Transcript Microeconomics MECN 430 Spring 2016

session seven
the oligopoly model(II): competition in prices
price competition: introduction ………….1
the ketchup “war” ………….3
first mover advantage ………….8
key points ………...10
“discount” strategy (I): the simultaneous game ..……….11
“discount” strategy (II): the sequential game ..……….14
own and cross elasticity …..…….16
spring
2016
microeconomi
the analytics of
cs
constrained optimal
microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
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 (Q1, Q2,…) to bring to the market
● the market price for that commodity is based on the total quantity brought to the market
Q = Q1 + Q2 + … Qn 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 …
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
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
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
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:
own price demand
sensitivity
market size for Heinz
cross price demand
sensitivity
demand for Heinz:
Q1 = 90 – 15P1 + 10P2
demand for Hunts:
Q2 = 90 – 15P2 + 10P1
cross price demand
sensitivity
market size for Hunts
own price demand
sensitivity
► 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
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lecture 7
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
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:
Q1 = 90 – 15P1 + 10P2
Q2 = 90 – 15P2 + 10P1
● 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 P1 Firm 1 will set given that Firm 2 is
supposed to set a price P2)
● the equilibrium will be determine by the pair of prices (P1,P2) 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)
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
the ketchup “war”
► Heinz profit is given by
Π1 = P1∙Q1 – MC1∙Q1 = (P1 – MC1)∙Q1
Now we use the market demand for Heinz Q1 = 90 – 15P1 + 10P2 and marginal cost MC1 = 2 to get the profit
Π1 = (P1 – 2)∙(90 – 15P1 + 10P2) = -15(P1)2 + (120 + 10P2)∙P1 – 180 – 20∙P2
► What about Hunts? Similar process to write profit first as
Π2 = P2∙Q2 – MC2∙Q2 = (P2 – MC2)∙Q2
► Now we use the market demand for Heinz Q2 = 90 – 15P2 + 10P1 and marginal cost MC2 = 2 to get the profit
Π2 = (P2 – 2)∙(90 – 15P2 + 10P1) = -15(P2)2 + (120 + 10P1)∙P2 – 180 – 20∙P1
► Steps in solving the model:
● choice for each firm is its own price: P1 for Heinz and P2 for Hunts…
● maximize their own profit given the other firm’s price → reaction functions
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
the ketchup “war”
► Heinz profit is given by
Π1 = (P1 – 2)∙(90 – 15P1 + 10P2) = -15(P1)2 + (120 + 10P2)∙P1 – 180 – 20∙P2
To maximize this profit function means to maximize a quadratic function: take the derivative and set it equal to zero:
-15∙2∙P1 + (120 + 10P2) = 0 → P1 = 4 + P2/3
(Heinz’s reaction function)
► Hunts profit is given by
Π2 = (P2 – 2)∙(90 – 15P2 + 10P1) = -15(P2)2 + (120 + 10P1)∙P2 – 180 – 20∙P1
To maximize this profit function means to maximize a quadratic function: take the derivative and set it equal to zero:
-15∙2∙P1 + (120 + 10P2) = 0 → P2 = 4 + P1/3
(Hunts’ reaction function)
► Let’s bring together these two reaction functions:
P1 = 4 + P2/3
P2 = 4 + P1/3
► The solution to this system of two equations with two unknowns gives the prices in equilibrium …
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
the ketchup “war”
P2
Firm 1
reaction function
P1 = 4 + P2/3
► Reaction functions:
P1 = 4 + P2/3
P2 = 4 + P1/3
► Plug the first equation into the second equation
P2 = 4 + (4 + P2/3)/3
12
Bertrand
equilibrium
with solution
P2 = 6
► Use this back into the first equation
P1 = 4 + 6/3 = 6
8
6
Firm 2
reaction function
P2 = 4 + P1/3
4
► equilibrium (prediction for market)
P1 = 6 and P2 = 6
4
► graphically the equilibrium is at the intersection of the
two reaction functions
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lecture 7
6
8
12
P1
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microeconomics
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the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
first mover advantage
(i) Simultaneous decision
1
0
● Both firms announce their price at time 0
● Equilibrium
● The behavior is dictated by the conjectures
(reaction functions) of both firms
P1 = 6, P2 = 6
P1 = 4 + P2/3
profits Π1 = Π2 = 240
P2 = 4 + P1/3
(ii) Sequential decision (leader-follower)
1
0
● Firm 1 announces price P1 at time 0,
Firm 2 learns this price at time 0
● Firm 1 perfectly anticipates P2 given P1
→ can “manipulate” Firm 2 into choosing
a certain P2 by carefully choosing P1
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● Firm 2 , knowing P1, decides on its
own price P2 based on its reaction
function
P2 = 4 + P1/3
● Firm 1 by announcing its price P1 at
time 0 knows exactly what price P2 will be
chosen by Firm 2 at time 1
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
first mover advantage
► For the sequential case Firm 1 profit is
Π1 = (P1 – MC1)∙Q1 = (P1 – 2)∙(90 – 15P1 + 10P2)
► But Firm 1, based on price P1, knows exactly what P2 will be, namely
P2 = 4 + P1/3
► So profit for Firm 1 really depends only on its own price P1 (plug equation for P2 into profit above):
Π1 = (P1 – 2)∙(90 – 15P1 + 10∙(4 + P1/3))
► Firm 1 has to pick P1 that maximizes its profit. If you plot this in Excel you’ll get P1 = 6.57 (precisely 46/7)
► Firm 2 responds with P2 = 4 + P1/3 = 6.19
► What are the profits?
