# DUOPOLY. MICROECONOMICS Principles and Analysis Frank Cowell. July 2015 Frank Cowell: Duopoly. Almost essential Monopoly

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1 Prerequisites Almost essential Monopoly Useful, but optional Game Theory: Strategy and Equilibrium DUOPOLY MICROECONOMICS Principles and Analysis Frank Cowell 1

2 Overview Duopoly Background How the basic elements of the firm and of game theory are used Price competition Quantity competition Assessment 2

3 Basic ingredients Two firms: issue of entry is not considered but monopoly could be a special limiting case Profit maximisation Quantities or prices? there s nothing within the model to determine which weapon is used it s determined a priori highlights artificiality of the approach Simple market situation: there is a known demand curve single, homogeneous product 3

4 Reaction We deal with competition amongst the few Each actor has to take into account what others do A simple way to do this: the reaction function Based on the idea of best response we can extend this idea in the case where more than one possible reaction to a particular action it is then known as a reaction correspondence We will see how this works: where reaction is in terms of prices where reaction is in terms of quantities 4

5 Overview Duopoly Background Introduction to a simple simultaneous move price-setting problem Price competition Competition Quantity competition Assessment 5

6 Competing by price Simplest version of model: there is a market for a single, homogeneous good firms announce prices each firm does not know the other s announcement when making its own Total output is determined by demand determinate market demand curve known to the firms Division of output amongst the firms determined by market rules Take a specific case with a clear-cut solution 6

7 Bertrand basic set-up Two firms can potentially supply the market each firm: zero fixed cost, constant marginal cost c if one firm alone supplies the market it charges monopoly price p M > c if both firms are present they announce prices The outcome of these announcements: if p 1 < p 2 firm 1 captures the whole market if p 1 > p 2 firm 2 captures the whole market if p 1 = p 2 the firms supply equal amounts to the market What will be the equilibrium price? 7

8 Bertrand best response? Consider firm 1 s response to firm 2 If firm 2 foolishly sets a price p 2 above p M then it sells zero output firm 1 can safely set monopoly price p M If firm 2 sets p 2 above c but less than or equal to p M then: firm 1 can undercut and capture the market firm 1 sets p 1 = p 2 δ, where δ >0 firm 1 s profit always increases if δ is made smaller but to capture the market the discount δ must be positive! so strictly speaking there s no best response for firm 1 If firm 2 sets price equal to c then firm 1 cannot undercut firm 1 also sets price equal to c If firm 2 sets a price below c it would make a loss firm 1 would be crazy to match this price if firm 1 sets p 1 = c at least it won t make a loss Let s look at the diagram 8

9 Bertrand model equilibrium p 2 Marginal cost for each firm Monopoly price level Firm 1 s reaction function Firm 2 s reaction function Bertrand equilibrium p M c B c p M p 1 9

10 Bertrand assessment Using natural tools prices Yields a remarkable conclusion mimics the outcome of perfect competition price = MC But it is based on a special case neglects some important practical features fixed costs product diversity capacity constraints Outcome of price-competition models usually very sensitive to these 10

11 Overview Duopoly The link with monopoly and an introduction to two simple competitive paradigms Background Price competition Quantity competition Collusion The Cournot model Leader-Follower Assessment 11

12 Quantity models Now take output quantity as the firms choice variable Price is determined by the market once total quantity is known: an auctioneer? Three important possibilities: 1. Collusion: competition is an illusion monopoly by another name but a useful reference point for other cases 2. Simultaneous-move competing in quantities: complementary approach to the Bertrand-price model 3. Leader-follower (sequential) competing in quantities 12

13 Collusion basic set-up Two firms agree to maximise joint profits what they can make by acting as though they were a single firm essentially a monopoly with two plants They also agree on a rule for dividing the profits could be (but need not be) equal shares In principle these two issues are separate 13

14 The profit frontier To show what is possible for the firms draw the profit frontier Show the possible combination of profits for the two firms given demand conditions given cost function Distinguish two cases 1. where cash transfers between the firms are not possible 2. where cash transfers are possible 14

