LECGE1330_24_01_ProductPrice.pdf

LECGE1330 – Industrial Organization 1. Product and price decisions Learning objectives Analyse basic models of oligopoly to understand how competing firms… Position their products; Choose at which price to sell their products. Choose how much to produce. Master the game-theory...

LECGE1330 – Industrial Organization 1. Product and price decisions Learning objectives Analyse basic models of oligopoly to understand how competing firms… Position their products; Choose at which price to sell their products. Choose how much to produce. Master the game-theory tools necessary to carry out this analysis. Apply the analysis to real-life cases. Complementary readings Belleflamme & Peitz (2015). Industrial Organization: Markets and Strategies 1.A. Chapter 5 (5.1, 5.2.1), Chapter 3 (3.1.1, 3.1.3) 1.B. Chapter 3 (3.1.1, 3.1.3), Chapter 5 (5.2.2, 5.2.3, 5.3.1) 1.C. Chapter 3 (3.2.1, 3.3, 3.4) Introduction Our running story 2 companies … sell “glints” … to a large mass of potential consumers ATREX BULVA 5 1. Product and price decisions A. Which sort of glint should I sell? B. At which price(s) should I sell my product? INTERDEPENDENT DECISIONS C. Which quantity should I produce? GAME THEORY ATREX BULVA 6 1.A Product positioning 1.A Product positioning A. Which sort of glint should I sell? How should I position my glints with DECISIONS ABOUT respect to those of my competitor? “PRODUCT Should I sell a ‘mass’ or a ‘niche’ DIFFERENTIATION” product? Should I sell a better product? ATREX BULVA 8 A model of product differentiation 9 Hotelling model - Representation Disutility from 𝑇(𝑥, 𝑙! ) “travelling” 𝑇(𝑥, 𝑙" ) l1 l2 0 1 x c p1 c p2 € Firm 1 € Firm 2 ATREX BULVA € € € € € Mass 1 of consumers, uniformly distributed 𝑟 − 𝑇 𝑥, 𝑙! − 𝑝! if buys from 1 𝑣 𝑥 = $𝑟 − 𝑇(𝑥, 𝑙" ) − 𝑝" if buys from 2 0 if doesn’t buy 10 Hotelling model - Setting Consumers Mass 1, uniformly distributed on 0,1 Uniform distribution → There are 0.25 “consumers” between 0 and 0.25 Location = location in geographical space OR ideal point in product space Unit demand (consumers buy at most one unit from one of the firms) Firms Constant marginal cost of production: 𝑐# = 𝑐 Choose their location: 𝑙# ∈ 0,1 Possibly, choose their price: 𝑝# 11 Geographical interpretation 6000 people live in Soluszowa, Poland on one single street. https://twitter.com/ianbremmer/status/1642559514737090561 https://twitter.com/RustyRoad/status/1572056389673000961?s=20 12 Product space interpretation Very hard texture Very soft texture Possible textures of glints (0% of Metomol) (100% of Metomol) l1 x 0 1 € ATREX produces € Miss x’s would glints with 30% of prefer glints with Metomol 60% of Metomol 𝑙! = 0.3 𝑥 = 0.6 If Miss x consumes glints from Atrex, her utility is reduced because she doesn’t get her preferred texture. (She is 0.6 − 0.3 = 0.3 “away from” her preferred product.) 13 Illustration Product space interpretation Ranking of breakfast cereals according to their sugar content https://www.elevate.in/?w=which-instant-cereal-is-best-for-you-life-cc-4dNvE6Cv 14 A simple location model Suppose that firms cannot choose their price Firms only choose their location. Prices are regulated. If duopolists choose product locations WATCH (but do not set prices), they offer the same products, that is, they choose not to differentiate their products. (Principle of minimum differentiation) This is detrimental from a social point of view as consumers as a whole would be better off if products were more differentiated. 