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Stackelberg Models of Duopoly�

Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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Outline�

Review
Stackelberg’s Model of Duopoly
Model of Duopoly with Advertising

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Review:�Perfect vs. Imperfect Information

Perfect information: All previous moves are observed before the next move is chosen and each player knows Who has moved Where before she makes a decision

Player 2 makes her choice after observing player 1’s choice

Imperfect information: A player may not know Who has moved Where before making a decision

Player 2 makes her choice at the same time as player 1 does

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Stackelberg Model of Duopoly�

A product is produced by two firms: firm 1 and firm 2

The quantities are denoted by q₁ and q₂, respectively.

The timing of the game is as follows: firm 1 chooses a quantity q₁ ≥0, then firm 2 observes q₁ and chooses a quantity q₂ ≥0

The payoff of each firm depends on the market price and the cost of production

The market price is P(Q)=a-Q, where a is a constant number and Q=q₁+q₂

The cost to firm i of producing quantity q_i is C_i(q_i)=cq_i

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Stackelberg Model of Duopoly�

Below is the extensive form of the game

Payoff functions at the bottom of the tree are:� u₁(q₁, q₂)=q₁{a-(q₁+q₂)}-q₁c� u₂(q₁, q₂)=q₂{a-(q₁+q₂)}-q₂c

How many subgames?

Infinite!

Firm 1

q₁

Firm 2

q₂

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Stackelberg Model of Duopoly�

Find the SPNE by backward induction:

We first solve firm 2’s problem for any q₁≥0 to get firm 2’s best response to q₁

That is, we first solve all the subgames beginning at firm 2

Then we solve firm 1’s problem

That is, solve the subgame beginning at firm 1

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Stackelberg Model of Duopoly�

Solve firm 2’s problem for any q₁≥0 to get firm 2’s best response to q₁

Solve

Maximize u₂(q₁, q₂) = q₂{a - (q₁+q₂)} - q₂c �Subject to 0 ≤ q₂≤ +∞��FOC: u₂'(q₁, q₂) = a - q₁ - 2q₂ - c = 0�Solution: R₂(q₁) = q₂ = (a - c - q₁)/2 if q₁ ≤ a - c and R₂(q₁) = 0 if q₁ > a – c

(best response of firm 2)

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Stackelberg Model of Duopoly�

Solve firm 1’s problem

Note that firm 1 can also solve firm 2’s problem
That is, firm 1 knows firm 2’s best response to any q₁

Solve �Maximize u₁(q₁,R₂(q₁)) = q₁{a - (q₁+R₂(q₁))} - q₁c = q₁(a - c - q₁)/2 �Subject to 0 ≤ q₁≤ +∞��FOC: u₁'(q₁, q₂) = (a - c - 2q₁)/2 = 0�Solution: q₁ = (a - c)/2

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Stackelberg Model of Duopoly�

SPNE is ( (a-c)/2, R₂(q₁) ), where R₂(q₁) = (a - c – q₁)/2 if q₁ ≤ a - c and R₂(q₁) = 0 if q₁ > a - c

That is, firm 1 chooses q₁= (a-c)/2, firm 2 chooses q₂= R₂(q₁) if firm 1 chooses a quantity q₁

The backward induction path is

(q₁, q₂) = ( (a-c)/2, (a-c)/4 )

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Stackelberg Model of Duopoly�

Production quantity:

Firm 1: q₁ = (a-c)/2
Firm 2: q₂ = (a-c)/4
Aggregate: Q* = q₁ + q₂ = 3(a-c)/4

Profit:

Firm 1: u₁(q₁, q₂) = (a-c)²/8
Firm 2: u₂(q₁, q₂) = (a-c)²/16
First-mover advantage

The market price is: P(Q*) = a - Q* = a - 3(a-c)/4

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Cournot Model of Duopoly�

Production quantity:

