EC930 Theory of Industrial Organisation Topic 2: Oligopoly theory

1. EC930 Theory ofIndustrial OrganisationTopic 2: Oligopoly theory 2013-14, spring term

2. Outline – Review and New Oligopoly : Homogeneous Products – For which market structures does price rise above marginal cost? Cournot-- concentration, prices and profits closely related Stackelberg– value of commitment – asymmetric outputs and profits, higher price Bertrand -- concentration, prices and profits may not be closely related (“Bertrand Paradox”) – a problem for regulation? Escaping the Bertrand Paradox: Capacity Constraints – Bertrand behaviour with Cournot outcomes – dampening “business stealing” Conjectural Variations – inferring behaviour: can we nest all the models in one? Measuring Market Power (the ability to raise price above costs) Monopoly – depends on price elasticity of demand Oligopoly -- concentration measures – CR, HHI Examples: Exam question; Salmon merger case Reading: Lecture notes 1, continued Problem Set 1; Problem Set 2 Support: Cabral ch. 7

3. Basic concepts of market structure • Monopoly: single firm supplying whole market • Perfect competition: large number of competing firms • Each believes that the market price is given, and is not influenced by its own actions (price takers) • Will not review these two explicitly…See Cabral 5,6 and problem set 1, q 1. • Oligopoly: competition among a few firms • Firms are interdependent: actions do affect price, and firms recognise this • Consumers are price-takers

4. Q: Thinking of the Cournot and Bertrand models, we can say: The Cournot model is more realistic than the Bertrand model since firms do use prices as a strategic variable more frequently than outputs. Neither model is realistic with homogeneous products, since all products are differentiated as a matter of fact. Neither model is realistic since both are static and real interactions are dynamic. Both models are good starting points for industry analysis, but only sometimes can be used without significant modification.

5. Oligopoly models • Static models: strategies chosen simultaneously • Cournot (1838): firms compete in quantities • Bertrand (1883): firms compete in prices • Dynamic models • Stackelberg leader-follower equilibrium • Initial capacity choices: Kreps and Scheinkman (1983)

6. Cournot Duopoly: static competition in quantities • 2 firms, 1 and 2, produce and demand is now • each firm “simultaneously” chooses q1 or so as to maximise: • Where we can add the outputs to generate demand when products homogeneous. • Each firm sets output to maximise profits given anticipated output of rival, where • we require that this expectation be consistent with outcome in equilibrium. • Neither firm can commit to its own output – ie set an output that is observed by the rival – before the other.

7. Cournot Duopoly: static competition in quantities First order condition: And this defines an implicit relation between the outputs of the two firms: In other words, the reaction or best response functions simply restate the first order conditions. The second order conditions are:

8. Cournot Duopoly: static competition in quantities To determine how the firms react to each other, we need to know the slope of the reaction function: whether 1 reacts aggressively or passively to an aggressive move by a rival. Implicitly differentiating the first order conditions, we obtain: And we know from the second order conditions that denominator is negative. Since MR is decreasing, we know reaction function has negative slope. If less than 1 in absolute value an increase in increases industry output (or stays the same if C’’ is zero).

9. Cournot Duopoly: static competition in quantities Notice: Any change in rival’s output does not affect own marginal cost, but does affect marginal revenue: When rival increases output, price falls. This lowers marginal revenue of firm 1…which tends to decrease firm 1’s output. If slope of demand not constant, then if it tends to become more negative, marginal revenue decreases further, strengthening this effect. If demand slope becomes more positive, it counteracts this effect. Hence, the reaction function slopes down if demand not too convex.

10. Figure 2: Residual Demand £ residual demand P = a-b( market demand • MC = c , Q

11. Figure 1: Cournot equilibrium Reaction function passes through monopoly output. The point of intersection is the point that satisfies all first order conditions, so all firms are maximising profits and their decisions are consistent with each other. This is the equilibrium, then.

