the analytics of constrained optimal decisions microeco nomics spring 2016 the oligopoly model(II): competition in prices ………….1price competition: introduction ………….3 the ketchup “war” session seven ………….8first mover advantage ………...10 key points..……….11“discount” strategy (I): the simultaneous game..……….14“discount” strategy (II): the sequential game …..…….16 own and cross elasticity

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |1 price competition: introduction ► We are interested in studying competition in markets in which there are at least two firms… ► Challenge : how do firms interact? How is the outcome (equilibrium price/output) determined? ► One answer is the Cournot model : ● all firms produce exactly the same good (a commodity) ● since we’re dealing with a commodity all firms will sells at the same price P * ● each firm decides on its own quantity ( Q 1, Q 2,…) to bring to the market ● the market price for that commodity is based on the total quantity brought to the market Q = Q 1 + Q 2 + … Q n we get the price as P = a – b ∙ Q ► Challenge: firms decide on their individual quantity based on their conjectures of what the other firms are going to produce… ● the equilibrium is defined as the case in which conjectures are mutually confirmed: … my decision is a best response to your decision which in turn is a best response to my decision …

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |2 price competition: introduction ► Is the Cournot model the answer to what we expect to see in a market competition? ● if the good is a commodity (perhaps) we are not that far away from reality ● but not all goods are commodities… ● think about (Coke-Pepsi, BMW-Mercedes, North Face – Colombia – Patagonia, etc.) ► How do we characterize these goods? ● they are substitutes : can use either one with similar satisfaction ► What does this imply? ● firms can now charge different prices since consumers can differentiate between goods ► Challenge : how do firms interact? How is the equilibrium price/output determined? ► This type of market (differentiated goods) is solved by the Bertrand model : ● firms compete in prices, i.e. their decision variable is the price they charge ● each firm faces an individual demand, i.e. a demand that is particular to the specific good that the firm sells

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |3 the ketchup “war” ► Consider the market for (a very popular good) ketchup: Heinz (Firm 1) versus Hunts (Firm 2)  Some simplifying assumptions: ● both firms have a common marginal cost of MC = $2 ● demand curves: Q 1 = 90 – 15 P P 2 Q 2 = 90 – 15 P P 1 ► These demand curves show that demand for one firm’s ketchup depends both on the price set by the firm and the price set by its competitor demand for Heinz : market size for Heinz demand for Hunts: own price demand sensitivity cross price demand sensitivity own price demand sensitivity market size for Hunts

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |4 the ketchup “war” ► How do we solve this model? We have to determine the price each firm will set…. ► We need an assumption on how firms interact, i.e. to model the pricing process… ► Back to the demands for the two goods: Q 1 = 90 – 15 P P 2 Q 2 = 90 – 15 P P 1 ● each firm wants to maximize its own profit given its own price and the price set by the other firm ● determine a reaction function of each firm (what is the best price P 1 Firm 1 will set given that Firm 2 is supposed to set a price P 2 ) ● the equilibrium will be determine by the pair of prices ( P 1, P 2 ) for which the conjectures are confirmed ► We encounter again this concept of equilibrium based on best responses and confirming conjectures … this is the famous Nash equilibrium concept applied to specific setups (choice variable is output/capacity or price for our simple models)

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |5 the ketchup “war” ► Heinz profit is given by Π 1 = P 1 ∙Q 1 – MC 1 ∙Q 1 = ( P 1 – MC 1 ) ∙Q 1 Now we use the market demand for Heinz Q 1 = 90 – 15 P P 2 and marginal cost MC 1 = 2 to get the profit Π 1 = ( P 1 – 2) ∙ (90 – 15 P P 2 ) = -15( P 1 ) 2 + ( P 2 )∙ P 1 – 180 – 20∙ P 2 ► What about Hunts? Similar process to write profit first as Π 2 = P 2 ∙Q 2 – MC 2 ∙Q 2 = ( P 2 – MC 2 ) ∙Q 2 ► Now we use the market demand for Heinz Q 2 = 90 – 15 P P 1 and marginal cost MC 2 = 2 to get the profit Π 2 = ( P 2 – 2) ∙ (90 – 15 P P 1 ) = -15( P 2 ) 2 + ( P 1 )∙ P 2 – 180 – 20∙ P 1 ► Steps in solving the model: ● choice for each firm is its own price : P 1 for Heinz and P 2 for Hunts… ● maximize their own profit given the other firm’s price → reaction functions