15 A simple location model: Formally Firms Firms choose how to position their product in the product space: Firm 𝑖 chooses 𝑙# ∈ 0,1 They take the price of the product as given. Let 𝑝̅ denote the imposed price. Consumers Mass 1 uniformly distributed on 0,1 Consumer’s location = Ideal point in product space Unit demand Each consumer buys at most one unit from one of the firms. → Total demand for a firm = # of consumers choosing to buy from this firm Utility for consumer 𝑥 if they buy from firm 𝑖: 𝑣# 𝑥 = 𝑟 − 𝜏 𝑥 − 𝑙# − 𝑝̅ Assumption: Linear “transportation” cost → The disutility from not having one’s preferred product is proportional to the distance separating the consumer’s ideal location and the location of the chosen product. Utility from not consuming = 0 16 A simple location model: Formally (2) Demands Suppose: 0 ≤ 𝑙! ≤ 𝑙" ≤ 1 Consumers’ decision: Buy from firm 1 iff 𝑣, 𝑥 ≥ 𝑣- 𝑥 ⇔ 𝑟 − 𝜏 𝑥 − 𝑙, − 𝑝̅ ≥ 𝑟 − 𝜏 𝑙- − 𝑥 − 𝑝̅ ⇔ 𝑥 ≤ ,- 𝑙, + 𝑙- So, the demands for the two firms are: 𝑄, 𝑙,, 𝑙- = ,- 𝑙, + 𝑙- 𝑄- 𝑙,, 𝑙- = 1 − !" 𝑙, + 𝑙- 17  THEORY EXCURSUS Normal form games Games in normal form A normal form game consists of Set of players, 𝑁 For each player 𝑖 in 𝑁, set of feasible actions 𝑋# For each player 𝑖 in 𝑁, profit function 𝜋. (𝑥. , 𝑥/. ) Player 𝑖’s action Other players’ actions profile Strategy Pure strategy: 𝑥# in 𝑋# Mixed strategy: 𝜎# in Σ# (probability distribution over 𝑋# ) 19 Games in normal form (2) Best-response (in pure strategies) 𝑥# such that 𝜋# (𝑥# , 𝑥$# ) ≥ 𝜋# (𝑥#% , 𝑥$# ) for all 𝑥#% ∈ 𝑋# Best-response correspondence 𝑅# 𝑥$# = 𝑥# ∈ 𝑋# 𝜋# (𝑥# , 𝑥$# ) ≥ 𝜋# (𝑥#% , 𝑥$# ) for all 𝑥#% ∈ 𝑋# Nash equilibrium (NE) in pure strategies Strategy profile 𝑥 ∗ such that for all players 𝑖 in 𝑁, 𝜋. (𝑥.∗ , 𝑥/. ∗ ∗ ) ≥ 𝜋. (𝑥. , 𝑥/. ) for all 𝑥. ∈ 𝑋. or 𝑥.∗ ∈ 𝑅. 𝑥/. ∗ Nash equilibrium (NE) in mixed strategies Mutual Strategy profile 𝜎 ∗ such that for all players 𝑖 in 𝑁, best-responses 𝜋. (𝜎.∗ , 𝜎/. ∗ ∗ ) ≥ 𝜋. (𝜎. , 𝜎/. ) for all 𝜎. ∈ Σ. 20 Finite games Definition Each player’s strategy set consists of a finite number of possible actions. Properties A pure-strategy NE may fail to exist. Any finite game must have at least one mixed-strategy equilibrium. Example The ‘Battle of the sexes’ (a coordination game) Watch an explanatory video on Moodle. You will see that, in this game, there are 2 pure-strategy NE and 1 mixed-strategy NE. Watch video “Battle of the sexes” 21 Continuous games Definition Each player’s strategy is a real number. Properties Suppose that there is a lower and an upper bound for each player’s strategy. A sufficient condition for the existence of a NE in pure strategies is that the payoff function 𝜋# is quasi-concave in 𝑥# and continuous in 𝑥 for all players 𝑖. Note: If a function of one variable is single-peaked, then it is quasi-concave. Example The price and quantity competition games that we will analyze in the next section. 22 A simple location model: Formally (3) Normal for game 2 players: firm 1 and firm 2 Feasible actions: choose 𝑙# ∈ 0,1 Profit functions: 23 A simple location model: Formally (3) Best-response Increases with 𝑙# → Choose 𝑙# = 𝑙$ − 𝜀 2 1 Decreases with 𝑙# → Choose 𝑙# = 𝑙$ + 𝜀 4 3 Nash equilibrium 24 1.B Price decisions 1.B Price decisions A. Which sort of glint should I sell? B. At which price(s) should I sell my product? INTERDEPENDENT DECISIONS C. Which quantity should I produce? GAME THEORY ATREX BULVA 26 What do we study in this section? 1. Competitive pricing of homogeneous products (Bertrand model) 2. Location-then-price model A. Linear transportation costs B. Quadratic transportation costs C. Main intuitions 3. Vertical differentiation 27 What if firms can choose prices? After choosing their location, firms set the price of their product. → Minimal differentiation is no longer observed. If both firms are located at the same spot, consumers see their products as undifferentiated (or ‘homogeneous’). Bertrand (1883) presents a model of price competition between firms producing homogeneous products. Main result. If firms have the same constant marginal cost of production, they all set their price equal to the marginal cost at equilibrium. As the equilibrium profit is nil, firms prefer not to locate at the same spot (to relax price competition). ⇒ The equilibrium exhibits product differentiation. 