12. Cournot Duopoly: static competition in quantities • 2 firms, 1 and 2, simultaneously choose q1 and q2 • Linear inverse demand fn: p = a – b(q1+q2) • Constant marginal cost c • Nash equilibrium: firm i chooses qi given choice of its rival • Set qi to max i = (a – bqi– bqj – c) qi for i = 1, 2, i  j • FOC: (a – bqi– bqj – c) – bqi = 0 • Rearrange: Reaction (BR) function for firm i • Solve simultaneous equations: • Price Profit

13. Figure 3: Cournot equilibrium, Example = =

14. n-firm Cournot oligopoly • n identical firms, each has cost c, linear demand • Cournot outcomes: using (a-bQ-c) – bqi = 0 for all i and symmetry • Quantity • Price • Profit • As n  : qi  0, Q  , p  c,   0

15. Figure 4: Isoprofit Contours Isoprofit contour of i: combinations of outputs for which profit of i remains constant. Profits of firm 1 increase in direction of arrow…

16. Stackelberg’s model of duopoly • Each firm’s strategic variable is output, as in Cournot’s model, but the firms make their decisions sequentially, rather than simultaneously: one firm chooses its output, then the other firm does so, knowing the output chosen by the first firm. • The “follower” behaves as it did in Cournot’s model, since the reaction function maps the profit maximising output given any output of the “leader”. • Leader knows its own behaviour will change the follower’s…and that follower behaves according to reaction function. • Hence, leader takes follower’s reaction function as given and maximises profit.

17. Figure 5: Stackelberg equilibrium Stackelberg equilibrium: 1 maximises profits subject to reaction function “constraint”: a point of tangency • •

18. Stackelberg’s model of duopoly • There are two firms in the market. • P = a-b( • MC = c for both players • Recall: • Max profit = [a-b( • FOC: • P = • Profit firm 2 = profits unequal… = price rises…

19. Bertrand: static competition in prices • What happens if firms instead choose prices? • Model • 2 firms, homogeneous product • Each sets price, assuming price set by its rival is given • Suppose firm 2 chooses p2 (c, pm] • Firm 1 would undercut by (tiny) , and steal whole market • Residual demand: /2 if

20. Bertrand: static competition in prices • Knowing this, 1 will undercut: 45◦ line £ • • c Firms indifferent among reactions to these prices c

21. Linear Costs and Demand • Ci (qi) = cqi, i=1, …, n. • D (p) = a – p if a > p, D (p) = 0 if a < p. • Let pj= min {p1, …, pn}. • Profit is: • pi(p1, …, pn) = (pi – c)(a - pi) if pi < pj, • pi(p) = (pi – c)(a - pi)/n if pi = pj, • pi(p) = 0 if pi > pj.

22. Bertrand paradox • Duopoly  competitive market outcome, zero profit • Is this realistic? • Extensions • Cost asymmetries: low cost firm just undercuts its rival’s cost and makes +ve profit • Capacity constraints • Repeated game: collusion • Differentiated goods

23. Bertrand with capacity constraints • Suppose each firm has maximum capacity k < D(c) • i.e. cost = c for q  k  for q > k • p1 = p2 = c is no longer an equilibrium • Firm 1 can increase p1 somewhat and still make positive sales, since 2 cannot supply D(c) • p1 (c, p(k)), p1 > p2  q1 = D(p1) – k2 • Firm 2 will raise p2 above c: but eqm not straightforward

24. Rationing rules • Prices alone do not determine each firm’s sales • Suppose p1 < p2 • Firm 1 supplies up to capacity: q1 = k1 • 2’s sales depend on which consumers buy from 1 • Rationing rules: which consumers are served first? • Lowest-valuation • These buy from (low p) firm 1 • Highest possible residual demand for firm 2 • Proportional: allocated randomly between firms • Efficient (surplus-max’ing): highest-valuation • Lowest residual demand for firm 2

25. Bertrand with capacity choices • Kreps & Scheinkman (1983): endogenise capacities • Stage 1: choose capacity k • Stage 2: compete in prices • Cost of capacity is given by a convex fn (i.e. incr in k) • Efficient rationing: unique eqm has Cournot q’s and p • Not robust to different rationing rules:can have multiple equilibria and higher outputs • Thus Cournot is a reasonable (and least optimistic) reduced form even when firms choose p, not q

26. Simple illustration of K&S 1983 • Inverse demand p = 1 – q1 – q2; • unit cost c = 0 for qi and c = ∞ for qi > • qi ≤ 1/3. • Stage 2: firms choose prices given q1 and q2 • Assume efficient rationing • Let firm 1 be low priced firm  q2 = 1- - p2 p P = 1- – q2 p1 q

27. Similarly, at prices below firm 1, firm 2 sells entire capacity Let p* = 1- . At this price, both sell to capacity and market clears. Also, p ≥ c = 0. Is p<p* profitable for either firm? At p* firm sells exactly As firm i cannot produce more than , so sells same quantity at lower price. This cannot increase profit. Is p>p* profitable for either firm? RD = 1-p2 -  PR2 = p2(1-p2- ) but p2 = 1-q2 -  PR2 = (1-q2- )q2. Note that PR2’ = 1-2q2 – q1 > 0 evaluated at capacity, and PR”<0 So that as reduce q2 from capacity, profits will fall.