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |6 the ketchup “war” ► Heinz profit is given by Π 1 = ( P 1 – 2) ∙ (90 – 15 P P 2 ) = -15( P 1 ) 2 + ( P 2 )∙ P 1 – 180 – 20∙ P 2 To maximize this profit function means to maximize a quadratic function: take the derivative and set it equal to zero: -15∙2∙ P 1 + ( P 2 ) = 0 → P 1 = 4 + P 2 /3 (Heinz’s reaction function ) ► Hunts profit is given by Π 2 = ( P 2 – 2) ∙ (90 – 15 P P 1 ) = -15( P 2 ) 2 + ( P 1 )∙ P 2 – 180 – 20∙ P 1 To maximize this profit function means to maximize a quadratic function: take the derivative and set it equal to zero: -15∙2∙ P 1 + ( P 2 ) = 0 → P 2 = 4 + P 1 /3 (Hunts’ reaction function ) ► Let’s bring together these two reaction functions: P 1 = 4 + P 2 /3 P 2 = 4 + P 1 /3 ► The solution to this system of two equations with two unknowns gives the prices in equilibrium …

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |7 the ketchup “war” ► Reaction functions: P 1 = 4 + P 2 /3 P 2 = 4 + P 1 /3 ► Plug the first equation into the second equation P 2 = 4 + (4 + P 2 /3)/3 with solution P 2 = 6 ► Use this back into the first equation P 1 = 4 + 6/3 = 6 ► equilibrium (prediction for market) P 1 = 6 and P 2 = 6 ► graphically the equilibrium is at the intersection of the two reaction functions 6 6 P2P P1P Firm 2 reaction function P 2 = 4 + P 1 /3 Firm 1 reaction function P 1 = 4 + P 2 /3 Bertrand equilibrium

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |8 first mover advantage 0 1 ● Both firms announce their price at time 0 ● The behavior is dictated by the conjectures (reaction functions) of both firms P 1 = 4 + P 2 /3 P 2 = 4 + P 1 /3 0 1 ● Firm 1 announces price P 1 at time 0, Firm 2 learns this price at time 0 (ii) Sequential decision (leader-follower) ● Firm 2, knowing P 1, decides on its own price P 2 based on its reaction function P 2 = 4 + P 1 /3 ● Firm 1 by announcing its price P 1 at time 0 knows exactly what price P 2 will be chosen by Firm 2 at time 1 ● Firm 1 perfectly anticipates P 2 given P 1 → can “manipulate” Firm 2 into choosing a certain P 2 by carefully choosing P 1 ● Equilibrium P 1 = 6, P 2 = 6 profits Π 1 = Π 2 = 240 (i) Simultaneous decision

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |9 first mover advantage ► For the sequential case Firm 1 profit is Π 1 = ( P 1 – MC 1 )∙ Q 1 = ( P 1 – 2)∙(90 – 15 P P 2 ) ► But Firm 1, based on price P 1, knows exactly what P 2 will be, namely P 2 = 4 + P 1 /3 ► So profit for Firm 1 really depends only on its own price P 1 (plug equation for P2 into profit above): Π 1 = ( P 1 – 2)∙(90 – 15 P ∙(4 + P 1 /3)) ► Firm 1 has to pick P 1 that maximizes its profit. If you plot this in Excel you’ll get P 1 = 6.57 (precisely 46/7) ► Firm 2 responds with P 2 = 4 + P 1 /3 = 6.19 ► What are the profits?

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |10 key points ► What does the Bertrand solution represent conceptually? ● notice that each company has to make its individual choice based on “conjectures” of what all other companies are going to choose… the choices are made simultaneously … ● the Bertrand solution calculates the “self-fulfilling conjectures”… “you will get what you expect” ► The steps in obtaining the Bertrand solution : ● for each player, based on individual demand and marginal cost, derive the profit function as it depends on prices ● each player maximizes its profit with respect to its own price and considering other player’s price as a parameter ● from the previous step each player will have its own reaction function of the form: own price as a function of other player’s price ● the Bertrand solution is found by solving the system of two equations and two unknowns (prices) as defined by the reaction functions ► What is the difference between the Bertrand and Stackelberg models? ● for Bertrand choices are made simultaneously … while for Stackelberg one player announces first its choice and the other player optimally (according to its reaction function) responds ● it is this forward thinking (or anticipation) – first player to announce knows how the second player will respond so it will strategically choose its announcement – that gives an advantage to the “first mover”