15 Frontier non-transferable profits Π 2 Take case of identical firms Constant returns to scale DRTS (1): MC always rising DRTS (2): capacity constraints IRTS (fixed cost and constant MC) Π 1 15

16 Frontier transferable profits Π 2 Π M Increasing returns to scale (without transfers) Now suppose firms can make side-payments Profits if everything were produced by firm 1 Profits if everything were produced by firm 2 The profit frontier if transfers are possible Joint-profit maximisation with equal shares Side payments mean profits can be transferred between firms Π J Cash transfers convexify the set of attainable profits Π J Π M Π 1 16

17 Collusion simple model Take the special case of the linear model where marginal costs are identical: c 1 = c 2 = c Will both firms produce a positive output? 1. if unlimited output is possible then only one firm needs to incur the fixed cost in other words a true monopoly 2. but if there are capacity constraints then both firms may need to produce both firms incur fixed costs We examine both cases capacity constraints first 17

18 Collusion: capacity constraints If both firms are active total profit is [a bq] q [C C cq] Maximising this, we get the FOC: a 2bq c = 0 Which gives equilibrium quantity and price: a c a + c q = ; p = 2b 2 So maximised profits are: [a c] 2 Π M = [C C 0 2 ] 4b Now assume the firms are identical: C 0 1 = C 0 2 = C 0 Given equal division of profits each firm s payoff is [a c] 2 Π J = C 0 8b 18

19 Collusion: no capacity constraints With no capacity limits and constant marginal costs seems to be no reason for both firms to be active Only need to incur one lot of fixed costs C 0 C 0 is the smaller of the two firms fixed costs previous analysis only needs slight tweaking modify formula for P J by replacing C 0 with ½C 0 But is the division of the profits still implementable? 19

20 Overview Duopoly Background Simultaneous move competition in quantities Price competition Quantity competition Collusion The Cournot model Leader-Follower Assessment 20

21 Cournot basic set-up Two firms assumed to be profit-maximisers each is fully described by its cost function Price of output determined by demand determinate market demand curve known to both firms Each chooses the quantity of output single homogeneous output neither firm knows the other s decision when making its own Each firm makes an assumption about the other s decision firm 1 assumes firm 2 s output to be given number likewise for firm 2 How do we find an equilibrium? 21

22 Cournot model setup Two firms labelled f = 1,2 Firm f produces output q f So total output is: q = q 1 + q 2 Market price is given by: p = p (q) Firm f has cost function C f ( ) So profit for firm f is: p(q) q f C f (q f ) Each firm s profit depends on the other firm s output (because p depends on total q) 22

23 Cournot firm s maximisation Firm 1 s problem is to choose q 1 so as to maximise Π 1 (q 1 ; q 2 ) := p (q 1 + q 2 ) q 1 C 1 (q 1 ) Differentiate Π 1 to find FOC: Π 1 (q 1 ; q 2 ) = p q (q 1 + q 2 ) q 1 + p(q 1 + q 2 ) C q1 (q 1 ) q 1 for an interior solution this is zero Solving, we find q 1 as a function of q 2 This gives us 1 s reaction function, χ 1 : q 1 = χ 1 (q 2 ) Let s look at it graphically 23

24 Cournot the reaction function q 2 χ 1 ( ) Firm 1 s Iso-profit curves Assuming 2 s output constant at q 0 firm 1 maximises profit If 2 s output were constant at a higher level 2 s output at a yet higher level The reaction function q 0 Π 1 (q 1 ; q 2 ) = const Π 1 (q 1 ; q 2 ) = const Π 1 (q 1 ; q 2 ) = const Firm 1 s choice given that 2 chooses output q 0 q 1 24