28. Stage 1: firms choose outputs (capacities) anticipating eqm in stage 2 • Stage 2 eqm is the market clearing price • Thus, stage 1 capacity choice is equivalent to Cournot output choice • Eqm choices are Cournot outputs, q1 = q2 = 1/3 • NB: this is not a complete proof • Have not checked that larger capacities are not an eqm • Dependent on rationing rule • Cournot result arises under efficient rationing, but not from other rules (as incentive to raise price in stage 2)

29. Conjectural Variations In Cournot/Bertrand models, each firm takes other’s action as given (not a function of own choice – and not collusive). But suppose this need not be so. Define: ie, the conjectural variation is the expected change in all other firms’ outputs when I change my output; or, for N firms:

30. Conjectural Variations In Cournot/Bertrand models, each firm takes other’s action as given (not a function of own choice) so the conjectural variation is zero. A positive conjectural variation leads to a less competitive market than Cournot since any increase in output is anticipated to generate an increase by the rival (so expansion discouraged and price rises). A negative conjectural variation leads to a more competitive market than Cournot since any increase in output is anticipated to generate a decrease by the rival (since expansion encouraged and price falls). perfect collusion “like” perfect competition – I can sell as much as I want at the given price since any increase leaves price unchanged.

31. Conjectural variations allow us to estimate “true” market behaviour compared to our modelsby nesting all behaviours in one model. Eg. FOC: For Cournot, we assume that . Consider the linear demand example given earlier P = a-bQ and suppose the conjectural variation is 1: a-bQ + -2b) – c = 0 and the same for firm 2  a-3b …so the industry perfectly colludes. • Conjectural Variations

32. Conjectural variations allow us to summarise easily changes in “true” market behaviour compared to our models…to be used sparingly… Because it is not theoretically well-grounded. Why? In the Cournot model, the expectations are fulfilled in equilibrium: this is the meaning of equilibrium, in fact, as the original presumptions are confirmed (so there is no reason to change the presumption that generates the equilibrium)…in other words, the reaction function I imagined would be the reaction function that reflects your true behaviour (even out of equilibrium). • Conjectural Variations

33. If we insert a conjectural variation, each firm should observe that the rival adjusts output in response to own behaviour so that the slope of the reaction function should not be the same as in the Cournot model. If it were, then we would have consistency between what is anticipated and what actually would happen (even out of equilibrium). But…We haven’t done this …nor have we really defined the conjectural variation fully in a game theoretic sense: does it make sense to define what you would do conditional on my action…in a simultaneous game? …Does it make sense to define a constant rule no matter whether we are in or out of equilibrium? Cournot assumption has neat game theoretic interpretation that makes it easy to embed in more complex models of multiple-stage behaviour. • Conjectural Variations

34. Measurement of Market Power (ability to raise price above marginal cost) For a monopolist notice that: ) where: ɛ = (-) price elasticity of demand = - = - and this measures the sensitivity of demand to price changes (the percentage change in quantity demanded for a one percent change in price) But using MR = MC, we have: MR = MC =)  MC/P = 1-  or Hence, the profit maximising monopolist exercises market power in proportion to the sensitivity of demanders to price increases. We can measure elasticity, so we can compute this measure. A perfect competitor, on the other hand, exercises no market power and sets P = MC, since the perfect competitor faces infinitely elastic demand.

35. Measuring market structure • Equivalent of this for industry with many firms: • A sensible industry measure of market power is, then: But we often see market “concentration” rather than “market power” measured. Why? It is easier to measure when we don’t know cost… We have seen that in the case of Bertrand behaviour, market “concentration” – ie how many firms produce – is not well-correlated with market power. But we have just seen a justification for using the Cournot model more generally to represent market behaviour…market concentration may fare better here as a measure of market power…

36. Measuring market structure • How “concentrated” is a market? • Measures of concentration (for same market size) • With n equally-sized firms the market share of each firm, s = 1/n, would be a reasonable measure • E.g. 3 firms: s = 1/3 • With asymmetric firms: no unique measure • (r firm) Concentration Ratio: CRr = • Herfindahl-Hirschman index: HHI or H = • Monopoly: CR = HHI = 1 • Perfect competition: both approx. 0

37. Measuring market structure • Luckily, these concepts are related for Cournot behaviour: • Recall FOC in Cournot model of firm i: • P(Q) - • or… • so that, summing over all i, L = /ε •  HHI = Lε • for ε = absolute value of price elasticity of demand. • Hence, here we see that market structure and market conduct are related.

