Industrial Organization Past Paper PDF

P1: SFK Trim: 246mm × 189mm Top: 10mm Gutter: 19.001mm CUUK2869-SOL CUUK2869/Belleflamme & Peitz 978 1 107 06997 8 February 18, 2015 20:12 P1: SFK Trim: 246mm × 189mm Top: 10mm Gutter: 19.001mm...

P1: SFK Trim: 246mm × 189mm Top: 10mm Gutter: 19.001mm CUUK2869-SOL CUUK2869/Belleflamme & Peitz 978 1 107 06997 8 February 18, 2015 20:12 P1: SFK Trim: 246mm × 189mm Top: 10mm Gutter: 19.001mm CUUK2869-02 CUUK2869/Belleflamme & Peitz 978 1 107 06997 8 February 18, 2015 16:20 Selected end-of-chapter exercises with solutions Taken from Belleflamme, Paul and Martin Peitz (2015). Industrial Organization: Games and Strategies (2nd Edition). Cambridge: Cambridge University Press 38 Firms, consumers and the market Solutions Prepared for to theend-of-chapter exercises course LECGE1330-Industrial Organization taught at UCLouvain by Paul Belleflamme 2. Assume pC = 50. What now is the effect on demand and welfare in country A if country DOwith A signs a free-trade agreement NOT CIRCULATE country B? PART I. GETTING STARTED 2.2 Monopoly What is Markets and Strategies? Chapter 1.problem No exercise. Consider a monopolist with a linear demand curve: q = a − bp, where a, b > 0. It produces at constant marginal cost c and has no fixed cost. Assume that 0 < c < a/b. 1.Chapter Find the2.monopoly Firms, consumers and price, quantity andthe market profits. 2.1 Free trade and competitive markets 2.In the Derive the inverse absence demand agreement of a free-trade curve P(q).withDraw thethe P(q),B, country MR curve andinthe consumers MC curve country in a buy A always diagram. Explain why we need the assumption c < a/b. shoes from country C as pC + t < 59.99 + t whether pC = 45 or pC = 50. If there is a free-trade 3.agreement with country Does it matter that theB, then consumers monopolist in country sets price compare pC + t with p B = 59.99. As A quantity? instead of t = 10, we have that consumers buy shoes from country C if pC = 45 but from country B if 4.pC Calculate the deadweight = 50. Comparing the twoloss of monopoly. situations, we see that the free-trade agreement with country B has no effect if pC = 45 (i.e., if country C is very inexpensive with respect to country B), as consumers 5. Derive the price elasticity of demand η for any price. How does η change with p? in country A continue to buy shoes from country C. However, if pC = 50, the free-trade agreement 6.makes Show mathematically consumers as well buy shoes fromascountry graphically that the B rather thanprice fromelasticity country of C; demand is strictly this allows them to pay greater a lower thanwhich price, one atincreases the monopoly price. their surplus. Solution 2.2 Two-period 2.3 Monopoly problemmonopoly problem 1. The monopoly Consider a monopolistchooses p to maximize that produces for twoπ= ( p − c)(a periods. The−demand bp). Thecurves first-order condition in both periodsyields are t a − bpt − bp + bc = 0, which is equivalent to p m = (a + bc) / (2b). 1We compute then q m = q = 1 − p for t = 1, 2. The marginal costs are c in the first and c − λq in the second period. Here,a λ−isbp m = (aand a small − bc) /2 and positive p m − cThere number. = (a is−abc) / (2b). factor discount It followsof δ that π m =the between − bc)2 / (4b). (aperiods. 2. The inverse demand curve P(q) is obtained by inverting q = a − bp: bp = a − q ⇔ p = 1. Explain a/b − q/b.briefly Thehow the monopolist’s intercept on the vertical problem changes axis (where compared price with is is measured) a situation a/b, and where the intercept the marginal cost is c in both periods. on the horizontal axis (where quantity is measured) is a. The MC curve is a horizontal line that cuts 2. Find the vertical the quantities q 1axis andatq 2c;that if cthe were larger than monopolist a/b, inverse chooses in the demand two periods.would be everywhere (Hint: Start below by the MC solving and no production the monopolist’s problem would in thebesecond profitable. periodTheandMR thencurve is M R continue to = first− 2q/b, thea/b which has the same vertical intercept as the inverse demand curve but cuts the horizontal axis period.) at a/2 instead of