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |11 “discount” strategy (I): a simultaneous game Simultaneous Game : Let’s consider the following setup: ► Firm 1 picks its price P 1 that maximizes its profit: Π 1 = ( P 1 – MC 1 )∙ Q 1 = ( P 1 – 2)∙(90 – 15 P P 2 ) ► But Firm 2, chooses to use a “discount” strategy pricing policy: whatever Firm 1 chooses, Firm 2 will set a price P 2 at a discount d relative to price P 1 : P 2 = P 1 – d ► The game is played according to the following rule: each firm “hands over” its chosen pricing strategy to a market maker that establishes the equilibrium prices at the intersection of the two pricing (reaction) functions. ► How would you set up the problem and find the equilibrium prices? Who gains market share (relative to the Bertrand solution)? ► Does the reaction function for Firm 1 change relative to the Bertrand situation? ► Do you expect Firm 2 to make more profit relative to the Bertrand situation?

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |12 “discount” strategy (I): a simultaneous game ► In the Bertrand solution Firm 2 will set a reaction function that maximizes its profit Π 2 = ( P 2 – MC 2 )∙ Q 2 = ( P 2 – 2)∙(90 – 15 P P 1 ) however Firm 2 uses the discount pricing strategy P 2 = P 1 – d. Firm 1 is not aware of this – does it matter for Firm 1 in setting its own reaction function? Hint: this is a simultaneous game. ► Firm 1 has the same reaction function as it did under the Bertrand solution. The logic is fairly simple: Firm 1 and Firm 2 simultaneously announces their price (they simultaneously hand over their pricing functions to a market maker). Thus Firm 1’s reaction function is the same: P 1 = 4 + P 2 /3 ► Firm 2 has a reaction function given by its discount pricing strategy: P 2 = P 1 – d ► Equilibrium: the market maker will seek the pair of prices ( P 1 *, P 2 *) that solves the system of equations represented by the reactions functions (graphically – the intersection of the reaction functions): P 1 * = 4 + P 2 */3 P 2 * = P 1 *– d Solution: P 1 * = 6 – d /2 P 2 * = 6 – 3 d /2

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |13 “discount” strategy (I): a simultaneous game The equilibrium derived according to the rules of the game: P 1 * = 6 – d /2 P 2 * = 6 – 3 d /2 ► Is Firm 2 really applying a “smart” strategy? What does it gain?  higher profit ? No. (Why?)  higher market share ? Yes (Why?) ► Can we find the new market shares (outputs)? Plug the prices in the demand functions…a bit of algebra will give you: Q 1 = 60 – 15 d /2 Q 2 = d /2 ► Compare this result with the Bertrand market shares of (60,60). 6 6 P2P2 P1P1 4 4 Firm 2 (Bertrand) reaction function P 2 = 4 + P 1 /3 Firm 1 (Bertrand) reaction function P 1 = 4 + P 2 /3 Bertrand equilibrium Firm 2 (discount pricing) reaction function P 2 = P 1 – d 6 – d /2 6 – 3 d /2 new equilibrium

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |14 “discount” strategy (II): a sequential game Sequential Game : Let’s consider the following setup: ► Firm 2, chooses to use a “discount” strategy pricing policy: whatever Firm 1 chooses, Firm 2 will set a price P 2 at a discount d relative to price P 1 : P 2 = P 1 – d ► Firm 2 announces this strategy and also announces the discount d before Firm 1 chooses its price P 1. ► Firm 1 picks its price P 1 that maximizes its profit knowing Firm 2’s strategy P 2 = P 1 – d and also the announced d : Π 1 = ( P 1 – MC 1 )∙ Q 1 = ( P 1 – 2)∙(90 – 15 P P 2 ) ► The game is played sequentially: Firm 2 announces its discount strategy and the discount d. Firm 1 chooses its price P 1 knowing Firm 2’s strategy and the discount d. As soon as Firm 1 chooses its price P 1, Firm 2’s price is determined according to the discount strategy P 2 = P 1 – d. ► How would you set up the problem and find the equilibrium prices? Who gains market share (relative to the Bertrand solution)? ► Does the reaction function for Firm 1 change relative to the Bertrand/Stackelberg situation? ► Do you expect Firm 2 to make more profit relative to the Bertrand/Stackelberg situation?

