# Duopoly - London School of Economics

Prerequisites Prerequisites Almost Almostessential essential Monopoly Monopoly Frank Cowell: Microeconomics Useful, Useful,but butoptional optional Game Theory: Strategy Game Theory: Strategy and andEquilibrium Equilibrium January 2007 Duopoly MICROECONOMICS Principles and Analysis Frank Cowell Overview... Duopoly

Frank Cowell: Microeconomics Background How the basic elements of the firm and of game theory are used. Price competition Quantity competition Assessment Basic ingredients Frank Cowell: Microeconomics Two firms: game between them Profit maximisation. Quantities or prices?

Issue of entry is not considered. But monopoly could be a special limiting case. 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. Reaction Frank Cowell: Microeconomics 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. Overview... Duopoly Frank Cowell: Microeconomics Background Introduction to a simple simultaneous move price-setting problem.

Price competition Quantity competition Assessment Competing by price Frank Cowell: Microeconomics 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. Lets take a specific model with a clear-cut solution Bertrand basic set-up Frank Cowell: Microeconomics

Two firms can potentially supply the market. Each firm: zero fixed cost, constant marginal cost c. If one firm alone supplied the market it would charge 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? Bertrand best response? Frank Cowell: Microeconomics Consider firm 1s response to firm 2

If firm 2 foolishly sets a price p2 above pM then it sells zero output. 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 also sets price equal to c . If firm 2 sets a price below c it would make a loss. 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 can safely set monopoly price pM . 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 Bertrand model equilibrium Frank Cowell: Microeconomics Marginal cost for each firm p 2 Monopoly price level Firm 1s reaction function pM Firm 2s reaction function Bertrand equilibrium c c

B pM p1 Bertrand assessment Frank Cowell: Microeconomics Using natural tools prices. Yields a remarkable conclusion. Mimics the outcome of perfect competition But it is based on a special case. Neglects some important practical features Price = MC.

Fixed costs. Product diversity Capacity constraints. Outcome of price-competition models usually very sensitive to these. Overview... Duopoly Frank Cowell: Microeconomics Background The link with monopoly and an introduction to two simple competitive paradigms. Price competition Quantity competition Assessment Collusion The Cournot model Leader-Follower

quantity models Frank Cowell: Microeconomics Now take output quantity as the firms choice variable. Price is determined by the market once total quantity is known: 1. Three important possibilities: Collusion: 2. Competition is an illusion. Monopoly by another name. But a useful reference point for other cases Simultaneous-move competing in quantities: 3. An auctioneer?

Complementary approach to the Bertrand-price model. Leader-follower (sequential) competing in quantities. Collusion basic set-up Frank Cowell: Microeconomics Two firms agree to maximise joint profits. This is what they can make by acting as though they were a single firm. They also agree on a rule for dividing the profits. Essentially a monopoly with two plants. Could be (but need not be) equal shares. In principle these two issues are separate. The profit frontier Frank Cowell: Microeconomics

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 Frontier transferable profits Frank Cowell: Microeconomics 2 ) Now suppose firms can make side-payments So profits can be transferred between firms Profits if everything were produced by firm 1 Profits if everything were produced by firm 2 M The profit frontier if transfers are possible Joint-profit maximisation with equal shares

J J M 1 Cash transfers convexify the set of attainable profits. Collusion simple model Frank Cowell: Microeconomics Take the special case of the linear model where marginal costs are identical: c1 = c2 = c. Will both firms produce a positive output?

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. Collusion: capacity constraints Frank Cowell: Microeconomics 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 M = [C 0 1 + C 0 2 ] . 4b Now assume the firms are identical: C01 = C02 = C0. Given equal division of profits each firms payoff is [a c]2 J = C0 . 8b Collusion: no capacity constraints Frank Cowell: Microeconomics With no capacity limits and constant marginal costs there 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 J by replacing C0 with C0. But is the division of the profits still implementable? Overview... Duopoly Frank Cowell: Microeconomics Background Simultaneous move competition in quantities Price competition Quantity competition Assessment

Collusion The Cournot model Leader-Follower Cournot basic set-up Frank Cowell: Microeconomics Two firms. Price of output determined by demand. Single homogeneous output. Neither firm knows the others decision when making its own. Each firm makes an assumption about the others decision

Determinate market demand curve Known to both firms. Each chooses the quantity of output. Assumed to be profit-maximisers Each is fully described by its cost function. Firm 1 assumes firm 2s output to be given number. Likewise for firm 2. How do we find an equilibrium? Cournot model setup Frank Cowell: Microeconomics Two firms labelled f = 1,2 Firm f produces output qf. So total output is: Market price is given by:

q = q1 + q2 p = p (q) Firm f has cost function Cf(). So profit for firm f is: p(q) qf Cf(qf ) Each firms profit depends on the other firms output (because p depends on total q). Cournot firms maximisation Frank Cowell: Microeconomics