a. 3.3. Derive the restriction No, because on λ that the monopoly ensures controls thethe profit function demand function,isthatstrictly concave. is, the relationship between p and q. To be sure, solve the monopoly problem in terms of quantity. That is, let the monopoly choose q to maximize π = (a/b − q/b) q − cq. p 2.4 Essay 4. Thequestion: first-best isMarket achieved structure for rating at marginal agencies cost pricing: p∗ = c; the corresponding quantity is In theq ∗recent = a −financial bc. Welfarecrisis,israting then agencies equal to have W ∗ become a focus = (1/2) (a/b of(aattention. − c) − bc) =The (a −market has bc)2 / (2b). traditionally been dominated by a few big agencies, Under monopoly, the consumer surplus is equal to C S = − bc) / currently Standard m (a & Poor, 2 Moody’s (8b); and adding the Fitch.monopoly’s In 2006, the Securities profit, and Exchange we compute Commission welfare under monopoly (SEC)as Wintroduced m = 3 (a −measures to speed bc)2 / (8b). Hence, up the approval process for rating agencies ∗ with the the deadweight loss of monopoly is W − W = (a − bc) / (8b).m aim of increasing 2 competition. However, in response to theelasticity 5. The price financial ofcrisis, the Fed demand introduced is defined as lending programmes for which – this is what it said initially – it accepts only collateral that has been appraised by one of the big three. Discuss the likely consequences of% such ap decisionbp on market structure. η ( p) = −q ( p) =. q ( p) a − bp p See ‘The wages of Sin’, The Economist, 25 April 2009, p. 76. CUUK2869-SOL CUUK2869/Belleflamme & Peitz 978 1 107 06997 8 February 18, 2015 20:12 P1: SFK Trim: 246mm × 189mm Top: 10mm Gutter: 19.001mm CUUK2869-SOL CUUK2869/Belleflamme & Peitz 978 1 107 06997 8 February 18, 2015 20:12 Solutions to end-of-chapter exercises 739 P1: SFK Trim: 246mm × 189mm We Top: 10mm compute Gutter: 19.001mm CUUK2869-03 CUUK2869/Belleflamme & Peitz 978 1 107 06997 8 ab February 18, Solutions to end-of-chapter exercises 739 2015 16:29 η" ( p) = > 0, (a − bp)2 We compute meaning that the price elasticity of demand increases with p. 6. We have that ab 74 Static imperfect competition η" ( p) = > 0, m 2 (a − bpbp) a + bc η ( pm ) = = > 1. meaning that the price elasticityaof−demand − a bp m bc increases with p. Exercises 6. We have that 2.3 Two-period monopoly problem 3.1 Price 1. Ifcompetition the marginal bp m a + bc then the two periods are unrelated (as the monopolist’s η ( p m )cost = is c in mboth = periods, > 1. decisions Consider a duopoly in thein whicha −period first bp doanot homogeneous − bc affect consumers the demandof massor the cost 1 have in the second unit demand. period). In Their valuation contrast, for good i = 1,when the second-period 2 is v({i}) = v i with v 1marginal > v 2. Marginalcost is cost c − λq of 1production , the monopolistis assumedlowers tohis cost in be zero. 2.3 Suppose Two-period the second that firmsmonopoly period competebyproblem producing in prices. more in the first period (this can result from some learning 1. If the marginal economies, forcost is c in both periods, then the two periods are unrelated (as the monopolist’s instance). 1. Suppose decisions thatin consumers the first makedo period a discrete not affect choice between or theq 2demand thethe two2cost products. in−the Characterize second 2. In the second period, the monopolist chooses to maximize π = (1 q 2 )q 2 − (cperiod). − λq 1 )qIn2. the Nash contrast, equilibrium. when the second-period marginal 2 cost is c − λq 1 ,1the monopolist lowers his cost2 in The optimum is easily found as q = (1/2)(1 − c + λq ), resulting in a profit of π = the second − cperiod + λq 1 )by2 producing more in the first period (this can 1result from some π1 + learning 2 (1/4)(1 2. Suppose that consumers. In can the now first period, also decide the monopolist to buy bothchooses products.q Iftothey maximize do so they areδπ = economies,1 1 for instance). (1 − q )qto have 1 − cqa valuation + (δ + 4)(1 − c2}) += 1 2 λqv)12.with The vfirst-order assumed v({1, 1 + v 2 > condition v 12 > v 1. yields Firms still compete 2. In the second period, the monopolist chooses