# Prerequisites Prerequisites Almost Almost essential essential Monopoly Monopoly

Prerequisites Prerequisites Almost Almost essential essential Monopoly Monopoly Useful, Useful,but butoptional optional Game GameTheory: Theory:Strategy Strategy and Equilibrium and Equilibrium DUOPOLY MICROECONOMICS Principles and Analysis Frank Cowell April 2018 Frank Cowell: Duopoly

1 Overview Duopoly Background How the basic elements of the firm and of game theory are used Price competition Quantity competition Assessment April 2018 Frank Cowell: Duopoly 2 Basic ingredients Two firms: issue of entry is not considered

but monopoly could be a special limiting case Profit maximisation Quantities or prices? theres nothing within the model to determine which weapon is used its determined a priori highlights artificiality of the approach Simple market situation: there is a known demand curve single, homogeneous product April 2018 Frank Cowell: Duopoly 3 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

April 2018 Frank Cowell: Duopoly 4 Overview Duopoly Background Introduction to a simple simultaneous move price-setting problem Price Price competition Competition Quantity competition Assessment April 2018 Frank Cowell: Duopoly 5

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 others 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 April 2018 Frank Cowell: Duopoly 6 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 pM > c if both firms are present they announce prices

The outcome of these announcements: if p1 < p2 firm 1 captures the whole market if p1 > p2 firm 2 captures the whole market if p1 = p2 the firms supply equal amounts to the market What will be the equilibrium price? April 2018 Frank Cowell: Duopoly 7 Bertrand best response? Consider firm 1s response to firm 2 If firm 2 foolishly sets a price p2 above pM then it sells zero output firm 1 can safely set monopoly price pM If firm 2 sets p2 above c but less than or equal to pM then: firm 1 can undercut and capture the market firm 1 sets p1 = p2 , where >0 firm 1s profit always increases if is made smaller but to capture the market the discount must be positive!

so strictly speaking theres 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 p1 = c at least it wont make a loss Lets look at the diagram April 2018 Frank Cowell: Duopoly 8 Bertrand model equilibrium Marginal cost for each firm Monopoly price level Firm 1s reaction function Firm 2s reaction function Bertrand equilibrium p2 pM

c c April 2018 B pM p1 Frank Cowell: Duopoly 9 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 sensitive to these

April 2018 Frank Cowell: Duopoly 10 Overview Duopoly Background The link with monopoly and an introduction to two simple competitive paradigms Price competition Quantity competition Collusion The Cournot model Leader-Follower Assessment

April 2018 Frank Cowell: Duopoly 11 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 April 2018 Frank Cowell: Duopoly 12 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 April 2018 Frank Cowell: Duopoly 13 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. 2. April 2018 where cash transfers between the firms are not possible where cash transfers are possible

Frank Cowell: Duopoly 14 Frontier non-transferable profits P2 Take case of identical firms Constant returns to scale DRTS (1): MC always rising DRTS (2): capacity constraints IRTS (fixed cost and constant MC) P1 April 2018 Frank Cowell: Duopoly 15

Frontier transferable profits Increasing returns to scale (without transfers) Now suppose firms can make side Profits payments if everything were produced by firm 1 Profits if everything were produced by firm 2 P2 The profit frontier if transfers are possible Joint-profit maximisation with equal shares PM Side payments mean profits can be transferred between firms Cash transfers convexify the set of attainable profits PJ PJ April 2018 P1

PM Frank Cowell: Duopoly 16 Collusion simple model Take the special case of the linear model where marginal costs are identical: c1 = c2 = c Will both firms produce a positive output? 1. 2. April 2018 if unlimited output is possible then only one firm needs to incur the fixed cost in other words a true monopoly 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 Frank Cowell: Duopoly 17 Collusion: capacity constraints If both firms are active total profit is [a bq] q [C01 + C02 + cq] Maximising this, we get the FOC: a 2bq c = 0 Which gives equilibrium quantity and price: ac q = ; 2b a+c p = 2 So maximised profits are: [a c]2 PM = 4b

[C01 + C02 ] Now assume the firms are identical: C01 = C02 = C0 Given equal division of profits each firms payoff is [a c]2 PJ = 8b April 2018 C0 Frank Cowell: Duopoly 18 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 C0 C0 is the smaller of the two firms fixed costs previous analysis only needs slight tweaking modify formula for PJ by replacing C0 with C0 But is the division of the profits still implementable? April 2018

Frank Cowell: Duopoly 19 Overview Duopoly Background Simultaneous move competition in quantities Price competition Quantity competition Collusion The Cournot model Leader-Follower Assessment April 2018 Frank Cowell: Duopoly