28 Bertrand model Consumers utilities if 𝑙, = 𝑙- = 𝑙 𝑟 − 𝑇 𝑥, 𝑙 − 𝑝! if buys from 1 𝑣 𝑥 = 6 𝑟 − 𝑇(𝑥, 𝑙) − 𝑝" if buys from 2 0 if doesn’t buy Demand functions (supposing 𝑟 large enough) Joseph Louis If 𝑝! < 𝑝" , all 𝑛% consumers buy product 1 François Bertrand If 𝑝" < 𝑝! , all 𝑛% consumers buy product 2 (1822-1900) If 𝑝! = 𝑝" , 𝛼! 𝑛% buy product 1 and 𝛼" 𝑛% = (1 − 𝛼! )𝑛% buy product 2. Profit function of firm 𝑖 (𝑝# − 𝑐)×𝑛% if 𝑝# < 𝑝$ 𝜋# = M(𝑝 − 𝑐)×𝛼# 𝑛% if 𝑝# = 𝑝$ = 𝑝 0 if 𝑝# > 𝑝$ 29 Bertrand model. More general presentation 2 firms Homogeneous products Identical constant marginal cost: 𝑐 Set price simultaneously to maximize profits Consumers Firm with lower price attracts all demand, 𝑄(𝑝) At equal prices, market splits at 𝛼! and 𝛼" = 1 − 𝛼! ® Firm 𝑖 faces demand ⎧ Q( pi ) if pi < p j ⎪ Qi ( pi ) = ⎨α iQ( pi ) if pi = p j ⎪ 0 if pi > p j ⎩ 30 Standard Bertrand model (2) Unique Nash equilibrium Both firms set price = marginal cost: 𝑝, = 𝑝- = 𝑐 Proof For any other (𝑝! , 𝑝" ), a profitable deviation exists. Or: unique intersection of firms’ best-response functions 31 Standard Bertrand model (3) “Bertrand paradox” Only 2 firms but perfectly competitive outcome No market power! Message: There exist circumstances under which duopoly competitive pressure can be very strong. What if cost asymmetries? Consider 𝑛 firms with 𝑐. < 𝑐.Q, Equilibrium Any price 𝑝. = 𝑝R = 𝑝 ∈ 𝑐,, 𝑐- and 𝑞, = 𝑞 𝑝 , 𝑞R = 0 (𝑗 ≠ 1) 32 The “commodity trap” A commodity trap is a situation where products and services have slipped into purely price-based competition Source: Roland Berger (2014), Escaping the commodity trap. https://www.rolandberger.com/publications/publication_pdf/ro land_berger_escaping_the_commodity_trap_20140422.pdf Definition of "commodities" and a "commodity trap" > "Commodities" are products and services that – Have a high level of standardization (quality, technical features, etc.) – Face intense competition with comparable, substitutable products/services – Are subject to major price transparency for customers, usually in a buyers' market > The "commodity trap" is a situation where providers of "commodity" products or services find themselves facing – Increasing price and margin pressure – New market players, often followed by production over- capacities – A downward spiral of purely price-based competition and no escape by traditional means of standing out from the competition 33 A location-then-price model 34 A location-then-price model. Setting Timing 1. Firms choose their location: 𝑙# ∈ 0,1 2. Firms choose their price: 𝑝# 3. Consumers decide which product to buy Solution concept Subgame-perfect equilibrium (we use backward induction) We already know that the equilibrium involves WATCH different locations! Equal locations ⇒ Bertrand competition with perfectly substitutable products ⇒ Zero profits ⇒ Incentive to move away from common location (to relax price competition) 35  THEORY EXCURSUS Extensive form games Games in extensive form The extensive form representation specifies … The players in the game; When each player has to make a decision; What decisions each player can make whenever it is their turn; What each player knows about the previous decisions by other players; The payoffs of each player for all potential combination of decisions by the players. Can be represented by a “game tree” Game starts at an initial decision node at which player i makes a decision. Each of the possible choices by player i is represented by a branch. At the end of each branch is another decision node at which some other player j has to make a choice (with choices appearing on different branches). The end of the game is reached, represented by terminal nodes, when no more choices can be made. At each terminal node, we list the players’ payoffs arising from the sequence of decisions leading to that terminal node. 37 Games of perfect information Player 1 L R Players do not move simultaneously. Player 2 Player 2 When moving, each player is aware of all the previous moves (perfect information). L R L R A (pure) strategy for player i is a mapping from player i’s Player 1 nodes to actions. 