q 2 to maximize π 2 = (1 − q 2 )q 2 − (c − λq 1 )q 2. in prices (each firm sets the price for its product – there 2 1 is no additional price for the The optimum (1 + 2(4 + δ)λ)(1 is easily − c) as found − 2(1 q2 = − (1/2)(1 (4 + δ)λ−)q c +=λq0,1 ), resulting in a profit of π 2 = bundle). Characterize the Nash equilibrium. (1/4)(1 − c + λq 1 )2. In the first period, the monopolist chooses 2 q 1 to maximize π 1 + δπ 2 = and the1 second-order condition yields −2(1 − (4 + δ)λ ) < 0, which is satisfied as long as 3. Compare 1 (1 − q )q2regimes 1 − cq from + (δ√ parts + 4)(1(1)−andc +(2) 1 λq with 2 ). The respect to consumer first-order condition surplus. yieldsComment on (4 + δ)λ < 1 or λ < 4 + δ (this is the answer to question 3). Then, the optimum is found as your results. (1 + 2(4 1 ++2(4 δ)λ)(1 + δ)λ − c)1− −c2(1 − (4 + δ)λ2 )q 1 = 0, 1 q =. 3.2 Asymmetric duopoly and the second-order + δ)λ2 2yields −2(1 − (4 + δ)λ2 ) < 0, which is satisfied as long as 1 − (4 condition Consider √ (4 + two δ)λ2quantity-setting It follows < that1 or λ < 4firms that is + δ (this produce the answer a homogeneous to question good 3). Then,and the choose their quantities optimum is found as simultaneously. The inverse demand function for the good is given by P = a − q1 − q2 , where 1 + 2(4 2+ λδ)λ 1 −1 c− c q1 and q2 are qthe 12 outputs = of+firms 1 and. 2.respectively. The cost functions of the two firms are 2(1 − (4 + 2 2 )2 2 1 − (4 + δ)λ δ)λ C1 (q1 ) = c1 q1 and C2 (q2 ) = c2 q2 , where c1 < a and c2 < (a + c1 )/2. √ It We follows checkthe that that λ = 0, q 1 =ofqthe for equilibrium 2 =game. (1 − c)/2, +the 1 q 2 > (1 − 1. Compute Nash Whatwhile are theformarket0 < λshares < 4of δ, qtwo>firms? c)/2. 2+λ 1−c 2. Given your q 2answer = to (1), compute 2 the.equilibrium profits, consumer surplus and social 2(1 − (4 + δ)λ ) 2 welfare. √ PARTWe II.check MARKET that for λ = 0, q 1 = q 2 = (1 − c)/2, while for 0 < λ < 4 + δ, q 1 > q 2 > (1 − POWER 3. Prove that if c2 decreases slightly, then social welfare increases if the market share of firm c)/2. 2 exceeds 1/6, but decreases if the market share of firm 2 is less than 1/6. Give an Chapter economic 3. Static imperfect interpretation of thiscompetition finding. 3.1 Price competition PART 1. TheII.Nash MARKET POWER equilibrium is given by p1 = capacity v 1 − v 2 , p2 = 0, π 1 = v 1 − v 2 and π 2 = 0. 3.3 Price-setting in a market with limited Solutions2. The Nash equilibrium is given Suppose that two identical firms in a homogeneous-product by p 1 = v 12 − v 2 , pmarket 2 = v 12compete − v 1 , π in 1 = v 12 −The prices. and π 2 = v 2 capacity Chapter v − v3.. Static imperfect competition of each12firm 1is 3. The firms have constant marginal cost equal to 0 up to the capacity constraint. 3.1 The Price competition 3. demand Consumer in surplus the market in (1) is C Saby=Q( is given v 2 ,p)whereas = 9 − p. it isIf given the firms by Cset = vsame Sb the 1 + v 2price, − v 12they in (2). splitAs 1. The v Nash > v byequilibrium assumption, is given consumer by p = welfare 1 v 1 − is v , p strictly 2 2 = 0, π greater the demand equally. If the firms set a different price, the demand of each one of the firms is 12 1 1 = in v (1)1 − v than 2 and in (2).π In 2 = 0. (2), the nature 2. The Nash of competition equilibrium changes isbecause givenrationing by p1 =rule. consumers − v 12have p2 =pvvaluation , that v 2positive 12 − v 1 , πto1buy= vboth 12 − products; v 2 and π 2this= calculated according to the efficient Show 1 = p2 = 3 can be sustained as an v − relaxes 12 v 1. competition. equilibrium. Calculate the equilibrium profits. 