1. Stackelberg Models of Duopoly
Lecture 10
2/18/2008
Stackelberg Models of Duopoly
Roman Sheremeta, Ph.D.
Professor, Weatherhead School of Management
Case Western Reserve University

2. Outline Review Stackelberg’s Model of Duopoly
Lecture 3
Outline
Review
Stackelberg’s Model of Duopoly
Model of Duopoly with Advertising

3. 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

4. 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 qi is Ci(qi)=cqi

5. 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

6. 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

7. 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)

8. 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

9. 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 )

10. 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

11. 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

12. Advertising and Competition
Lecture 3
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)
(2) Highlighting the disadvantages of the competing products (negative advertisements)

13. 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/ ac/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

14. 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/ u2(q1, q2)=q2{a-(q1+q2)}-q2c
How many subgames?
Infinite!
Firm 2
Firm 1 q2 q1 a

15. 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

16. 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/ ac/9
u2* = (a-c)2/9

17. 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/ ac/9
FOC: ∂u1*/∂a = 2(a-c)/9 - 3a2/ c/9 = 0
Solution: a* = 20
SPNE is given by a* = 20, q1*(a) = (a-c)/3, and q2*(a) = (a- c)/3

18. Two Experiments
Next time!

19. Thank you! Roman Sheremeta, Ph.D.
Professor, Weatherhead School of Management
Case Western Reserve University

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