Session 4: Homogeneous Products. What is a homogeneous product? Examples: 1. vegetables 2. fruit 3. grain 4. oil 5. construction hardware (lumber)

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1 Session 4: Homogeneous Products What is a homogeneous product? Examples: 1. vegetables 2. fruit 3. grain 4. oil 5. construction hardware (lumber) In this part of the course, we assume that buyers are competitive, but that firms behave according to some market structure. Many court cases involving collusion are in markets with homogeneous products. 1

2 Market Structures 2

3 Outline: Cournot (Nash-in-quantities) Cournot with Sequential Moves Bertrand (Nash-in-prices) Bertrand with capacity constraints Cournot versus Bertrand Cournot and Bertrand and generally referred to as conjectural variation models of firm behavior. 3

4 1. Cournot Cournot wrote in 1838 well before John Nash! He proposed an oligopoly-analysis method that is (under many conditions) in fact the Nash equilibrium w.r.t. quantities. Firms choose... production levels. A. Two-seller game Cost: T C i (q i ) = c i q i Demand: p(q) = a bq where Q = q 1 + q 2 4

5 Define a game: Players: firms. Action/Strategy set: production levels/quantities. Payoff function: profits, defined: π i (q 1, q 2 ) = p(q 1 + q 2 )q i T C i (q i ), i = 1, 2 Now we need an equilibrium concept. 5

6 {p c, q1 c, qc 2 } is a Cournot equilibrium ( Nashin-quantities equilibrium) if: 1. a) given q 2 = q c 2, qc 1 solves max q 1 π 1 (q 1, q c 2 ) b) given q 1 = q c 1, qc 2 solves max q 2 π 2 (q c 1, q 2) 2. p c = a b(q c 1 + qc 2 ), for pc, q c 1, qc 2 0. No firm could increase its profit by changing its output level, given that other firms produced the Cournot output levels. 6

7 π 1 (q 1, q 2 ) = (a bq 1 bq 2 )q 1 c 1 q 1 F.O.C. (for firm 1): π 1 (q 1, q 2 ) q 1 = a 2bq 1 bq 2 c 1 = 0 As always, set marginal revenue equal to marginal cost. How do we know this is a maximum and not a minimum? 2 π 1 (q 1, q 2 ) (q 1 ) 2 = 2b < 0 q 1, q 2 7

8 Note: q 1 is a function of q 2. It is the bestresponse function, a.k.a. reaction function of firm 1. Call this R 1 (q 2 ). q 1 = R 1 (q 2 ) = a c 1 2b 1 2 q 2 8

9 q 2 R 1 (q 2 ) q c 2 R 2 (q 1 ) q c 1 q 1 9

10 Note both rest-response functions are downward sloping. For each firm: if the rival s output increases, I lower my output level. (i.e., an increase in a rival s output shifts the residual demand facing my firm inward, leading me to produce less.) Now we can compute Cournot equilibrium output levels. q c 1 = a 2c 1 + c 2 3b q c 2 = a 2c 2 + c 1 3b Equilibrium quantity supplied on the market is Q c = q c 1 + qc 2 = 2a c 1 c 2 3b 10

11 We can also find equilibrium price. p c = a bq c = a + c 1 + c 2 3 What are the pay-off functions in equilibrium? π c i = (pc c i )(q c i ) = b(qc i )2 > 0 11

12 Extensions (we ll see some later in the course). Role of cost functions (own and rivals). Note there are no fixed costs in this example! Raising rival s costs R&D to reduce own costs 12

13 Extending this to N firms. It s harder to see the reaction functions, but the story is exactly the same. Now each firm maximizes profits according to: π i (q 1, q 2,...q N ) = p(q)q i T C i (q i ) We would derive the best response function for all N firms. For firm 1, q 1 = R 1 (q 2,..., q N ) = a c 1 2b 1 2 ( N j=2 q j ) 13

14 We need N of these equations. However, if we assume that firms costs are the same (T C i (q i ) = c i), it s a lot easier. Each firm has the same reaction function, which is q i = R i (q i ) = a c 2b 1 2 ( N j i q j ) 14

15 Look at the 2-firm example. When costs (c 1 and c 2 ) are the same, output levels of the two firms are the same. So, we guess in this case that all the output levels (q 1...q N ) are going to be the same too. If so, denote q i = q i. WATCH OUT! We cannot substitute in q i = q before we re derived the best response function. This is a common mistake. Why is it a mistake? Assuming that q i = q before we take first order conditions implies that firm i has control over all firms output decisions (the q i s). Here, we substitute q i = q into the already-derived best-response functions. We do this to make the process of solving N equations for N unknowns easier. And, hopefully because we believe the assumption that firms cost functions are truly the same. 15