3. Consumer surplus in (1) is C Sa = v 2 , whereas it is given by C Sb = v 1 + v 2 − v 12 in (2). As 3.4 Essay > v 1 by assumption, v 12question: Industries consumer with price welfare or is strictly greater quantity competitionin (1) than in (2). In (2), the nature of competition changes because consumers Which model, the Cournot or the Bertrand model, would you think provides have positive valuation to buy both first a better products; approx-this relaxes competition. imation to each of the following industries/markets: the oil refining industry, farmers’ markets, cleaning services. Discuss. CUUK2869-03 CUUK2869/Belleflamme & Peitz 978 1 107 06997 8 February 18, 2015 16:29 740 74 Solutions to end-of-chapter Static imperfect competitionexercises 3.2 Asymmetric duopoly Exercises 1. The Nash equilibrium of the game is obtained by solving the following system of equations: P1: SFK Trim: 246mm × 189mm Top: 1 10mm 3.1 ∂π Price competition∂q1 = (a − q1Gutter: − q2 )19.001mm − q1 − c1 = 0, CUUK2869-SOL CUUK2869/Belleflamme ∂π 1 & Peitz 978 1 107 06997 8 February 18, 2015 20:12 Consider a duopoly ∂q2 = (a − q1 −homogeneous in which q2 ) − q2 − c2consumers= 0. of mass 1 have unit demand. Their valuation for good i = 1, 2 is v({i}) = v i with v 1 >∗ v 2. 1Marginal cost of production is assumed to be zero. The equilibrium is given by the pair q1 = 3 (a − 2c1 + c2 ) and q2∗ = 13 (a − 2c2 + c1 ). Given Suppose that firms compete in prices. q1∗ and q2∗ , the equilibrium market shares are: 740 1.Solutions Supposeα tothat consumers q1∗ end-of-chapter 1 = q ∗ +q ∗ = 2a−c1 −c2 , make a−2c +c2 a discrete choice between the two products. Characterize 1exercises the Nash equilibrium. 1 ∗ q2 2 a−2c2 +c1 a2 = q ∗ +q ∗ = 2a−c1 −c2. 1 2 2. Suppose 3.2 Asymmetric duopolythat consumers can now also decide to buy both products. If they do so they are 2. Itassumed is easilytoseen from have a valuation the first-order conditions = v 12 with that1 +thev 2equilibrium > v 1profits are stillequal to the 1. The Nash equilibrium game2}) of thev({1, is ∗obtained 1 byvsolving >thev 12 2 following ∗. Firms compete 1 system of equations: 2 square of the in prices (each equilibrium quantities: π = (a − 2c + c ) and π = (a − 2c + c ). The ∂π 1 firm sets the price for its 1 product – there is no additional 9 price for the 9 1 2 2 2 1 consumer surplus = (a is − q1 − given by q2 ) − q1 − c1 = 0, bundle). Characterize ∂q 1 ∂π 1 ! ∗ ∗ the Nash equilibrium. ∂q2 = (aq1− +qq 2 1 − q2 ) − q2 − c2 = 0. 3. Compare S ∗ = from(a C regimes parts (1) and − q)dq − (a(2) ∗ q1∗ −respect −with q2∗ )(q1∗to+consumer q2∗ ) ∗ surplus. 1 Comment on The your equilibrium results. 0is given by the pair q 1 = 3 (a − 2c 1 + c 2 ) and q2 = 3 (a − 2c2 + c1 ). Given q1∗ and q2∗ , the = equilibrium (q + q2∗ )2market 1 ∗ = 18 1 shares are: 2 (2a − c1 − c2 ). 2 ∗1 q1 a−2c +c 3.2 Asymmetric αduopoly 1 = q ∗ +q ∗ = 2a−c1 −c2 , 1 2 Adding the three together, 1 2 social welfare is given by Consider two quantity-setting q2∗ firms a−2c 2 +cthat produce a homogeneous good and choose their quantities a∗ 2 = q ∗ +q ∗ ∗ = ∗ 2a−c1 −c∗ 2. 1 ∗ 1 ∗ 2 ∗ 2 ∗ 2 simultaneously. W = The C Sinverse 1 + 1 + π 2 =function 2 π demand (q + qfor 2 1 + (q 2 ) the 1) + good is(qgiven 2 ). by P = a − q1 − q2 , where q 2.and are theseen q2 easily It is outputs fromofthe firms 1 and 2conditions first-order respectively. thatThethe cost functionsprofits equilibrium of theare twoequal firmstoare the 3.1 Differentiating W ∗ with respect to c2 yields ∗ 1 2 ∗ 1 2 1 ) = c1of C1 (qsquare q1 the C2 (q2 ) = c2quantities: andequilibrium q2 , where cπ11 3. It is easy to 1 2 2 1 2 6 the demand show that (6 equally. − p) p isIfdecreasing the firms set in pa fordifferent all p price, > 3. This, the demand in turn, of each one implies that of the firms deviating to is a ∗ ∗ Hence, calculated price a small aboveaccording reduction in to the efficient 3 is not profitable. c Hence, increases 2 rationing p1 = p2rule.W if = 3 Show α > 1/6 and that p1 = p2The is 2an equilibrium. decreases W if = 3equilibrium can be sustained α 1/6. 2 profitsas < Note an are: N E that the result πequilibrium. 