38. A B Firm 1 -- .35 Firm 1 -- .45 Firm 2 -- .14 Firm 2 -- .10 Firm 3 -- .10 Firm 3 -- .35 Firm 4 -- .31 Firm 4 -- .05 Firm 5 -- .10 Firm 5 -- .05 CR(3) = .35+.31+.14=.80 CR(3) = .45+.10+.35=.90 HHI=.352+.142+.102+.312 +.12 HHI = .452+.12+.352+.052+.052 =.1225+.0196+.01+.0961+.01 = .2025+.01+.1225+.0025+.0025 = .2582 = .34 What do these two indexes say about intensity of competition? Not clear, as we don’t know much about the strategic variable… Firms 1 and 2 in market A merge. New intensity of competition? Again, not clear as we don’t know much about strategic variable…A very large firm might have tendency to become a market leader, suggesting Stackelberg tendencies might emerge. Clearly, HHI changes (=.3562) as does CR3 (=.90)

39. Example: Merger of salmon producers (Pan Fish/Marine/Fjord) – MS1 = 18% MS2 = 6% MS3= 6% (total = 30% world harvest; 45% Scottish harvest) Others: 8%; 3%; 3%; 3%; 3%; rest of market = 49%, all tiny firms. Scottish salmon command small premium; production costs higher than Norway Prices well correlated over time (.89), and relative prices stationary over time. Previous studies usually group all Atlantic salmon in single market as close substitutes (transitory changes in production lead to significant market substitution in UK).

40. Capacity constraints significant (3 year maturation cycle)  strong Cournot characteristics HHI no merger: .1036 HHI, with merger: .1576  investigate given price elasticity estimates, impact of merger is about 1% price increase, assuming no cost efficiencies. Conduct robustness testing with alternative market definitions and demand elasticities (using historical estimates) MC £/tonne 1940 1920 Residual demand MR 200,000 240,000 Merged firm output

41. French Competition Commission allows for *no* rival response vs. Cournot due to capacity constraints and time to reaction (3 year cycle). (ie, in Cournot, reaction functions slope down so if I cut my output when I merge, you normally react by increasing…) With no rival response modelled, price effect of merger is higher (5%). Larger Lerner index effect if Scottish salmon is independent of Norwegian (HHI changes by over 1000 and has higher starting value).

42. Summary Market Structure, Conduct and Performance linked, but measuring market structure to infer performance not clear Cournot, Stackelberg, Bertrand – used duopoly in each case, different price and hence, consumer surplus outcomes. Are all models equally reasonable? Depends on capacity constraints. Capacity constraints in Bertrand setting *may* generate Cournot outcomes In this case, structure and performance tightly linked and measurable. Inferring strategic behaviour: Conjectural Variations are a “quick and dirty” method Measuring Market Structure: CR, HHI may generate different conclusions Using the model: Salmon Merger issues – Market definition, magnitude of price effects, nature of strategic response.

43. References Kreps, D. and Scheinkman, J. (1983) “Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes” Bell Journal of Economics 14(2), 326-337.

44. Q: Thinking of the Cournot and Bertrand models, we can say: The Cournot model is more realistic than the Bertrand model since firms do use prices as a strategic variable more frequently than outputs. Neither model is realistic with homogeneous products, since all products are differentiated as a matter of fact. Neither model is realistic since both are static and real interactions are dynamic. Both models are good starting points for industry analysis, but only sometimes can be used without significant modification.

45. Multiple Choice Comments Bertrand can be adapted to account for this observation by adding capacity constraints, which tends to yield Cournot- like outcomes. That being said, capacity constraints are not always applicable, as in an economic downturn. What matters is what the salient characteristic of consumer choice is. Sometimes, consumers view products as “essentially” undifferentiated (despite the best efforts of the firms), sometimes differences are salient. We gave an example of a case where differences not salient in lecture. Real interactions tend to be dynamic, although sometimes firms do enter as short term players. That being said, the static models are good “reversions” for dynamic models and so are useful to derive equilibrium. We’ll note (eg in tacit collusion analysis) that even in dynamic settings, equilibrium may not move from the static case. See later. This is generally an accurate statement, although sometimes other elements (like sequencing decisions) make other models more applicable (such as Stackelberg).