20 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 others decision when making its own Each firm makes an assumption about the others decision firm 1 assumes firm 2s output to be given number likewise for firm 2 How do we find an equilibrium? April 2018 Frank Cowell: Duopoly 21 Cournot model setup Two firms labelled f = 1,2 Firm f produces output qf So total output is: q = q 1 + q2

Market price is given by: p = p (q) Firm f has cost function Cf() So profit for firm f is: qf Cf(qf ) Each firms profit depends on the other firms output p(q) (because p depends on total q) April 2018 Frank Cowell: Duopoly 22 Cournot firms maximisation Firm 1s problem is to choose q1 so as to maximise P1(q1; q2) := p (q1 + q2) q1 C1 (q1) Differentiate P1 to find FOC: P1(q1; q2) = pq(q1 + q2) q1 + p(q1 + q2) Cq1(q1) q1 for an interior solution this is zero Solving, we find q1 as a function of q2

This gives us 1s reaction function, c1 : q1 = c1 (q2) Lets look at it graphically April 2018 Frank Cowell: Duopoly 23 Cournot the reaction function Firm 1s Iso-profit curves Assuming 2s output constant at q0 firm 1 maximises profit If 2s output were constant at a higher level 2s output at a yet higher level q2 c1() The reaction function P1(q1; q2) = const

q0 P1(q1; q2) = const P1(q1given ;given q2) =that const Firm 22 Firm1s 1schoice choice that chooses choosesoutput outputqq0 0 April 2018 q1 Frank Cowell: Duopoly 24

Cournot solving the model c1() encapsulates profit-maximisation by firm 1 Gives firms reaction 1 to fixed output level of competitor: q1 = c1 (q2) Of course firm 2s problem is solved in the same way We get q2 as a function of q1 : q2 = c2 (q1) Treat the above as a pair of simultaneous equations Solution is a pair of numbers (qC1 , qC2) So we have qC1 = c1(c2(qC1)) for firm 1 and qC2 = c2(c1(qC2)) for firm 2 This gives the Cournot-Nash equilibrium outputs April 2018 Frank Cowell: Duopoly

25 Cournot-Nash equilibrium (1) Firm 2s Iso-profit curves If 1s output is q0 firm 2 maximises profit q2 c1() P2(q2; q1) = const Repeat at higher levels of 1s output Firm 2s reaction function Combine with firm s reaction function Consistent conjectures Firm Firm2s 2schoice choicegiven giventhat

that11 chooses choosesoutput outputqq0 0 C c2() P1(q2; q1) = const P2(q2; q1) = const q0 April 2018 q1 Frank Cowell: Duopoly 26

Cournot-Nash equilibrium (2) q2 Firm 1s Iso-profit curves Firm 2s Iso-profit curves Firm 1s reaction function Firm 2s reaction function c1() Cournot-Nash equilibrium Outputs with higher profits for both firms Joint profit-maximising solution (qC1, qC2) c2() (q1J, q2J) 0 April 2018 q1

Frank Cowell: Duopoly 27 The Cournot-Nash equilibrium Why Cournot-Nash ? It is the general form of Cournots (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 others strategy (action) To see more, take a simplified example April 2018 Frank Cowell: Duopoly 28

Cournot a linear example Take the case where the inverse demand function is: p = b0 bq And the cost function for f is given by: Cf(qf ) = C0f + cf qf So profits for firm f are: [b0 bq ] qf [C0f + cf qf ] Suppose firm 1s profits are P Then, rearranging, the iso-profit curve for firm 1 is: b0 c 1 C01 + P q2 = q1 b b q1 April 2018 Frank Cowell: Duopoly 29 Cournot solving the linear example Firm 1s profits are given by P1(q1; q2) = [b0 bq] q1 [C01 + c1q1]

So, choose q1 so as to maximise this Differentiating we get: P1(q1; q2) = 2bq1 + b0 bq2 c1 q1 FOC for an interior solution (q1 > 0) sets this equal to zero Doing this and rearranging, we get the reaction function: q1 = max April 2018 { b0 c1 q2 , 0 2b } Frank Cowell: Duopoly 30 The reaction function again Firm 1s Iso-profit curves q2

Firm 1 maximises profit, given q2 The reaction function c1() P1(q1; q2) = const q1 April 2018 Frank Cowell: Duopoly 31 Finding Cournot-Nash equilibrium Assume output of both firm 1 and firm 2 is positive Reaction functions of the firms, c1(), c2() are given by: 1 a c q1 = q2 ; 2b