3)/2 that = 3(9 −Calculate = 9.athe cost reduction profits. equilibrium may be socially undesirable was first demonstrated in Lahiri and Ono (1988).1 Solution 3.4 Chapter Essay question: Industries 4. Dynamic aspectswith price or quantity of imperfect competition competition Which 3.3 Sequential 4.1 model, Price-setting the in aCournot price-setting market orwith with the Bertrand model, limited capacity differentiated productswould you think provides a better first approx- imation At the to each prices p of 1 =thep following = 3, both 1. Nash equilibrium of the simultaneous Bertrand game. 2 industries/markets: firms produce at thecapacity. full oil Firmrefining 1 We industry, conclude chooses p1 tofarmers’ that markets, the firms maximize π 1have = cleaning services. no(aincentives to Discuss. deviate to a lower price. This would result in the − 2 p1 + p2 ) p1. Solving the first-order condition for p1 , one finds firm 1’s reaction function same amount of sales but at a lower price. The efficient rationing rule implies that a firm as p1 = (a + p2 ) /4. Firm 2 chooses p2 to maximize π 2 = (a − 2 p2 + p1 )( p2 − c). Solving that deviates upwards faces a demand of D( p) = 6 − p. Thus, the profits are (6 − p) p when deviating to some p > 3. It is easy to show S. 1 Lahiri, that (6 − p) p is decreasing in p for all p > 3. This, in turn, implies that deviating to a and Ono, Y. (1988). Helping minor firms reduces welfare. Economic Journal 98: 1199–1202. price above 3 is not profitable. Hence, p1 = p2 = 3 is an equilibrium. The equilibrium profits are: π N E = 3(9 − 3)/2 = 9. Chapter 4. Dynamic aspects of imperfect competition 4.1 Sequential price-setting with differentiated products 1. Nash equilibrium of the simultaneous Bertrand game. Firm 1 chooses p1 to maximize π 1 = (a − 2 p1 + p2 ) p1. Solving the first-order condition for p1 , one finds firm 1’s reaction function as p1 = (a + p2 ) /4. Firm 2 chooses p2 to maximize π 2 = (a − 2 p2 + p1 )( p2 − c). Solving 1 Lahiri, S. and Ono, Y. (1988). Helping minor firms reduces welfare. Economic Journal 98: 1199–1202.  q1 +q2 2a−c1 −c2 2. It is easily seen from the first-order conditions that the equilibrium profits are equal to the Exercises 103 square of the equilibrium quantities: π ∗1 = 19 (a − 2c1 + c2 )2 and π ∗2 = 19 (a − 2c2 + c1 )2. The consumer surplus is given by Exercises∗ ! q1∗ +q2∗ CS = (a − q)dq − (a − q1∗ − q2∗ )(q1∗ + q2∗ ) 0 4.1 Sequential price-setting with differentiated products = 12 (q1∗ + q2∗ )2 = 181 (2a − c1 − c2 )2. Consider a market with two horizontally differentiated products and inverse demands given by Adding Pi (q i, qj) = thea three − bqitogether, social − dq j. Set b =welfare is dgiven 2/3 and by The system of demands is then given by = 1/3. Q i ( pi , p j ) = ∗ − 2 pi ∗+ p j.∗Suppose W = C S + π 1 + π 2 = 2 (q1 + q2 ) + cost a ∗ that 1 ∗firm 1 ∗ has 2 (q1∗ )2c1+=(q02∗ and )2. firm 2 has cost c2 = c (with 7c < 5a). The two firms compete in prices. Compute the firms’ profits: 3. Differentiating W ∗ with respect to c2 yields 1. at the Nash ∗equilibrium of the ∗simultaneous ∗ Bertrand game; ∂W ∗ ∗ ∂(q1 + q2 ) ∗ ∂q1 ∗ ∂q2 2. at the subgame-perfect= (q1 + qequilibrium 2) + 2q1 + 2q ∂c2 ∂c2 of the sequential ∂c2 game 2 ∂c2 1 ∗ ∗ ∗ 2 ∗ " ∗ 1 # (a) with firm = (q + q − 6(q2 ) ) = 2(q1 + q2 ) 6 − α 2. 1 being 3 1 the 2leader, and (b) with firm 2 being the leader. Hence, a small reduction in c2 increases W ∗ if α 2 > 1/6 and decreases W ∗ if α 2 < 1/6. Note 3. that Showthe that result that2 aalways firm cost reduction may be socially has a second-mover undesirable advantage, whereaswas firm first demonstrated in Lahiri 1 has a first-mover 1 and Ono (1988). advantage if c is large enough. 4.2 Price-setting 3.3 Timing game in a market with limited capacity Usethe At resultsp1of=the theprices = 3, bothexercise p2 previous to solve at firms produce forfull thecapacity. Nash equilibria of the that We conclude endogenous the firmstiming have game no in whichtofirms incentives deviate simultaneously to a lower price. choose Thiswhether to playin‘early’ would result or toamount the same play ‘late’. If they of sales but both at a make price. lower the same Thechoice efficient (either ‘early’ rationing ruleorimplies ‘late’), that the simultaneous Bertrand a firm that deviates game faces upwards follows; if they a demand make of D( p) = 6 − choices, different p. Thus,athe sequential profits are game − p) p when (6 follows with the firm having deviating to some chosenp >‘early’ 3. It isbeing easy the to show that (6 − p) p is decreasing in p for all p > 3. This, in turn, implies that deviating to a leader. Discuss the economic intuition behind your result. price above 3 is not profitable. Hence, p1 = p2 = 3 is