Firm 1: q₁ = (a-c)/3
Firm 2: q₂ = (a-c)/3
Aggregate: Q* = q₁ + q₂ = 2(a-c)/3

Profit:

Firm 1: u₁(q₁, q₂) = (a-c)²/9
Firm 2: u₂(q₁, q₂) = (a-c)²/9

The market price is: P(Q*) = a - Q* = a - 2(a-c)/3

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Advertising and Competition�

Background: To generate profits, firms must do more than just produce goods or services; they must also market their products to consumers

Firms advertise to increase the demand for their products

(1) Highlighting own products’ advantages (positive advertisements)
https://www.youtube.com/watch?v=p69HFTRgXIM
(2) Highlighting the disadvantages of the competing products (negative advertisements)
https://www.youtube.com/watch?v=bWByAPK9gzM

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Model of Duopoly with Advertising�

A simple model incorporating advertising:

Stage 1:

Firm 1 selects an advertising level a (where a > 0) and pays an advertising cost of C₁(a)=a³/270 - 2ac/9
Advertising has a positive effect on the demand for the goods sold in the industry, enhancing the price that consumers are willing to pay for the output of both firms
The market price is P(Q)=a-(q₁+q₂)

Stage 2:

The amount of advertising a is observed by firm 2, and both firms simultaneously and independently select their production levels q₁ and q₂
Firms produce at the cost: C₁(q₁)=cq₁ and C₂(q₂)= cq₂

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Model of Duopoly with Advertising�

Below is the extensive form of the game

Payoff functions at the bottom of the tree are :� u₁(q₁, q₂)=q₁{a-(q₁+q₂)}-q₁c-a³/270+2ac/9� u₂(q₁, q₂)=q₂{a-(q₁+q₂)}-q₂c

How many subgames?

Infinite!

Firm 2

Firm 1

q₂

q₁

Firm 1

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Model of Duopoly with Advertising�

Stage 2: Find the equilibrium quantities of firms 1 and 2 given any advertisement level a selected by firm 1

Firm 1 maximizes u₁(q₁, q₂)=q₁{a-(q₁+q₂)}-q₁c-a³/270+2ac/9

FOC: ∂u₁/∂q₁ = 0 and then solve for q₁

Solution: q₁ = R₁(q₂) = (a - c - q₂)/2

Firm 2 maximizes u₂(q₁, q₂)=q₂{a-(q₁+q₂)}-q₂c

FOC: ∂u₂/∂q₂ = 0 and then solve for q₂

Solution: q₂ = R₂(q₁) = (a - c - q₁)/2

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Model of Duopoly with Advertising�

Solving best response functions:

Best response for firm 1 is q₁ = (a - c - q₂)/2
Best response for firm 2 is q₂ = (a - c - q₁)/2
Solving them simultaneously, gives q₁^*=q₂^*= (a-c)/3

The equilibrium price is: P^* = a - q₁^* - q₂^*= a - 2(a-c)/3

The equilibrium payoffs are:

u₁^* = (a-c)²/9 - a³/270 + 2ac/9

u₂^* = (a-c)²/9

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Model of Duopoly with Advertising�

Stage 1: Evaluate firm 1’s advertisement level at the beginning of the game

Firm 1 knows that choosing a will induce a subgame equilibrium with a payoff of u₁^* = (a-c)²/9 - a³/270 + 2ac/9

FOC: ∂u₁^*/∂a = 2(a-c)/9 - 3a²/270 + 2c/9 = 0

Solution: a^* = 20

SPNE is given by a^* = 20, q₁^*(a) = (a-c)/3, and q₂^*(a)= (a-c)/3

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Two Experiments�

Next time!

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Thank you!

Roman Sheremeta, Ph.D.

Professor, Weatherhead School of Management

Case Western Reserve University

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References�

Watson, J. (2013). Strategy: An Introduction to Game Theory (3rd Edition). Publisher: W. W. Norton & Company. (Chapters 15 & 16)