2, 4 5, 3 3, 2 L R Terminal nodes of the tree show player 1’s payoff first, 1, 0 0, 1 then player 2’s payoff. 38 Games of perfect information (2) Player 1 L R SUBGAME Begins at a decision node Player 2 Player 2 that is the only decision node of the information set L R L R it belongs to. Includes those and only those decision and terminal Player 1 nodes that follow the 2, 4 5, 3 3, 2 decision node at which the L R subgame starts. 1, 0 0, 1 39 Games of perfect information (3) Player 1 BACKWARD INDUCTION When we know what will L R happen at each of a node’s branches, we can decide the best action for the player who Player 2 Player 2 is moving at that node. L R L R Starting from the last nodes, we “fold back the tree”. SUBGAME-PERFECT NASH Player 1 EQUILIBRIUM 2, 4 5, 3 3, 2 Objective. Exclude “non- L R credible threats” How? A Nash equilibrium of the whole game must induce 1, 0 0, 1 a Nash equilibrium in each of its subgames. 40 Extensive vs. Normal form Simultaneous-move game Sequential-move game Suppose she can make a choice before him He claims that he’ll go to the football game no matter what. Is this credible? 41 A. Linear transportation costs Assumption: Transportation costs increase linearly with distance: 𝑇 𝑥, 𝑙# = 𝜏 𝑥 − 𝑙# Price stage A price equilibrium fails to exist when firms are located too close to each other. Intuition: Demand functions are discontinuous; all consumers on firm 𝑗’s turf switch to firm 𝑖 for a small reduction in 𝑝#. If products are too similar, it is profitable for both firms to undercut the other. Location stage Where a price equilibrium exists, firms want to move towards a zone where a price equilibrium does not exist. ⟹ Instability in competition 42 Linear transportation costs (2) Lesson Although product differentiation relaxes price competition, firms may have an incentive to offer better substitutes to generate more demand, which may lead to instability in competition. 43 B. Quadratic transportation costs Alternative assumption: Transportation costs increase with the square of distance: 𝑇 𝑥, 𝑙# = 𝜏(𝑥 − 𝑙# )" The cost of ‘traveling’ to the nearest shop (in geographical space) or to the nearest product specification (in product space) increases with distance at an increasing rate. Under this assumption, demand functions are continuous in prices. They are linear in both prices, for all location pairs. ⟹ A unique price equilibrium exists in all subgames 44  Quadratic transportation costs - Demands Indifferent consumer 𝑝" + 𝜏(𝑥 − 𝑙" )" 𝑝! + 𝜏(𝑥 − 𝑙! )" p2 p1 l1 l2 0 € 1 € Firm 1 Firm 2 𝑥O € € Q1 ( p1, p2 ) Q2 ( p1, p2 ) 45 Quadratic transportation costs – Price stage Nash equilibrium in prices for a given pair of locations max p1 ( p1 − c) x̂( p1, p2 ) and max p2 ( p2 − c) [1− x̂( p1, p2 )] p1* = c + τ3 (l2 − l1 )(2 + l1 + l2 ) → * (unique price equilibrium) p = c + τ3 (l2 − l1 )(4 − l1 − l2 ) 2 46  Quadratic transportation costs – Price stage Detailed computations Indi↵erent consumer 1 p 1 + ⌧ (x 2 b l1 ) = p 2 + ⌧ ( x b l2 ) 2 , p1 = c + ⌧ l22 l12 + p2 ⌘ R1 (p2 ) (8) 2 2 2 2 p1 + ⌧ x b 2⌧ x bl1 + ⌧ l1 = p2 + ⌧ x b bl2 + ⌧ l22 , 2⌧ x b(l2 l1) = ⌧ (l22 l12) (p1 p2) , 2⌧ x l 1 + l2 p1 p2 Problem of firm 2 b (p1, p2) = x. p2 c 2 2⌧ (l2 l1 ) max ⇡ 1 = (⌧ (2 l1 l2 ) (l2 l1 ) + p 1 p2 ) p2 2⌧ (l2 l1 ) Problem of firm 1 p1 c First-order condition max ⇡ 1 = ⌧ l12 l22 p1 + p2 p1 2⌧ (l2 l1 ) @⇡ 2 = 0 , ⌧ (2 l1 l2 ) (l2 l1 ) + p 1 2p2 + c = 0 @p2 First-order condition 1 , p2 = (c + ⌧ (2 l1 l2 ) (l2 l1 ) + p1 ) ⌘ R2 (p1 ) @⇡ 1 2 = 0 , ⌧ l22 l12 2p1 + p2 + c = 0 (9) @p1 47  Quadratic transportation costs – Price stage Detailed computations (continued) Nash equilibrium Substitute into (9): (p⇤1 , p⇤2 ) such that p⇤1 = R1 (p⇤2 ) and p⇤2 = R2 (p⇤1 ) ✓ ◆ 1 c + ⌧ (2 l 1 l 2 ) (l 2 l 1 ) p⇤2 (l1 , l2 ) = ⌧ 2 +c + 3 2 (l