an equilibrium. The equilibrium profits are: NE P1: SFK 4.3 × 189mm Trim: 246mm πInformation-sharing = 3(9 − 10mm Top: in Cournot 3)/2 = 9.Gutter: 19.001mmduopoly Solutions Consider the Cournot duopoly with linear demand P(q) = 1 − q, with q = q1 + q2 and constant CUUK2869-SOL CUUK2869/Belleflamme & Peitz 978 1 107 06997 8 February 18, 2015 20:12 marginal cost. Firm 1 has marginal Chapter 4. Dynamic aspects of imperfect competition cost of zero. This is commonly known. The marginal cost of firm 2 isprice-setting 4.1 Sequential privately known withtodifferentiated firm 2; firm 1products only knows that it is either prohibitively high or zero 1. and equilibrium Nash that both events of theare equally likely simultaneous (this isgame. Bertrand commonly Firm known). 1 choosesHigh p1 tomarginal maximize costs π 1 are = assumed (a − 2to p1be +prohibitively p2 ) p1. Solving high thesuch that firm first-order 2 does not condition forproduce. Consider p1 , one finds firm 1’sthereaction three-stage game function Solutions to end-of-chapter exercises 741 in which, as p1 =at(astage+ p1, firmFirm 2 ) /4. 2 draws its marginal 2 chooses costs; then,π 2at=stage p2 to maximize (a −2,2 itp2decides + p1 )( pwhether to share 2 − c). Solving its information with its competitor; finally, at stage 3, both firms compete in quantity. the S. 1 Lahiri, first-order and Ono, Y.condition for pminor 2 , one finds firmwelfare. 2’s reaction function Journal as 2 = (a + 2c + p1 ) /4. 98: p1199–1202. 1. Characterize the(1988). Helping equilibrium of this firms reduces game. Economic The solution to the system made up of the two reaction functions is p1 = (5a + 2c) /15 and 2 2 = (5a 2. pDoes firm+28c)have an We /15. can then incentive compute to share the equilibrium its private information?profits as π 1B = 225 (5a + 2c)2 and B 2 2 π 2 = 225 (5a − 7c). 3. Would firm 1 be better off under information-sharing? 2. Sequential game. (a) If firm 1 is the leader, it takes firm 2’s price reaction into account when maximizing profits; 4.4 Essay question: First-mover (dis)advantage that is, it chooses p1 to maximize π 1 = (a − 2 p1 + 14 (a + 2c + p1 )) p1. The optimal Explain in your own words the relationship between the concept of strategic complements vs. sub- price is found as p1 = (5a + 2c) /14. Firm 2 then reacts by setting p2 = (19a + 30c) /56. stitutes and the advantage or disadvantage of choosing one’s strategic variable first in a sequential Plugging the prices into the profit functions, one obtains π 1L = (5a + 2c)2 /112 and π 2F = game. 2 (19a − 26c) /1568. (b) If firm 2 is the leader, it takes firm 1’s price reaction into account when maximizing profits; that is, it chooses p2 to maximize π 2 = (a − 2 p2 + 14 (a + p2 ))( p2 − c). The optimal price is found as p1 = (5a + 7c) /14. Firm 1 then reacts by setting p1 = (19a + 7c) /56. Plugging the prices into the profit functions, one obtains π 2L = (5a − 7c)2 /112 and π 1F = (19a + 7c)2 /1568. F 2 L 3. Firm 2 always has a second-mover 2 √ advantage: √ π 2 = (19a − 26c) /1568 > π 2 = (5a − √ /112 is equivalent to√(19 − 5 14)a + (7 14 − 26)c > 0, which is satisfied as 19 − 7c) 5 14 = 0.292 > 0 and 7 14 − 26 = 0.192 > 0. Firm 1 has a first-mover advantage if π 1L = (5a + 2c)2 /112 > π 1F = (19a + 7c)2 /1568, which is equivalent to 7c2 + 14ac − 11a 2 > 0 or c > 0.604a. (Note: 0.604 < 5/7, i.e., there is a positive interval of c where the above condition is fulfilled.) 4.2 Timing game The following matrix represents the normal form of the game: Firm 1 / Firm 2 Early Late Early π 1B , π 2B π 1L , π 2F α Late π 1F , π 2L π 1B , π 2B 1−α β 1−β  7c) 2 √ √ 2 2 /112 is equivalent to (19 − 5 14)a + (7 14 − 26)c > 0, which is satisfied as 19 − 2 √ 14 = 0.292 > 0 and 7√ 14 − 26 = 0.192 > 0. Firm 1 has a first-mover advantageas 7c)√ /112 is equivalent to√(19 − 5 14)a + (7 14 − 26)c > 0, which is satisfied 19 − L 5 5 14 = 0.292 > 0 and 7 14 − 26 = 0.192 > 0. Firm 1 has a first-mover advantage if π 1L1 = if π = 2 F 2 2 2 (5a + 2c)2 /112 > πF1 = (19a + 7c)2 /1568, which is equivalent to 7c2 + 14ac − 11a2 > 0 or (5a + 2c) /112 > π 1 = (19a + 7c) /1568, which is equivalent to 7c + 14ac − 11a > 0 or cc > 0.604a. (Note: > 0.604a. (Note: 0.604 0.604 < 5/7, i.e., < 5/7, i.e., there there is is aa positive positive interval interval of of cc where where the the above above condition condition is fulfilled.) is fulfilled.) 