand and cost functions): individual equilibrium output qq(nn) is decreasing in nn; aggregate output qq nn qq(nn) increases in nn; equilibrium price pp(nn qq(nn)) remains above marginal costs regardless of the number of entrants nn. Advanced Microeconomic Theory 115
Endogenous Entry Equilibrium number of firms: The equilibrium occurs when no more firms have incentives to enter or exit the market, i.e., ππ jj (nn ee ) = 0. Note that individual profits decrease in nn, i.e., + ππ nn = pp nnnn nn cc (nn) qq nn + qq nn pp [nnnn nn ] nnnn nn + < 0 Advanced Microeconomic Theory 116
Endogenous Entry Social optimum: The social planner chooses the number of entrants nn oo that maximizes social welfare max nn nnnn(nn) WW nn pp ss dddd nn cc qq nn nn FF 0 Advanced Microeconomic Theory 117
Endogenous Entry nnnn(nn) pp ss dddd = 0 AA + BB + CC + DD p nn cc qq nn = CC + DD Social welfare is thus AA + BB minus total entry costs nn FF p(n q(n)) n c (q(n)) A B C D n q(n) n c (q) p(q) Q Advanced Microeconomic Theory 118
Endogenous Entry FOC wrt nn yields pp nnnn nn nn or, re-arranging, nn + qq nn cc qq nn nncc qq nn ππ nn + nn pp nnnn nn cc qq nn (nn) = 0 nn FF = 0 Hence, marginal increase in nn 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 119
Endogenous Entry The business stealing effect is represented by: nn pp nnnn nn cc qq nn (nn) < 0 which is negative since (nn) < 0 and nn pp nnnn nn cc qq nn > 0 by definition. Therefore, an additional entry induces a reduction in aggregate output by nn (nn), which in turn produces a negative effect on social welfare. Advanced Microeconomic Theory 120
Endogenous Entry Given the negative sign of the business stealing effect, we can conclude that WW nn = ππ nn + nn pp nnnn nn cc qq nn nn < ππ(nn) and therefore more firms enter in equilibrium than in the social optimum, i.e., nn ee > nn oo. Advanced Microeconomic Theory 121
Endogenous Entry Advanced Microeconomic Theory 122
Endogenous Entry Example: Consider a linear inverse demand pp QQ = 1 QQ and no marginal costs. The equilibrium quantity in a market with nn firms that compete a la Cournot is qq nn = 1 nn+1 Let s check if the three assumptions from above hold. Advanced Microeconomic Theory 123
Endogenous Entry Example (continued): First, individual output decreases with entry nn = 1 nn+1 2 < 0 Second, aggregate output nnnn(nn) increases with entry nnnn nn = 1 > 0 nn+1 2 Third, price lies above marginal cost for any number of firms pp nn cc = 1 nn 1 = 1 nn+1 nn+1 > 0 for all nn Advanced Microeconomic Theory 124
Endogenous Entry Example (continued): Every firm earns equilibrium profits of ππ nn = 1 1 1 FF = nn + 1 nn + 1 nn + 1 2 FF qq(nn) pp(nn) Since equilibrium profits after entry, nn+1 2, is smaller than 1 even if only one firm enters the industry, nn = 1, we assume that entry costs are lower than 1, i.e., FF < 1. 1 Advanced Microeconomic Theory 125
Endogenous Entry Example (continued): Social welfare is nn nn+1 WW nn = (1 ss)dddd nn FF 0 = ss ss 2 0 nn nn+1 nn FF = nn nn + 2 2 1 nn + 1 2 nn FF Advanced Microeconomic Theory 126
Endogenous Entry Example (continued): The number of firms entering the market in equilibrium, nn ee, is that solving ππ nn ee = 0, 1 nn ee + 1 2 FF = 0 nnee = 1 FF 1 whereas the number of firms maximizing social welfare, i.e., nn oo solving WW nn oo = 0, WW nn oo 1 = nn oo + 1 3 = 0 nnoo = 1 3 1 FF where nn ee < nn oo for all admissible values of FF, i.e., FF 0,1. Advanced Microeconomic Theory 127
Endogenous Entry Example (continued): Number of firms n e = 1 F ½ 1 (Equilibrium) n o = 1 F ⅓ 1 (Soc. Optimal) 0 Entry costs, F Advanced Microeconomic Theory 128
Repeated Interaction Advanced Microeconomic Theory 129
Repeated Interaction In all previous models, we considered firms interacting during one period (i.e., one-shot game). However, firms compete during several periods and, in some cases, during many generations. In these cases, a firm s actions during one period might affect its rival s behavior in future periods. More importantly, we can show that under certain conditions, the strong competitive results in the Bertrand (and, to some extent, in the Cournot) model can be avoided when firms interact repeatedly along time. That is, collusion can be supported in the repeated game even if it could not in the one-shot game. Advanced Microeconomic Theory 130
Repeated Interaction: Bertrand Model Consider two firms selling homogeneous products. Let pp tt jj denote firm jj s pricing strategy at period tt, which is a function of the history of all price choices by the two firms, HH tt 1 = pp tt tt 1, pp tt 1 2 tt=1. Conditioning pp tt jj on the full history of play allows for a wide range of pricing strategies: setting the same price regardless of previous history retaliation if the rival lowers its price below a threshold level increasing cooperation if the rival was cooperative in previous periods (until reaching the monopoly price pp mm ) Advanced Microeconomic Theory 131
Repeated Interaction: Bertrand Model Finitely repeated game: Can we support cooperation if the Bertrand game is repeated for a finite number of TT rounds? No! To see why, consider the last period of the repeated game (period TT): Regardless of previous pricing strategies, every firms optimal pricing strategy in this stage is to set pp ii,tt = cc, as in the one-shot Bertrand game. Advanced Microeconomic Theory 132
Repeated Interaction: Bertrand Model Now, move to the previous to the last period (TT 1): Both firms anticipate that, regardless of what they choose at TT 1, they will both select pp ii,tt = cc in period TT. Hence, it is optimal for both to select pp ii,tt 1 = cc in period TT 1 as well. Now, move to period (TT 2): Both firms anticipate that, regardless of what they choose at TT 2, they will both select pp ii,tt = cc in period TT and pp ii,tt 1 = cc in period TT 1. Thus, it is optimal for both to select pp ii,tt 2 = cc in period TT 2 as well. The same argument extends to all previous periods, including the first round of play tt = 1. Hence, both firms behave as in one-shot Bertrand game. Advanced Microeconomic Theory 133
Repeated Interaction: Bertrand Model Infinitely repeated game: Can we support cooperation if the Bertrand game is repeated for an infinite periods? Yes! Cooperation (i.e., selecting prices above marginal cost) can indeed be sustained using different pricing strategies. For simplicity, consider the following pricing strategy pp jjjj HH tt 1 = ppmm if all elements in HH tt 1 are pp mm, pp mm or tt = 1 cc otherwise In words, every firm jj sets the monopoly price pp mm in period 1. Then, in each subsequent period tt > 1, firm jj sets pp mm if both firms charged pp mm in all previous periods. Otherwise, firm jj charges a price equal to marginal cost. Advanced Microeconomic Theory 134
Repeated Interaction: Bertrand Model This type of strategy is usually referred to as Nash reversion strategy (NRS): firms cooperate until one of them deviates, in which case firms thereafter revert to the Nash equilibrium of the unrepeated game (i.e., set prices equal to marginal cost) Let us show that NRS can be sustained in the equilibrium of the infinitely repeated game. We need to demonstrate that firms do not have incentives to deviate from it, during any period tt > 1 and regardless of their previous history of play. Advanced Microeconomic Theory 135