Firm 1s problem is to choose q1 so as to maximise 1(q1; q2) := p (q1 + q2) q1 C1 (q1) Differentiate 1 to find FOC: 1(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, 1 : q1 = 1 (q2) Lets look at it graphically Cournot the reaction function Frank Cowell: Microeconomics Firm 1s Iso-profit curves Assuming 2s output constant at q0 firm 1 maximises profit q2 If 2s output were constant at a higher level 2s output at a yet higher level 1() The reaction function 1(q1; q2) = const

q0 1(q1; q2) = const 1 1 (q ;given q2) =that const Firm Firm1s 1schoice choice given that 22chooses choosesoutput outputqq0 0 q1 Cournot solving the model Frank Cowell: Microeconomics 1() encapsulates profit-maximisation by firm 1. Gives firms reaction 1 to a fixed output level of the competitor firm:

Of course firm 2s problem is solved in the same way. We get q2 as a function of q1 : q2 = 2 (q1) Treat the above as a pair of simultaneous equations. Solution is a pair of numbers (qC1 , qC2). q1 = 1 (q2) So we have qC1 = 1(2(qC1)) for firm 1 and qC2 = 2(1(qC2)) for firm 2. This gives the Cournot-Nash equilibrium outputs. Cournot-Nash equilibrium (1) q2 firm 2 maximises profit

Repeat at higher levels of 1s output Firm 2s reaction function 1() Combine with firm s reaction function 2(q2; q1) = const Consistent conjectures Firm Firm2s 2schoice choicegiven giventhat that11 chooses choosesoutput outputqq0 0 C

Frank Cowell: Microeconomics Firm 2s Iso-profit curves If 1s output is q0 2() 1(q2; q1) = const 2(q2; q1) = const q0 q1 Cournot-Nash equilibrium (2) Frank Cowell: Microeconomics q2 Firm 1s Iso-profit curves Firm 2s Iso-profit curves Firm 1s reaction function Firm 2s reaction function 1() Cournot-Nash equilibrium Outputs with higher profits for both firms Joint profit-maximising solution (qC1, qC2)

2() (q1J, q2J) 0 q1 The Cournot-Nash equilibrium Frank Cowell: Microeconomics Why Cournot-Nash ? It is the general form of Cournots (1838) solution. But it also is the Nash equilibrium of a simple quantity game: The players are the two firms. Moves are simultaneous.

Strategies are actions the choice of output levels. The functions give the best-response of each firm to the others strategy (action). To see more, take a simplified example Cournot a linear example Frank Cowell: Microeconomics Take the case where the inverse demand function is: p = q And the cost function for f is given by: Cf(qf ) = C0f + cf qf So profits for firm f are: [ q ] qf [C0f + cf qf ] Suppose firm 1s profits are Then, rearranging, the iso-profit curve for firm 1 is: c1 C01 + q2 = q1

q1 Cournot solving the linear example Frank Cowell: Microeconomics Firm 1s profits are given by 1(q1; q2) = [ q] q1 [C1 + c1q1] So, choose q1 so as to maximise this. Differentiating we get: 1(q1; q2) = 2q1 + q2 c1 q1 FOC for an interior solution (q1 > 0) sets this equal to zero. Doing this and rearranging, we get the reaction function: { }

c1 q1 = max q2 , 0 The reaction function again Frank Cowell: Microeconomics Firm 1s Iso-profit curves q2 Firm 1 maximises profit, given q2 . The reaction function 1() 1(q1; q2) = const q1 Finding Cournot-Nash equilibrium Frank Cowell: Microeconomics

Assume output of both firm 1 and firm 2 is positive. Reaction functions of the firms, 1(), 2() are given by: a c1 q = q2 ; 2b 1 a c2 q = q1 . 2b 2 Substitute from 2 into 1: a c1 a c2 1 q = qC . 2b 2b 1 C Solving this we get the Cournot-Nash output for firm 1: a + c2 2 c1 q = . 3b

1 C By symmetry get the Cournot-Nash output for firm 2: a + c1 2 c2 q = . 3b 2 C Cournot identical firms Frank Cowell: Microeconomics Reminder Reminder Take the case where the firms are identical. This is useful but very special. Use the previous formula for the Cournot-Nash outputs. a + c2 2c1 a + c1 2c2 2 q = ; qC = .