4.2 Timing 4.2 Timing game game The following matrix represents The following matrix represents the the normal normal form form of of the the game: game: Firm 11 // Firm Firm Early Firm 22 Early Late Late B B L F Early Early ππ1B1 ,,ππ2B2 ππ1L1,,ππ2F2 αα F L B B Late Late ππF1 ,,ππL2 − αα ππ1B1 ,,ππ2B2 11 − 1 2 ββ 11 −− ββ As far As far as as firm firm 22 is is concerned, concerned, we we already already know know from from Exercise Exercise 4.1 that ππF2F > 4.1 that > ππL2L.. ItIt is is also also L 2 B 2 2 2 F easy to see that π L2 = 2(5a − 7c)2 /224 is larger than πB2 = 2(5a − 7c)2 /225. We thus have πF2 > easy to see that π 2 = 2(5a − 7c) /224 is larger than π 2 = 2(5a − 7c) /225. We thus have π 2 > L > ππB2B.. As ππ2L2 > As for for firm firm 1, 1, itit is is easy easy to to see that ππL1L = see that = 2(5a 2(5a + 2c)22/224 + 2c) > ππB1B = /224 > = 2(5a 2(5a + 2c)22/225. + 2c) /225. 2 F B 1 1 We can also show that We can also show that π > π : πF1 > πB1 : 1 1 F 2 2 − ππB1B = ππ1F1 − 1 = 1568 2 1 (19a + 7c)2 − 2 (5a + 2c)2 (19a + 7c) − 225225(5a + 2c) 1 1568 = 35211800 (5a = − 7c) (5a − (565a + 7c)(565a + 217c) 217c),, 352 800 which is which is positive positive as as we we assume assume that that 7c 7c < 5a. Hence, < 5a. Hence, there there are are two two pure-strategy pure-strategy NE:NE: (Early, (Early, Late) Late) and (Late, Early). There also exists a mixed-strategy NE. To characterize and (Late, Early). There also exists a mixed-strategy NE. To characterize it, let firm 1 choose it, let firm 1 choose the probability α, the α, probability with with which which itit plays plays (Early), (Early), such such that that firm firm 22 is is indifferent indifferent between between (Early) (Early) B L F B L and (Late); that is, απ B2 + (1 − α)πL2 = απF2 + (1 − α)πB2 , which is equivalent to α = (πL2 − and (Late); that is, απ 2 + (1 − α)π 2 = απ 2 + (1 − α)π 2 , which is equivalent to α = (π 2 − B )/(πF2F + ππ2B2 )/(π + ππL2L − 2πB2B).). We − 2π We proceed proceed in in the the same same way way for for firm firm 2,2, which which chooses chooses β, the probability β, the probability 2 2 2 with which it chooses (Early), to make firm 1 indifferent between with which it chooses (Early), to make firm 1 indifferent between (Early) and (Late); that (Early) and (Late); that is, is, B L F B L B F L B βπ B1 + (1 − β)πL1 = βπF1 + (1 − β)πB1 , which implies that β = (πL1 − πB1 )/(πF1 + πL1 − 2πB1 ). βπ 1 + (1 − β)π 1 = βπ 1 + (1 − β)π 1 , which implies that β = (π 1 − π 1 )/(π 1 + π 1 − 2π 1 ). Replacing the Replacing the various various profit profit level level by by their their respective respective value, value, oneone finds finds 2 2 14(5a−7c)2 14(5a+2c)2 = 3175a14(5a−7c) αα = 2 −3760ac−878c2 and β = 3175a 214(5a+2c) and β = 3175a 2 −2590ac−1463c −2590ac−1463c22.. 3175a 2 −3760ac−878c2  Exercises 135 Exercises 5.1 Horizontal product differentiation Hong Kong Island features steep, hilly terrain, as well as hot and humid weather. Travelling up and down the slopes therefore causes problems; this has led the city authorities to imagine rather unusual methods of transport. One famous example can be found in the Western District, where one of the busiest commercial areas of Hong Kong can be found. This area stretches from Des Voeux Road in Central (which is at sea level) up to Conduit Road in the Mid-Levels (which is the mid-section of the hill of Hong Kong Island). Because the street is so steep, sidewalks are made of stairs. To make travelling up the slope easier for pedestrians, the Mid-Levels escalators were opened to the public in October 1993. (See http://www.12hk.com/area/Central/MidLevelEscalators.shtml for some pictures of the escalators and the stairs of this area). For the sake of this problem set, imagine the following story. Suppose that the street is 1 km long (kilometre 0 is down at the crossroads with Des Voeux Road and kilometre 1 is up at the crossroads with Conduit