l 1 ) (2 + l 1 + l2 ) 1⌧ Substitute (9) into (8): = c+ (l2 l1 ) (3 (2 l1 l2 ) + (2 + l1 + l2 )) 23 1 1⌧ p1 = c + ⌧ l22 l12 = c+ (l2 l1 )2 (4 l1 l2 ) 2 23 ⌧ 1 = c + (l2 l1 ) (4 l1 l2 ) + (c + ⌧ (2 l1 l2 ) (l2 l1 ) + p 1 ) 3 4 1 , p1 = 2c + 2⌧ l12 l22 + c + ⌧ (2 l1 l2 ) (l2 l1 ) + p 1 4 Note ⌧ , 3p1 = 3c + ⌧ (l2 l1 ) (2l2 + 2l1 + 2 l1 l2 ) p⇤1 (l1 , l2 ) p⇤2 (l1 , l2 ) = 2 (l2 l1 ) (l1 + l2 1) 3 ⌧ , p1 = c + (l2 l1 ) (2 + l1 + l2 ) ⌘ p⇤1 (l1 , l2 ) 3 48 Quadratic transportation costs – Location stage Firms choose their location, anticipating the effect on prices and quantities → 2 forces at play Competition effect ® Drives competitors apart Firms want to increase differentiation to relax price competition ↓ if 𝑙! ↑ (firm 1 moves to the right) ↓ if 𝑙" ↓ (firm 2 moves to the left) Market size effect ® Brings competitors closer Firms want to decrease differentiation to capture more consumers. ↑ if 𝑙! ↑ (firm 1 moves to the right) ↑ if 𝑙" ↓ (firm 2 moves to the left) 49 Quadratic transportation costs – Location stage Balance between the 2 forces? Equilibrium profits at the Nash equilibrium of the price game Profit-maximizing choice of location 50  Quadratic transportation costs – Equilibrium ocation stage @⇡ 1 (l1 , l2 ) 1 @l1 At the 18 = subgame-perfect ⌧ (2 + 3l1 l2 ) (l1 + l2 +equilibrium 2) < 0 @⇡ 2 (l1 , l2 ) 1 = ⌧ (4 + l1 3l2 ) (4 l1 l2 ) > 0 @l2 18 Firms acquire market power by ⇤ = 0 and l⇤ = 1 l1 2 differentiating their products. Market power increases with 𝜏. p⇤1 = p⇤2 = c + ⌧ > c 𝜏 measures the consumers’ perception of the differentiation between the two products. b p⇤ x , p ⇤ = 1 The equilibrium degree of differentiation 1 2 2 depends on the specifics of the model. ⌧ Quadratic transportation costs + Uniform ⇡ ⇤1 = ⇡ ⇤2 = > 0 distribution of consumers → Maximum 2 differentiation. Potentially less differentiation for different distributions of consumers, transportation costs functions, feasible product ranges, etc. 51 C. Location & price model – General lessons With endogenous product differentiation, the degree of differentiation is determined by balancing … the competition effect (which drives firms to increase differentiation in order to relax price competition) The market size effect (which drives firms to decrease differentiation in order to capture a larger share of the market). 52  THEORY EXCURSUS Product differentiation Views on product differentiation Product differentiation depends on consumers’ preferences How do consumers view the products/services they can choose from? “Characteristics approach” → Preferences are specified on the underlying space of product characteristics. 2 types of models Discrete choice approach Consumers have heterogeneous preferences and choose one (and only one) product among the available products. See the models covered this week (Hotelling model, Musa-Rosen model) Representative consumer approach Consumers are assumed to be identical and have a variable demand for all products. See Class 3 54 Discrete choice approach Horizontal product differentiation Each product would be preferred by some consumers. Vertical product differentiation Everybody would prefer one over the other product. Formal way to separate the two forms of differentiation If, at equal prices, consumers do not agree on which product is the preferred one ® Products are horizontally differentiated. If, at equal prices, all consumers prefer one over the other product ® Products are vertically differentiated. Notes To account for supply side characteristics, modify the definition by replacing “at equal prices” by “at prices set equal to marginal costs”. Not easy to draw the distinction in practice 55 Vertical differentiation - Setting Difference with horizontal differentiation All consumers agree that one product is preferable to another (that is, this product has a higher quality). Consumers Quality is described by 𝑠# ∈ 𝑠, 𝑠 ⊂ ℝ' Preference parameter for quality: 