3b 3b 1 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 C = C0 . 9b Symmetric Cournot Frank Cowell: Microeconomics A case with identical firms Firm 1s reaction to firm 2 q2

Firm 2s reaction to firm 1 The Cournot-Nash equilibrium 1() qC C 2() qC q1 Cournot assessment Frank Cowell: Microeconomics Cournot-Nash outcome straightforward. 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. Usually have continuous reaction functions. Contrast with Bertrand model. Absence of time in the model may be unsatisfactory. Overview... Duopoly Frank Cowell: Microeconomics Background Sequential competition in quantities Price competition Quantity competition

Assessment Collusion The Cournot model Leader-Follower Leader-Follower basic set-up Frank Cowell: Microeconomics Two firms choose the quantity of output. Both firms know the market demand curve. But firm 1 is able to choose first. Single homogeneous output. 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 Leader-follower model Frank Cowell: Microeconomics Firm 1 (the leader) knows firm 2s reaction. If firm 1 produces q1 then firm 2 produces 2(q1). Firm 1 uses 2 as a feasibility constraint for its own action. Building in this constraint, firm 1s profits are given by p(q1 + 2(q1)) q1 C1 (q1) In the linear case firm 2s reaction function is a c2 q2 = q1 . 2b Reminder Reminder

So firm 1s profits are [a b [q1 + [a c2]/2b q1]]q1 [C01 + c1q1] Solving the leader-follower model Frank Cowell: Microeconomics Simplifying the expression for firm 1s profits we have: [a + c2 bq1] q1 [C0 1 + c1q 1] The FOC for maximising this is: [a + c2] bq1 c1 = 0 Solving for q1 we get: Using 2s reaction function to find q2 we get: a + c2 2c1 1 q S = . 2b a + 2c 2c1 3c2 2 q S = . 4b

Leader-follower identical firms Frank Cowell: Microeconomics Of course they still differ in terms of their strategic position firm 1 moves first. Reminder Reminder Again assume that the firms have the same cost function. Take the previous expressions for the Leader-Follower outputs: a + c 2 2c 1 q = ; 2b 1 S Put c1 = c2 = c; then we get the following outputs: a c q = ; 2b 1 S

a + 2c 1 3 c 2 q = . 4b 2 S a c q = . 4b 2 S Using the demand curve, market price is [a + 3c]. So profits are: [a c]2 = C0 ; 8b 1 S [a c]2 = C0 . 16b 2 S Leader-Follower Frank Cowell: Microeconomics

Firm 1s Iso-profit curves q2 Firm 2s reaction to firm 1 Firm 1 takes this as an opportunity set and maximises profit here Firm 2 follows suit qS2 Leader has higher output (and follower less) than in Cournot-Nash C qS1 S 2() q1 S stands for von Stackelberg Overview...

Duopoly Frank Cowell: Microeconomics Background How the simple price- and quantitymodels compare. Price competition Quantity competition Assessment Comparing the models Frank Cowell: Microeconomics 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. Output under different regimes Frank Cowell: Microeconomics q2 Reaction curves for the two firms. Joint-profit maximisation with equal outputs Cournot-Nash equilibrium Leader-follower (Stackelberg) equilibrium qM qC qJ J qJ C qC

qM S q1 Profits under different regimes Frank Cowell: Microeconomics Attainable set with transferable profits 2 Joint-profit maximisation with equal shares M Profits at Cournot-Nash equilibrium Profits in leader-follower (Stackelberg) equilibrium J J Cournot and leader-follower

models yield profit levels inside the frontier. . C S J M 1 What next? Frank Cowell: Microeconomics Dynamic versions of Cournot competition Dynamic versions of Bertrand Competition

## Recently Viewed Presentations

• Relationships (of all varieties): THERE ONCE WAS A TIME WHEN A THREE-MINUTE PHONE CALL WOULD HAVE AVOIDED SETTING OFF THE DOWNWARD SPIRAL THAT RESULTED IN A COMPLETE RUPTURE. I believe this is true 100% of the time. Most of my...
• Golgi Apparatus (or Golgi Body) Golgi body is like a UPS center. It packs different carbohydrates, proteins, and lipids into containers called vesicles to be shipped to other parts of the cell. Named after its discoverer, Camillo Golgi
• CPC and EMC, NCEP/NWS/NOAA 32nd Climate Diagnostics and Prediction Workshop October 22-26, 2007 Tallahassee, Florida Examples of Storm Tracks for 4 NH Basins Examples of Storm Tracks for 4 NH Basins CFS Test and Evaluation Team - One of three...
• Appendages of the Skin. Sebaceous (oil) glands. Located all over the skin except for palms and soles. Produce sebum (oil) Makes skin soft and moist . Prevents hair from becoming brittle. Kills bacteria. Most have ducts that empty into hair...
• Weight values in second matrix calculated by normalizing importance ratings in first matrix (i.e.,divide by columntotal), then the mean value of each row's normalized values in second matrix found. AHP Consistency Ratio (CR), calculated from series of standardized equations (next...
• Tricare Health System. U.S. Naval Observatory. and others. Department of Defense research. Developed by: U-MIC 32 CFR 219. Department of Defense research. Developed by: U-MIC funding. cooperation, collaboration, or other agreements. ... University of Michigan Medical School ...
• Non-universalizable exception. People should respect property, except for me. This is the exception that morality condemns. The immoral person (Kant's immoralist) wants to return to animal individuality
• The Lana'i volcano reached the subaerial sheild stage, has a single volcanic rock member Lana'i Member, tholeiitic basalt, 1.46 to 0.78 Ma pahoehoe and a'ā, and and spatter cones on the rift zones caldera partly filled by lava, but is...