16 Back to the reaction function. q i = R i (q i ) = a c 2b 1 2 ( N j i q j ) Substitute q i = q i: q = a c 2b 1 (N 1)q 2 Thus, q c = (a c) (N + 1)b Q c = N(a c) (N + 1)b Now it is straightforward to solve for p c (the market price) and πi c (profits for each firm). 16

17 Sanity checks... Do we get the monopoly result for N = 1? Do we get the duopoly result for N = 2? What is the Cournot solution for N =? Take the limit as N for q c and Q c and p c. They are: lim N qc = 0 lim N Qc = a c b lim N pc = c Under the assumption of Cournot competition, market supply approaches the competitive supply as N. Individual firms output levels approach zero as N. 17

18 Convince yourself that you can find the equilibrium quantity supplied and price in the market under the assumption of perfect competition. Simple way: let number of firms = 1, but assume that p Q = 0 and take F.O.C. to find the Q that maximizes profits for the firm under assumptions of perfect competition. This is what we did in the first class. 18

19 We have obtained producer surplus under the assumption that N firms in a market set outputs levels simultaneously according to a Nash-in-quantities equilibrium. What about consumer surplus? We can compute consumer surplus as the area under the demand curve up to the output level that is supplied on the market. CS c (N) = N 2 (a c) 2 2b(N + 1) 2 Does consumer surplus go up or down as the number of firms increases in this model? 19

20 As for social welfare, we want to compare the sum of firm profits and consumer surplus as the number of firms increases. (Firms are owned by consumers in the end...) Shy computes this as a function of the number of firms in the market. The bottom line: firm profits go down as more firms enter, but consumer surplus goes up, and goes up by more. Shy gives you the algebra. CS c (N)+Nπ c (N) = (a c)2 2b N 2 + 2N N 2 + 2N + 1 The important point: W c (N) N > 0 20

21 We have been assuming that all firms are identical, which implies symmetric equilibrium outcomes. Take a look at the results when firms have different levels of marginal cost. Let marginal cost of firm i be c i. The F.O.C. is only a slight modification from before. π i = [(a bq i b j i q j ) c i ](q i ) π i = a 2bqi q b qj c i = 0 i j i 21

22 Rewrite the F.O.C. as a bq bq i = c i Substitute in for price: Rearrange... p c i = bq i p c i p p c i p Q Q = bq i Q Q p = dp dq qi Q Q p The term q i Q is the market share of firm i. Denote this simply as s i. We now have the inverse elasticity rule of Cournot oligopoly: p c i p = s i ɛ d 22

23 2. Cournot with Sequential Moves: We could also think about this in a game where firm 1 moves first, firm 2 moves second, etc. We call this a leader-follower market structure, or a Stackelberg game. The sequential moves game is a natural way to think about situations with incumbent firms and potential entrants (for example). Work backwards: Suppose firm 1 sets output level to q 1. What would firm 2 do? R 2 (q 1 ) = a c 2b 1 2 q 1 23

24 Firm 1 can figure this out. Firm 1 do in response? What will max q1 p(q 1 + R 2 (q 1 ))q 1 cq 1 What s different between this profit function than Cournot profit function? Turns out leader output level is: q S 1 = a c 2b = 3 2 qc. Follower output level is: q S 2 = a c 3 4 qc. 4b = Equilibrium price is lower than Cournot, output is larger than Cournot hence more consumer surplus. First-mover advantage: leader makes more profit than follower. Leader is better off than in Cournot. 24

25 3. Bertrand ( Nash in Prices ) When do you think price setting makes more sense than setting quantity? In general, economists may believe that different assumptions hold for different settings. Then we have to argue about which one is more consistent with the data. Bertrand reviewed Cournot s work 45 years later. Go through a two-firm example again. Now firms set prices. We need two assumptions. 1. Consumers always purchase from the cheapest seller (recall defn of homogeneous goods). 2. If two sellers charge the same price, consumers are split 50/50. 25

26 The second assumption is that q i takes the values: 0 if p i > a 0 if p i > p j a p 2b if p i = p j = p < a a p i b if p i < min(a, p j ) Equilibrium: {p b 1, pb 2, qb 1, qb 2 } is a Bertrand equilibrium ( Nashin-prices equilibrium) if: 1. a) given p 2 = p b 2, p b 1 solves max p 1 π 1 (p 1, p b 2 ) = (p 1 c 1 )q 1 b) given p 1 = p b 1, p b 2 solves max p 2 π 2 (p b 1, p 2) = (p 2 c 2 )q 2 2. q 1 and q 2 are determined as above. No firm can its profit by unilaterally changing its price. 26

27 Do you see a difference between Cournot and Bertrand? Hint: One of them has an important discontinuity in the game (more specifically, there is a discontinuity in the pay-off functions.) Solving for the equilibrium required us to say something about costs. 27