𝜃 ∈ 𝜃, 𝜃 ⊂ ℝ' Larger 𝜃 → consumer more sensitive to quality changes Uniform distribution on 𝜃, 𝜃 ; mass 𝑀 = 𝜃 − 𝜃 Each consumer chooses 1 unit of one of the products Utility for consumer 𝜃 from one unit of product 𝑖 𝑟 + 𝜃𝑠. − 𝑝. 56 Vertical differentiation – Setting (2) Firms (duopolists) 1. Choose their quality: 𝑠# ∈ 𝑠, 𝑠 2. Choose their price: 𝑝# Constant marginal cost of production: 𝑐# = 0 Solution concept Subgame-perfect equilibrium (we use backward induction) Anticipated results The equilibrium must involve different qualities. Same qualities ⇒ Bertrand competition ⇒ Zero profits Tension between same 2 effects as with horizontal differentiation Competition effect ® differentiate to enjoy market power ® drives competitors to produce different qualities Market size effect ® meet consumers preferences ® brings competitors to produce high quality 57 Price stage Assumption (without loss of generality) Firm 2 produces high quality: 𝑠" > 𝑠! Demand functions Indifferent consumer is determined by the ratio of price and quality differences: p −p r − p1 + θˆs1 = r − p2 + θˆs2 ⇔ θˆ = 2 1 for θˆ ∈ [θ , θ ] s2 − s1 Profit function of low-quality firm (firm 1) ì 0 if p1 > p2 - q (s2 - s1 ), ïï ( p2 - p1 p 1 ( p1 p2 ; s1 , s2 ) = í 1 s2 - s1 - q p ) if q (s2 - s1 ) £ p2 - p1 £ q (s2 - s1 ), ï ïî p1 (q - q ) if p1 < p2 - q (s2 - s1 ). 58 Price stage (2) Equilibrium in prices (for given qualities) Solving the system of first-order conditions: p1* = 13 (q - 2q )(s2 - s1 ) p2* = 13 (2q - q )(s2 - s1 ) (parameter restriction: q > 2q ) → Even the price of the low-quality firm increases with the quality difference! 59 Quality stage Equilibrium in qualities Substitute for second-stage equilibrium prices in profit function: π! 1 (s1, s2 ) = 19 (θ − 2θ )2 (s2 − s1 ) π! 2 (s1, s2 ) = 19 (2θ − θ )2 (s2 − s1 ) → Both profits ↑ in the quality difference → Equilibrium quality choices: Simultaneous choices: 𝑠!, 𝑠" = 𝑠, 𝑠 or 𝑠, 𝑠 Sequential choices: 1st chooses highest quality and 2nd chooses lowest quality 60 Vertical differentiation - General lesson In markets in which products can be vertically differentiated, firms offer different qualities in equilibrium so as to relax price competition. 61 1.C Quantity decisions 1.C. Quantity decisions A. Which sort of glint should I sell? B. At which price(s) should I sell my product? INTERDEPENDENT DECISIONS C. Which quantity should I produce? GAME THEORY ATREX BULVA 63  Limited capacity and price competition Edgeworth’s critique (1897) Bertrand model: no capacity constraint Same in Hotelling model But capacity may be limited in the short run Potential rationing of consumers Examples Retailers order supplies well in advance Tour operators book rooms in hotels and seats in flight one year in advance Flights more expensive around Xmas To account for this: two-stage model 1. Firms pre-commit to capacity of production Watch video on rationing in 2. Price competition practice 64 Capacity-then-price model Setting Firms produce a homogeneous product (no differentiation) Stage 1: firms set capacities 𝑞W# and incur cost of capacity 𝑐 Stage 2: firms set prices 𝑝# ; cost of production is 0 up to capacity (and infinite beyond capacity); demand is 𝑄 𝑝 = 𝑎 − 𝑝. Subgame-perfect equilibrium: firms know that capacity choices may affect equilibrium prices Potential rationing If quantity demanded to firm 𝑖 exceeds its supply...... some consumers must be rationed...... and possibly buy from more expensive firm 𝑗 Crucial question: Who will be served at the low price? 65 Capacity-then-price model (2) Efficient rationing First served → Consumers with larger willingness to pay Efficient because it maximizes consumer surplus Takes place in real life through queuing, secondary markets (see video) Consumers with unit demand, ranked by decreasing willingness to pay There is a positive residual demand for firm 2 Consumers with highest willingness to pay are served at firm 1’s low price Excess demand for firm 1 66 Subgame-perfect equilibrium Stage 2 If 𝑝! < 𝑝" and excess demand for firm 1, then demand for firm 2 is ⌢ ⎧⎪ Q( p2 ) − q1 if Q( p2 ) − q1 ≥ 0 Q( p2 ) = ⎨ ⎩⎪ 0 else ( Claim: if 𝑐 < 𝑎 < 𝑐, then both firms set the market-clearing price: ) 𝑝! = 𝑝" = 𝑝∗ = 𝑎 − 𝑞W! − 𝑞W" Intuition: Suppose 𝑝! = 𝑝∗; can firm 2 increase its profit by setting 𝑝" ≠ 𝑝∗? 