25 Cournot solving the model χ 1 ( ) encapsulates profit-maximisation by firm 1 Gives firm s reaction 1 to a fixed output level of the competitor firm: q 1 = χ 1 (q 2 ) Of course firm 2 s problem is solved in the same way We get q 2 as a function of q 1 : q 2 = χ 2 (q 1 ) Treat the above as a pair of simultaneous equations Solution is a pair of numbers (q C 1, q C2 ) So we have q C 1 = χ 1 (χ 2 (q C1 )) for firm 1 and q C 2 = χ 2 (χ 1 (q C2 )) for firm 2 This gives the Cournot-Nash equilibrium outputs 25

26 Cournot-Nash equilibrium (1) q 2 χ 1 ( ) Π 2 (q 2 ; q 1 ) = const Firm 2 s choice given that 1 chooses output q 0 Firm 2 s Iso-profit curves If 1 s output is q 0 firm 2 maximises profit Repeat at higher levels of 1 s output Firm 2 s reaction function Combine with firm s reaction function Consistent conjectures C q 0 Π 1 (q 2 ; q 1 ) = const Π 2 (q 2 ; q 1 ) = const χ 2 ( ) q 1 26

27 Cournot-Nash equilibrium (2) q 2 χ 1 ( ) Firm 1 s Iso-profit curves Firm 2 s Iso-profit curves Firm 1 s reaction function Firm 2 s reaction function Cournot-Nash equilibrium Outputs with higher profits for both firms Joint profit-maximising solution (q 1 C, q 2 C ) χ 2 ( ) (q1 J, q2 J ) 0 q 1 27

28 The Cournot-Nash equilibrium Why Cournot-Nash? It is the general form of Cournot s (1838) solution It also is the Nash equilibrium of a simple quantity game: players are the two firms moves are simultaneous strategies are actions the choice of output levels functions give the best-response of each firm to the other s strategy (action) To see more, take a simplified example 28

29 Cournot a linear example Take the case where the inverse demand function is: p = β 0 βq And the cost function for f is given by: C f (q f ) = C 0 f + c f q f So profits for firm f are: [β 0 βq ] q f [C f 0 + c f q f ] Suppose firm 1 s profits are Π Then, rearranging, the iso-profit curve for firm 1 is: β 0 c 1 C Π q 2 = q 1 β β q 1 29

30 Cournot solving the linear example Firm 1 s profits are given by Π 1 (q 1 ; q 2 ) = [β 0 βq] q 1 [C c 1 q 1 ] So, choose q 1 so as to maximise this Differentiating we get: Π 1 (q 1 ; q 2 ) = 2βq 1 + β 0 βq 2 c 1 q 1 FOC for an interior solution (q 1 > 0) sets this equal to zero Doing this and rearranging, we get the reaction function: β 0 c 1 q 1 = max ½ q 2, 0 2β { } 30

31 The reaction function again q 2 Firm 1 s Iso-profit curves Firm 1 maximises profit, given q 2 The reaction function χ 1 ( ) Π 1 (q 1 ; q 2 ) = const q 1 31

32 Finding Cournot-Nash equilibrium Assume output of both firm 1 and firm 2 is positive Reaction functions of the firms, χ 1 ( ), χ 2 ( ) are given by: a c 1 a c q 2 1 = ½q 2 ; q 2 = ½q 2b 2b 1 Substitute from χ 2 into χ 1 : 1 a c 1 a c 2 1 q C = ½ ½q C 2b 2b Solving this we get the Cournot-Nash output for firm 1: 1 a + c 2 2c 1 q C = 3b By symmetry get the Cournot-Nash output for firm 2: 2 a + c 1 2c 2 q C = 3b 32

33 Cournot identical firms Reminder Take the case where the firms are identical useful but very special Use the previous formula for the Cournot-Nash outputs 1 a + c 2 2c 1 2 a + c 1 2c 2 q C = ; q C = 3b 3b Put c 1 = c 2 = c. Then we find q 1 C = q 2 C = q C where a c q C = 3b From the demand curve the price in this case is ⅓[a+2c] Profits are [a c] 2 Π C = C 0 9b 33