28 1. If costs are the same, a Bertrand equilibrium is price = marginal cost, with quantity supplied on the market equal to the perfectly competitive outcome (equally split between the two firms). 2. If costs differ, (say firm 1 has cost = c 1 where c 1 < c 2 ), then the low-cost firm 1 charges p 1 = c 2 ɛ and sells quantity given by q b 1 = a c 2+ɛ b and the high-cost firm 2 charges p 2 = c 2 and sells zero. That is, if costs are the same, undercutting reduces price to marginal cost. If costs differ, undercutting reduces price to just below the cost of the high-cost firm. 28

29 Results (when costs are the same): 1. price competition reduces prices to unit costs 2. firms earn zero profits. Do we believe these results? Maybe, maybe not. We might worry about the fact that the number of firms makes no difference. Undercutting reduces price to marginal cost with only 2 firms. (Consider the policy implications of this for the anti-trust authorities.) 29

30 4. Bertrand with Capacity Constraints: In 1925 Edgeworth had the idea that in the short run firms may be constrained by capacity that limits their production levels. Example: There are two hair salons, both charging an initial price of $35. Under the basic Bertrand model, Salon 2 could take all customers if it set its price at $34. This, in turn, would lead Salon 1 to undercut to $33, etc., until p 1 = p 2 = c. Now suppose that initial business is 12 haircuts per day per firm, and that the maximum capacity for either firm is 15 haircuts in a day. The basic Bertrand result of maginal cost pricing won t arise. 30

31 If p 2 > p 1 = c, Salon 2 loses some customers, but not all of them, because Salon 1 cannot serve the entire demand. π 2 > 0 because p 2 > p 1 = c. In other words, there exists a unilateral profitable deviation from p 1 = p 2 = c. p 1 = p 2 = c is not a Nash equilibrium of the Bertrand game with capacity constraints (of the type described above). 31

32 In order the determine the Nash equilibrium of the price-setting game with capacity constraints, we can consider a two-stage game: 1. Firms invest in (choose) capacities; then 2. Firms choose prices As it turns out, the solution to this twostage game will give us a level of total market output (and hence price) that is identical to the equilibrium in Cournot model. So if we believe that capacity constraints are an important aspect to any particular industry, then the Cournot model is a useful approach to analyzing behavior of firms in that market. But if any one firm can satisfy total market demand, then we still get the competitive equilibrium. 32

33 Example: Let c = 0, p = 1 Q, or p = 1 q 1 q 2. Capacity of each firm: k 1, k 2 Capacity is acquired at unit cost c 0 > 3 4. Claim 1: Firms won t invest in capacities above 1 3. Proof 1: Monopoly profit is 1 4. Net profit is negative for k i > 1/3. 33

34 Claim 2: If k i 1/3, both firms charging p = 1 (k 1 + k 2 ) is an equilibrium, and this equilibrium is unique Proof 2, step 1: Suppose firm 2 is charging p and has capacity k 2. What should firm 1 do? Proof 2, step 2: Can firm 1 lower price? No, because at p, both firms are at capacity, so there s no use in lowering price. Proof 2, step 3: Can firm 1 raise price? If she charges p 1 > p, then profit is p 1 (1 k 2 p 1 ). The derivative is 1 k 2 2p 1. This is negative when p 1 = p = 1 (k 1 + k 2 ), i.e., raising price is not profitable. 34

35 Claim 3: Given Claim 2, it is optimal for firms to set capacities equal to Cournot levels in the first stage. Proof 3, step 1: In second stage, p = 1 (k 1 + k 2 ). Proof 3, step 2: Hence, firm 1 s profit function is: (1 k 1 k 2 )k 1, with derivative (or best-response function), 1 2k 1 k 2. Hence, k 1 = k 2 = 1 3. CRUCIAL ASSUMPTION: setting capacities above 3 1 is not profitable. This was due to the high investment costs (there s a bit of a paradox in the construction here, since firms appear to be making a loss). 35

36 We made this rather problematic assumption since it turns out that if firms build capacity above the Cournot level, then there is no pure strategy equilibrium in the second stage. Mixed strategies are pretty tough to derive; you can look at Kreps and Scheinkman (1983) for a full derivation. Main message of the above model, however, is that capacity constraints can soften Bertrand competition. Sometimes regulation/government can help firms coordinate on capacity limits. Example: Zoning laws in communities prevent large stores from being set up. 36

37 Comparison of Equilibrium in Models Covered So Far: Monopoly (M), Cournot (C), Stackelberg (S), Perfect competition and Bertrand (PC) CS M < CS C < CS S < CS P C P S M > P S C > P S S > P S P C = 0 DW L M > DW L C > DW L S > DW L P C = 0 37

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