𝑝" < 𝑝∗ → Not profitable Firm 2 already sell its capacity at 𝑝" = 𝑝∗ So it would sell the same but at a lower price 𝑝" > 𝑝∗ → Not profitable either Given 𝑝! = 𝑝∗ , the profit-maximizing price of firm 2, (𝑎 − 𝑞2!)/2, is lower than 𝑝∗ So, setting 𝑝" > 𝑝∗ further decreases profits 67 Subgame-perfect equilibrium (2) Stage 1 Same reduced-form profit functions as in the Cournot game 𝜋! (𝑞! , 𝑞W" ) = 𝑎 − 𝑞W! − 𝑞W" 𝑞W! − 𝑐 𝑞W! Results In the capacity-then-price game with efficient consumer rationing (and with linear demand and constant marginal costs), the chosen capacities are equal to those in a standard Cournot market. So, under certain conditions, the Cournot model can be seen as a simplified version of a more complex (and realistic) model in which firms commits first to their capacity of production and then choose the price of their product. 68 Cournot model Competition in quantities Firms choose the quantity of their good (and then, prices adjust). They have a constant marginal cost of production (𝑐5 ) and no fixed cost. To express prices as a function of quantities, we use the inverse demand functions. Duopoly: 𝑝! 𝑞! , 𝑞" and 𝑝" 𝑞! , 𝑞" The maximization problem of firm 1 can be written as: Antoine Augustin Cournot 𝑚𝑎𝑥Y! 𝑝, 𝑞,, 𝑞- − 𝑐, 𝑞, (1801-1877) 69 An example with linear demand functions Consumer chooses the quantities 𝑞, and 𝑞- to maximize 1 - 𝑎 𝑞, + 𝑞- − 𝑞, + 2𝑑𝑞,𝑞- + 𝑞-- − 𝑝,𝑞, − 𝑝-𝑞- + 𝑦 2 FOCs give: Z Z = 0 ⇔ 𝑎 − 𝑞, − 𝑑𝑞- = 𝑝, and = 0 ⇔ 𝑎 − 𝑞- − 𝑑𝑞, = 𝑝- ZY! ZY" → Inverse demand functions: 𝑝, 𝑞,, 𝑞- = 𝑎 − 𝑞, − 𝑑𝑞- M 𝑝- 𝑞,, 𝑞- = 𝑎 − 𝑞- − 𝑑𝑞, 0 ≤ 𝑑 ≤ 1 ⟶ Measures link between price and quantity of 𝑎 ⟶ Maximum price different goods ⟶ degree of consumer is willing to product substitutability pay for each good 𝑑 = 0 ⟶ independent goods 𝑑 = 1 ⟶ homogenous goods 70  Best-response functions max ⇡ (q1 , q2 ) = [p1 (q1 , q2 ) c 1 ] q 1 = (a c1 q1 dq2 ) q1 q1 First-order @⇡ (q1 , q2 ) = 0 , (a c1 q1 dq2 ) q1 = 0 condition @q1 1 max ⇡ (q1 , q2 ) = [p1 (q1 , q2 ) c 1 ] q 1 = (a c1 , q 1 = (a c1 q1dq2 ) ⌘ R1 (q2 ) 2 @R1 (q2 ) d @⇡ (q1, q2) = 0 , (a c1 q1 dq2) q1 =  0@q1 @q2 Firm 1’s best-response (or reaction) function 2 Downward-sloping ✓𝑞! to choose◆to → Tells firm 1 which quantity 2⇡ q , q → If firm11 expects 𝑞" to increase, @ @⇡ maximize its profit, ( q , 1 2q ) @ ( 1 2 ) , q1it= (aby cdecreasing reacts 1 dq2 ) ⌘𝑞 R1 (q2 ) = = d  20 ! for any quantity 𝑞" that@q firm 2 2 could @q1choose @q1 @q2 @R1 (q2 ) d 1 = 0 q2 = (a c2 dq1 ) ⌘ R2 (q1 ) @q2 2 2 ✓ ◆ 71 @ @⇡ (q1 , q2 ) @ 2 ⇡ (q 1 , q 2 )  THEORY EXCURSUS 1. Strategic substitutability 2. Strategic complementarity  Strategic substitutability max ⇡ (q1 , q2 ) = [p1 (q1 , q2 ) c ] q = (a c q dq ) q 1 1 1 1 2 1 q1 @⇡ (q1 , q2 ) In the Cournot = 0 , (model, a c1 the q1 decisions of0 dq2 ) q1 = @q1 the two firms are strategic substitutes because 1 mutually offset one they 𝑞, , q1 = (a c1 dq2 ) ⌘ R1 (q2 ) another. 2 @R1 (q2its Firm 2 increases ) quantity d = 0 ⟹ marginal profit @q2 of firm 2 1 decreases 1 > 𝑑, > 𝑑- > 𝑑] = 0 ✓ ◆ 2 @ @⇡ (q1 , q2 ) @ ⇡ (q 1 , q 2 ) = = d0 @q2 @q1 @q1 @q2 𝑑' = 0 1 Equivalent q2 = to(a say c2 that dq1best-response ) ⌘ R 2 (q 1 ) 2 functions are downward-sloping. 