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

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

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

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

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

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

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)

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

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 )

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

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

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

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

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 a

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

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

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

Two Experiments Next time!

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

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