2 a c q2 = q1 2b Substitute from c2 into c1: a c1 a c2 1 qC = qC 2b 2b 1 Solving this we get the Cournot-Nash output for firm 1: a + c2 2c1 q = 3b 1 C By symmetry get the Cournot-Nash output for firm 2: a + c1 2c2

q = 3b 2 C April 2018 Frank Cowell: Duopoly 32 Cournot identical firms Take the case where the firms are identical useful but very special Use the previous formula for the Cournot-Nash outputs a + c2 2c1 q = 3b 1 C Reminder Reminder a + c1 2c2

; q = 3b 2 C Put c1 = c2 = c. Then we find qC1 = qC2 = qC where ac qC = 3b From the demand curve the price in this case is [a+2c] Profits are [a c]2 PC = C0 9b April 2018 Frank Cowell: Duopoly 33 Symmetric Cournot

q2 A case with identical firms1s reaction to firm Firm 2 Firm 2s reaction to firm 1 The Cournot-Nash equilibrium c1() qC C c2() qC April 2018 q1 Frank Cowell: Duopoly 34

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 by-product contrast with Bertrand model Absence of time in the model may be unsatisfactory April 2018 Frank Cowell: Duopoly 35 Overview Duopoly Background Sequential competition in quantities Price competition Quantity competition

Collusion The Cournot model Leader-Follower Assessment April 2018 Frank Cowell: Duopoly 36 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 2s reaction as a menu for choosing its own output April 2018

Frank Cowell: Duopoly 37 Leader-follower model Firm 1 (the leader) knows firm 2s reaction if firm 1 produces q1 then firm 2 produces c2(q1) Firm 1 uses c2 as a feasibility constraint for its own action Building in this constraint, firm 1s profits are given by p(q1 + c2(q1)) q1 C1 (q1) In the linear case firm 2s reaction function is Reminder Reminder 2 a c q2 = q1 2b So firm 1s profits are [a b [q1 + [a c2]/2b q1]]q1 [C01 + c1q1] April 2018

Frank Cowell: Duopoly 38 Solving the leader-follower model Simplifying the expression for firm 1s profits we have: [a + c2 bq1] q1 [C01 + c1q1] The FOC for maximising this is: [a + c2] bq1 c1 = 0 Solving for q1 we get: a + c2 2c1 q = 2b 1 S Using 2s reaction function to find q2 we get: a + 2c1 3c2 q = 4b 2 S April 2018 Frank Cowell: Duopoly

39 Leader-follower identical firms Of Of course course they they still still differ differ in in terms terms of of their their strategic strategic position position firm firm 11 moves moves first first Reminder Reminder Again assume that the firms have the same cost function Take the previous expressions for the Leader-Follower

outputs: a + c2 2c1 qS = ; 2b 1 a + 2c1 3c2 qS = 4b 2 Put c1 = c2 = c; then we get the following outputs: a c qS = ; 2b 1 a c qS = 4b 2 Using the demand curve, market price is [a + 3c] So profits are: [a c]2 2[a c]2 PS = C0 ; PS = C0 8b

16b 1 April 2018 Frank Cowell: Duopoly 40 Leader-Follower Firm 1s Iso-profit curves Firm 2s reaction to firm 1 Firm 1 takes this as an opportunity set q2 and maximises profit here Firm 2 follows suit qS2 Leader has higher output (and follower less) than in Cournot Nash S stands for von Stackelberg

C S c2() qS1 April 2018 q1 Frank Cowell: Duopoly 41 Overview Duopoly Background How the simple price- and quantitymodels compare Price competition Quantity competition Assessment April 2018

Frank Cowell: Duopoly 42 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 April 2018 Frank Cowell: Duopoly 43 Output under different regimes q2 Reaction curves for the two firms Joint-profit maximisation with equal outputs Cournot-Nash equilibrium

Leader-follower (Stackelberg) equilibrium qM qC qJ April 2018 J qJ C qC qM S q1 Frank Cowell: Duopoly 44

Profits under different regimes Attainable set with transferable profits Joint-profit maximisation with equal shares Profits at Cournot-Nash equilibrium P2 PM Profits in leader-follower (Stackelberg) equilibrium Cournot and leader-follower models yield profit levels inside the frontier PJ J C .

S PJ April 2018 P P1 M Frank Cowell: Duopoly 45 What next? Introduce the possibility of entry General models of oligopoly Dynamic versions of Cournot competition April 2018 Frank Cowell: Duopoly 46

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