34 Symmetric Cournot q 2 A case with identical firms Firm 1 s reaction to firm 2 Firm 2 s reaction to firm 1 The Cournot-Nash equilibrium χ 1 ( ) q C C χ 2 ( ) q C q 1 34

35 Cournot assessment Cournot-Nash outcome straightforward usually have continuous reaction functions Apparently suboptimal from the selfish point of view of the firms could get higher profits for all firms by collusion Unsatisfactory aspect is that price emerges as a byproduct contrast with Bertrand model Absence of time in the model may be unsatisfactory 35

36 Overview Duopoly Background Sequential competition in quantities Price competition Quantity competition Collusion The Cournot model Leader-Follower Assessment 36

37 Leader-Follower basic set-up Two firms choose the quantity of output single homogeneous output Both firms know the market demand curve But firm 1 is able to choose first It announces an output level Firm 2 then moves, knowing the announced output of firm 1 Firm 1 knows the reaction function of firm 2 So it can use firm 2 s reaction as a menu for choosing its own output 37

38 Reminder Leader-follower model Firm 1 (the leader) knows firm 2 s reaction if firm 1 produces q 1 then firm 2 produces c 2 (q 1 ) Firm 1 uses χ 2 as a feasibility constraint for its own action Building in this constraint, firm 1 s profits are given by p(q 1 + χ 2 (q 1 )) q 1 C 1 (q 1 ) In the linear case firm 2 s reaction function is a c q 2 2 = ½q 2b 1 So firm 1 s profits are [a b [q 1 + [a c 2 ]/2b ½q 1 ]]q 1 [C c 1 q 1 ] 38

39 Solving the leader-follower model Simplifying the expression for firm 1 s profits we have: ½ [a + c 2 bq 1 ] q 1 [C c 1 q 1 ] The FOC for maximising this is: ½ [a + c 2 ] bq 1 c 1 = 0 Solving for q 1 we get: 1 a + c 2 2c 1 q S = 2b Using 2 s reaction function to find q 2 we get: 2 a + 2c 1 3c 2 q S = 4b 39

40 Reminder Leader-follower identical firms Again assume that the firms have the same cost function Take the previous expressions for the Leader-Follower outputs: 1 a + c 2 2c 1 2 a + 2c 1 3c 2 q S = ; q S = 2b 4b Put c 1 = c 2 = c; then we get the following outputs: 1 a c 2 a c q S = ; q S = 2b 4b Using the demand curve, market price is ¼ [a + 3c] So profits are: 1 [a c] 2 2[a c] 2 Π S = C 0 ; Π S = C 8b 16b 0 Of course they still differ in terms of their strategic position firm 1 moves first 40

41 Leader-Follower q 2 Firm 1 s Iso-profit curves Firm 2 s reaction to firm 1 Firm 1 takes this as an opportunity set and maximises profit here Firm 2 follows suit Leader has higher output (and follower less) than in Cournot-Nash q S 2 C S χ 2 ( ) S stands for von Stackelberg q S 1 q 1 41

42 Overview Duopoly Background How the simple price- and quantitymodels compare Price competition Quantity competition Assessment 42

43 Comparing the models The price-competition model may seem more natural But the outcome (p = MC) is surely at variance with everyday experience To evaluate the quantity-based models we need to: compare the quantity outcomes of the three versions compare the profits attained in each case 43

44 Output under different regimes q 2 Reaction curves for the two firms Joint-profit maximisation with equal outputs Cournot-Nash equilibrium Leader-follower (Stackelberg) equilibrium q M q C q J J C S q J q C q M q 1 44

45 Profits under different regimes Π 2 Π M Attainable set with transferable profits Joint-profit maximisation with equal shares Profits at Cournot-Nash equilibrium Profits in leader-follower (Stackelberg) equilibrium Cournot and leader-follower models yield profit levels inside the frontier Π J J. C S Π J Π M Π 1 45

46 What next? Introduce the possibility of entry General models of oligopoly Dynamic versions of Cournot competition 46

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