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |15 “discount” strategy (II): a sequential game ► Firm 1 has a different reaction function as it did under the Bertrand/Stackelberg solution. The logic is fairly simple: Firm 1 knows Firm 2’s strategy, P 2 = P 1 – d, thus Firm 1’s reaction function is the solution to the profit maximization function: Π 1 = ( P 1 – MC 1 )∙ Q 1 = ( P 1 – 2)∙(90 – 15 P P 2 ) where P 2 = P 1 – d ► Let’s plug Firm 2’s reaction function (strategy) in Firm 1’s profit function to get: Π 1 = ( P 1 – MC 1 )∙ Q 1 = ( P 1 – 2)∙(90 – 15 P  ( P 1 – d )) = – 5( P 1 ) 2 + (100 – 10 d ) P 1 – d ► The profit maximizing price P 1 for Firm 1 satisfies (from setting the first derivative equal to zero): – 10 P 1 + (100 – 10 d ) = 0 that is P 1 = 10 – d ► Given that P 1 = 10 – d Firm 2’s price is P 2 = P 1 – d = 10 – 2 d and the profit function for Firm 2 is: Π 2 = ( P 2 – MC 2 )∙ Q 2 = ( P 2 – 2)∙(90 – 15 P P 1 ) where P 1 = 10 – d and P 2 = 10 – 2 d ► Plugging these expressions for P 1 and P 2 : Π 2 = 40  (4 – d )  (2 + d ) = 40  (8 +2 d – d 2 ) ► The optimal discount is found from the condition 2 – 2 d = 0 which is d * = 1. The prices in equilibrium will be P 1 = 9 and P 2 = 8.

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |16 own and cross elasticity ► Consider again the Bertrand model with two firms: Q 1 = a 1 – b 1 P 1 + d 1 P 2 Q 2 = a 2 – b 2 P 2 + d 2 P 1 ► The marginal costs are MC 1 and MC 2 ► We can derive the reaction functions as: P 1 = 0.5 ∙ a 1 / b ∙ MC ∙ d 1 ∙ P 2 / b 1 P 2 = 0.5 ∙ a 2 / b ∙ MC ∙ d 2 ∙ P 1 / b 2 demand for Heinz : market size for Heinz demand for Hunts: own price demand sensitivity cross price demand sensitivity own price demand sensitivity market size for Hunts

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |17 own and cross elasticity ► For the general model: Q 1 = a 1 – b 1 P 1 + d 1 P 2 Q 2 = a 2 – b 2 P 2 + d 2 P 1 ► For good 1 the “own” elasticity of demand is defined as the elasticity of Q 1 with respect to P 1 as the “percent change in quantity demanded when the own price changes by one percent”. As a formula this is written as: ► For good 1 the “cross” elasticity of demand is defined as the elasticity of Q 1 with respect to P 2 as the “percent change in quantity demanded when the cross price changes by one percent”. As a formula this is written as: Remark: The own and cross elasticity is defined similarly for good 2.

microeconomic s the analytics of constrained optimal decisions lecture 7 the oligopoly model (II): competition in prices  2016 Kellogg School of Management lecture 7 page |18 own and cross elasticity ► Demand functions demand for firm 1 : Q 1 = a 1 – b 1 P 1 + d 1 P 2 demand for firm 2 : Q 2 = a 2 – b 2 P 2 + d 2 P 1 ► We can describe the “market” (consumers) as three categories: ● loyalists to firm 1 (they stick with firm 1 even if P 1 is extremely high relative to P 2 ) ● loyalists to firm 2 (they stick with firm 1 even if P 2 is extremely high relative to P 1 ) ● switchers (they can buy from firm1 or firm 2 depending on the relative prices) firm 1’s “loyalists” firm 2’s “loyalists” switchers Q 1 = a 1 – b 1 P 1 + d 1 P 2 Q 2 = a 2 – b 2 P 2 + d 2 P 1 ( a 1, b 1, P 1 ) ( a 2, b 2, P 2 ) ( b 1, d 1, P 1, P 2 )( b 2, d 2, P 2, P 1 )