Road). Suppose that 100 000 inhabitants are uniformly distributed along the street. Without loss of generality, we can approximate the consumer distribution by a continuum on [0, 1] with a mass set equal to 1 (i.e., we redefine all quantities by dividing them by 100 000). There are only two shops selling hot and sour soup in this area. For simplicity, we set their marginal cost of production to zero. As it happens, one shop (named ‘Won-Ton’ and indexed by 1) is located at point 0, while the other shop (named ‘Too-Chow’ and indexed by 2) is located at point 1. Every day, each inhabitant of the street may consume at most one bowl of hot and sour soup, bought either from Won-Ton or from Too-Chow. The prices per bowl of the two shops are respectively denoted by p1 and p2. The net utility for a consumer located at x on the interval [0, 1] is given by  r − τ1 (x) − p1 if consumer buys at Won-Ton, r − τ2 (1 − x) − p2 if consumer buys at Too-Chow,  0 if consumer does not buy, where it is assumed that r is large enough so that every consumer buys one bowl of soup. 1. Before 1993 and the installation of the Mid-Levels escalators, walking up the street with an empty stomach was much more painful than walking down. This is translated by the following assumptions: τ1 (x) = t x and τ2 (1 − x) = (t + τ ) (1 − x), with t, τ > 0. (a) Derive the identity of the consumer who is indifferent between the two shops. (b) Compute the equilibrium prices and profits of the two shops. (c) Show that Too-Chow’s profits increase if walking up the street becomes more costly for consumers, that is if τ increases (e.g., because the temperature has risen). Explain the intuition behind this result. 2. After 1993, the Mid-Levels escalators made going up and down equally painful for consumers. However, consumers had to pay a fixed fee f (independent of distance) to use the escalators. This is translated by the following assumptions: τ1 (x) = t x and τ2 (1 − x) = t (1 − x) + f , with f > 0.  π 1 = 12 (1 − q1 − q2 )q1 + 12 (1 − q1 )q1 , π 2 = (1 − q1 − q2 )q2. 136 Product differentiation First-order conditions of profit maximization at stage 2 can be written as q1 = (2 − q2 ) /4 and q2 = (1 − q1 ) /2. Solving this system we obtain q2 = 2/7 and q1 = 3/7. Note that q1m = 1/2 (a) > 3/7 >the Derive 1/3 = q1d. of identity Equilibrium the consumer profits who at is stage 2 under between indifferent information sharing the two are computed shops. s sL = 9/49 and as(b)π 1Compute π = the equilibrium 2 4/49. Because of strategic substitutes, prices and profits of the two shops. firm 1’s profit is larger than firm 2’s profit. (c) Express the condition (in terms of f and t) under which the previous answers are 2. We now validturn to the (i.e., the condition decision offor firm 2 at stageto1.setIfathe Too-Chow costs price of firm above 2 aremarginal its zero high, it obtains cost). zero profit independent of whether it shares information. If the costs of (d) Show that Too-Chow’s profits increase if taking the escalator becomes less expensive, firm 2 are low, then its profit will be higher if it shares information (1/9 rather than 4/49), because that is if f decreases. Explain the intuition behind this result and contrast with your then firm 1 will produce less. Thus, firm answer at (1c). 2 wants to share its information. Note that this is certainly true ex ante (i.e., if firm 2 has to decide before knowing its cost). In the example, it is also true interim (after 3. learning Comparing your because its cost), answers the for high (1) and cost(2), establish type and explain is indifferent whetherintuitively to share the followingor not. information results. 3. What about firm 1’s profit? If firm 2 shares information, firm 1’s expected profit is (1/2)π m 1 + (1/2)π d1 = (1/2)(1/4) + (1/2)(1/9) = 13/72. If firm 2 does not share information, its profit (a) Too-Chow suffers from the installation of the escalators (even when its access is free, has been calculated to be 9/49. Since 9/49 > 13/72, firm 1 would be better off if firm 2 did i.e., for f = 0). not share information. (b) Won-Ton benefits from the installation of the escalators, unless the extra tr