𝑅,(𝑞-) The smaller 𝑑 (the more products are 𝑑" differentiated) the less the strategy of 𝑑! one firm is sensitive to a change in the 𝑞- strategy of the other firm. 73  Strategic1 complementarity max ⇡ 1 (p1, p2 ) = (p 1 c 1 ) ( a (1 d) p1 + dp2 ) p1 1 d2 @⇡ 1 (In p1,the p2 ) Bertrand model (with horizontally 𝑝,0 = 0 ( ( 1 ) + differentiated products), the decisions of , a d p 1 dp 2 ) ( p 1 c 1 ) = @p1 the two firms are strategic 1 1 > 𝑑, > 𝑑- > 𝑑] = 0 complements , p 1 = (a (1 because d) + cthey mutually 1 + dp2 ) ⌘ R1 (p2 ) reinforce one2 another. Firm 2 increases @R1its(p2price ) d 𝑅,(𝑝-) = ⟹ >0 𝑑! marginal profit of @pfirm 2 1 increases 2 ✓ ◆ 𝑑" @ @⇡ 1 (p1, p2 ) @ 2 ⇡ 1 (p1, p2 ) d = = >0 @p2 @p1 @p1 @p2 1 d2 𝑑' = 0 1 to say that best-response Equivalent p 2 = (a (1 d) + c2 + dp1 ) ⌘ R2 (p1 ) functions2 are upward-sloping. The smaller 𝑑 (the more products are differentiated) the less the strategy of one firm is sensitive to a change in the 𝑝- strategy of the other firm. 74  @R1 (q2 ) d = 0 Nash equilibrium @q2 2 ✓ ◆ @ @⇡ (q1 , q2 ) @ 2 ⇡ (q 1 , q 2 ) = = d0 We find firm @q 2’s2 best-response @q1 function @q1 @q2 in a similar way 1 q2 = (a c2 dq1 ) ⌘ R2 (q1 ) 2 Nash equilibrium = Intersection of best-response functions 𝑞!∗ , 𝑞"∗ such that 𝑞!∗ = 𝑅! (𝑞"∗ ) & 𝑞"∗ = 𝑅" (𝑞!∗ ) Equilibrium 1 2q1 = a c1 d (a c2 dq1 ) 2 , 4q1 = 2a 2c1 da + dc2 + d2 q1 , 4 d 2 q 1 = (2 d) a 2c1 + dc2 (2 d) a 2c1 + dc2 ⇤ , q1 = ⌘ q 1 4 d2 (2 d) a 2c2 + dc1 q2⇤ = 4 d2 75  2 , 4q1 = 2a 2c1 da + dc2 + d2 q1 Nash equilibrium2 (2) , 4 d q = (2 d) a 2c1 + dc2 1 Effects of costs (2 d) a 2c1 + dc2 ⇤ , q1 = 2 ⌘ q 1 4 d The firm with a lower marginal cost produces a larger quantity at equilibrium. It also has a larger market ⇤ share (2 dand ) a a larger 2c2 + profit dc1 at equilibrium. q2 = prefers to leave A firm that is too inefficient 4 d2 the market. (2 d) a 2c1 + dc2 (2 d) a 2c2 + dc1 c2 c1 q1⇤ q2⇤ = = 4 d2 4 d 2 2 d (2 d) a + dc1 q2⇤ 0 , (2 d) a 2c2 +dc1 0 , c2  2 76 Nash equilibrium (3) Graphically 𝑞, 𝑅! (𝑞" ) 𝑞,∗ , 𝑞-∗ ↓ as 𝑑 ↑ 𝑅" (𝑞! ) 𝑞- 77 Symmetric firms Suppose 𝑐, = 𝑐- = 𝑐 a c c 1 = c2 = c ) q1⇤ = q2⇤ = ⌘ q⇤ 2+d a + (1 + d ) c p⇤1 = p⇤2 = a (1 + d ) q ⇤ = ⌘ p⇤ 2+d ⇤ a c p c= = q⇤ 2+d ⇤ 2 1 ⇡ ⇤1 = ⇡ ⇤2 = (p ⇤ ⇤ c ) q = (q ) = 2( a c )2 ⌘ ⇡ Q (2 + d ) 1 d = 0 ! ⇡Q = (a c )2 4 4 Q d = 1/2 ! ⇡ = (a c )2 25 1 d = 1 ! ⇡ = (a c )2 Q 9 78  Epilogue. Price or quantity competition?  What is the appropriate modelling choice? Monopoly? No difference: Monopolist controls the relationship between price and quantity. Oligopoly Price and quantity competitions lead to different residual demands. Price competition pj fixed ® rival willing to serve any demand at pj i’s residual demand: market demand at pi < pj; zero at pi > pj So, residual demand is very sensitive to price changes. Quantity competition qj fixed ® irrespective of price obtained, rival sells qj i’s residual demand: “what’s left” (i.e., market demand - qj) So, residual demand is less sensitive to price changes. 80  How do firms behave in the market place? Stick to a price and sell any Stick to a quantity and sell this quantity at this price? quantity at any price? → Price competition is the → Quantity competition is the appropriate model appropriate model Applies when Applies when Unlimited capacity of production Limited capacity of production Prices are more difficult to adjust in the Quantities are more difficult to adjust in short run than quantities. the short run than prices. Examples Examples Cable television services Automobiles Entertainment Pharmaceuticals Mass media Aluminium & steel Software Hardware (computers, smartphones) Insurance Oil and gas Caveat: Influence of technology (e.g., print-on-demand vs. batch printing) 81 This work was made for you to share, reuse, remix, rework... It is licensed under the Creative Commons BY-NC-SA license to allow for further contributions by experts and users. Free-to-use photos from Pexels.com. Unlimited license on icons acquired from Noun Project You are free to share and remix/adapt the work. You may not use this work for commercial purposes. You must cite this document: You may distribute a modified work Belleflamme, P. (2024). LECGE1330 – Industrial under the same or similar license. Organization / Product and price decisions, Slide presentation, UCLouvain