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
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”
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
“discount” strategy (I): a simultaneous game
Simultaneous Game: Let’s consider the following setup:
► Firm 1 picks its price P1 that maximizes its profit:
Π1 = (P1 – MC1)∙Q1 = (P1 – 2)∙(90 – 15P1 + 10P2)
► But Firm 2, chooses to use a “discount” strategy pricing policy: whatever Firm 1 chooses, Firm 2 will set a price P2 at a
discount d relative to price P1:
P2 = P1 – 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?
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
“discount” strategy (I): a simultaneous game
► In the Bertrand solution Firm 2 will set a reaction function that maximizes its profit
Π2 = (P2 – MC2)∙Q2 = (P2 – 2)∙(90 – 15P2 + 10P1)
however Firm 2 uses the discount pricing strategy P2 = P1 – 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:
P1 = 4 + P2/3
► Firm 2 has a reaction function given by its discount pricing strategy:
P2 = P1 – d
► Equilibrium: the market maker will seek the pair of prices (P1*,P2*) that solves the system of equations represented by the
reactions functions (graphically – the intersection of the reaction functions):
P1* = 4 + P2*/3
Solution:
P2* = P1*– d
P1* = 6 – d/2
P2* = 6 – 3d/2
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
“discount” strategy (I): a simultaneous game
P2
The equilibrium derived according to the rules of
the game:
Firm 1
(Bertrand)
reaction function
P1 = 4 + P2/3
P1* = 6 – d/2
P2* = 6 – 3d/2
Firm 2
(discount pricing)
reaction function
P2 = P1 – d
► 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:
Q1 = 60 – 15d/2
Q2 = 60 + 35d/2
Bertrand
equilibrium
6
4
6 – 3d/2
new
equilibrium
► Compare this result with the Bertrand market
shares of (60,60).
4
6
Firm 2
(Bertrand)
reaction function
P2 = 4 + P1/3
P1
6 – d/2
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
“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 P2 at a discount d
relative to price P1:
P2 = P1 – d
► Firm 2 announces this strategy and also announces the discount d before Firm 1 chooses its price P1.
► Firm 1 picks its price P1 that maximizes its profit knowing Firm 2’s strategy P2 = P1 – d and also the announced d:
Π1 = (P1 – MC1)∙Q1 = (P1 – 2)∙(90 – 15P1 + 10P2)
► The game is played sequentially: Firm 2 announces its discount strategy and the discount d. Firm 1 chooses its price P1
knowing Firm 2’s strategy and the discount d. As soon as Firm 1 chooses its price P1, Firm 2’s price is determined according to
the discount strategy P2 = P1 – 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?
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microeconomics
lecture 7
the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
“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, P2 = P1 – d, thus Firm 1’s reaction function is the solution to the profit maximization function:
Π1 = (P1 – MC1)∙Q1 = (P1 – 2)∙(90 – 15P1 + 10P2) where P2 = P1 – d
► Let’s plug Firm 2’s reaction function (strategy) in Firm 1’s profit function to get:
Π1 = (P1 – MC1)∙Q1 = (P1 – 2)∙(90 – 15P1 + 10(P1 – d)) = – 5(P1)2 + (100 – 10d)P1 – 180 + 20d
► The profit maximizing price P1 for Firm 1 satisfies (from setting the first derivative equal to zero):
– 10P1 + (100 – 10d) = 0 that is P1 = 10 – d
► Given that P1 = 10 – d Firm 2’s price is P2 = P1 – d = 10 – 2d and the profit function for Firm 2 is:
Π2 = (P2 – MC2)∙Q2 = (P2 – 2)∙(90 – 15P2 + 10P1) where P1 = 10 – d and P2 = 10 – 2d
► Plugging these expressions for P1 and P2:
Π2 = 40(4 – d)(2 + d) = 40(8 +2d – d2)
► The optimal discount is found from the condition 2 – 2d = 0 which is d* = 1. The prices in equilibrium will be P1 = 9 and P2 = 8.
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microeconomics
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the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
own and cross elasticity
► Consider again the Bertrand model with two firms:
own price demand
sensitivity
market size for Heinz
cross price demand
sensitivity
demand for Heinz:
Q1 = a1 – b1P1 + d1P2
demand for Hunts:
Q2 = a2 – b2P2 + d2P1
cross price demand
sensitivity
market size for Hunts
own price demand
sensitivity
► The marginal costs are MC1 and MC2
► We can derive the reaction functions as:
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P1 = 0.5 ∙ a1/b1 + 0.5 ∙ MC1
+ 0.5 ∙ d1 ∙ P2 / b1
P2 = 0.5 ∙ a2/b2
+ 0.5 ∙ d2 ∙ P1 / b2
+ 0.5 ∙ MC2
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microeconomics
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the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
own and cross elasticity
► For the general model:
Q1 = a1 – b1P1 + d1P2
Q2 = a2 – b2P2 + d2P1
► For good 1 the “own” elasticity of demand is defined as the elasticity of Q1 with respect to P1 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 Q1 with respect to P2 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.
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microeconomics
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the oligopoly model (II): competition in prices
the analytics of constrained optimal
decisions
► Demand functions
demand for firm 1:
demand for firm 2:
own and cross elasticity
Q1 = a1 – b1P1 + d1P2
Q2 = a2 – b2P2 + d2P1
► We can describe the “market” (consumers) as three categories:
● loyalists to firm 1 (they stick with firm 1 even if P1 is extremely high relative to P2)
● loyalists to firm 2 (they stick with firm 1 even if P2 is extremely high relative to P1)
● switchers (they can buy from firm1 or firm 2 depending on the relative prices)
(a2 , b2, P2)
(a1 , b1, P1)
Q1 = a1 – b1P1 + d1P2
firm 1’s
“loyalists”
switchers
(b1 , d1, P1, P2)
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firm 2’s
“loyalists”
Q2 = a2 – b2P2 + d2P1
(b2 , d2, P2, P1)
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