3.2 — Stackelberg Competition

ECON 326 • Industrial Organization • Spring 2023

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/ioS23
ioS23.classes.ryansafner.com

Stackelberg Competition: Moblab

Each of you is one Airline competing against another in a duopoly
- Each pays same per-flight cost
- Market price determined by total number of flights in market
LeadAir first chooses its number of flights, publicly announced
FollowAir then chooses its number of flights

Stackelberg Competition

Henrich von Stackelberg

1905-1946

“Stackelberg competition”: Cournot-style competition, two (or more) firms compete on quantity to sell the same good
Again, firms’ joint output determines the market price faced by all firms
But firms set their quantities sequentially
- Leader produces first
- Follower produces second

Stackelberg Competition: Example

Return to Coke and Pepsi again, with a constant marginal cost of $0.50 and the (inverse) market demand:

$\begin{aligned} P & = 5 - 0.05 Q \\ Q & = q_{c} + q_{p} \end{aligned}$

Stackelberg Competition: Example

$\begin{aligned} q_{c}^{*} & = 45 - 0.5 q_{p} \\ q_{p}^{*} & = 45 - 0.5 q_{c} \end{aligned}$

Suppose now that Coke is the leader and produces $q_{c}$ first

Stackelberg Competition: Example

$\begin{aligned} q_{c}^{*} & = 45 - 0.5 q_{p} \\ q_{p}^{*} & = 45 - 0.5 q_{c} \end{aligned}$

Suppose now that Coke is the leader and produces $q_{c}$ first
Coke knows exactly how Pepsi will respond to its output: $q_{p}^{*} = 45 - 0.5 q_{c}$

Stackelberg Competition: Example

$\begin{aligned} q_{c}^{*} & = 45 - 0.5 q_{p} \\ q_{p}^{*} & = 45 - 0.5 q_{c} \end{aligned}$

Suppose now that Coke is the leader and produces $q_{c}$ first
Coke knows exactly how Pepsi will respond to its output: $q_{p}^{*} = 45 - 0.5 q_{c}$
Coke, as leader, in theory faces entire market demand
- But not rational to act like a monopolist!
- knows that Pepsi (the follower) will still produce afterwards, which pushes down market price for both firms!

Stackelberg Competition as Sequential Game

This is a sequential game, so we should solve this via backward induction
Though Pepsi will move second (last), it will be responding to Coke's output
So Coke must know how Pepsi will react in order to choose its optimal output

Stackelberg Competition: ExampleSubstitute follower's reaction function into (inverse) market demand function faced by leader

  

Stackelberg Competition: Example

Substitute follower's reaction function into (inverse) market demand function faced by leader

$\begin{aligned} P & = 5 - 0.05 q_{c} - 0.05 p_{p} \\ P & = 5 - 0.05 q_{c} - 0.05 (45 - 0.5 q_{c}) \\ P & = 2.75 - 0.025 q_{c} \end{aligned}$

Stackelberg Competition: Example

Substitute follower's reaction function into (inverse) market demand function faced by leader

$\begin{aligned} P & = 5 - 0.05 q_{c} - 0.05 p_{p} \\ P & = 5 - 0.05 q_{c} - 0.05 (45 - 0.5 q_{c}) \\ P & = 2.75 - 0.025 q_{c} \end{aligned}$

Now find $M R (q)$ for Coke from this by doubling the slope:

$M R_{c} = 2.75 - 0.05 q_{c}$

Stackelberg Competition: Example

Now Coke can find its optimal quantity:

$\begin{aligned} M R_{c} & = M C \\ 2.75 - 0.05 q_{c} & = 0.50 \\ 45 & = q_{c}^{*} \end{aligned}$

Stackelberg Competition: Example

Now Coke can find its optimal quantity:

$\begin{aligned} M R_{c} & = M C \\ 2.75 - 0.05 q_{c} & = 0.50 \\ 45 & = q_{c}^{*} \end{aligned}$

Pepsi will optimally respond by producing:

$\begin{aligned} q_{p}^{*} & = 45 - 0.5 q_{c} \\ q_{p}^{*} & = 45 - 0.5 (45) \\ q_{p}^{*} & = 22.5 \end{aligned}$

Stackelberg Competition: Example

Stackelberg Nash Equilibrium: $(q_{c}^{*} = 45, q_{p}^{*} = 22.5)$

Stackelberg Competition: Example

With $q_{c}^{*} = 45$ and $q_{p}^{*} = 22.5$ , this sets a market-clearing price of:

$\begin{aligned} P & = 5 - 0.05 (67.5) \\ P & = $ 1.625 \end{aligned}$

Stackelberg Competition: Example

With $q_{c}^{*} = 45$ and $q_{p}^{*} = 22.5$ , this sets a market-clearing price of:

$\begin{aligned} P & = 5 - 0.05 (67.5) \\ P & = $ 1.625 \end{aligned}$

Coke's profit would be:

$\begin{aligned} π_{c} & = (1.625 - 0.50) 45 \\ π_{c} & = $ 50.625 \end{aligned}$

Stackelberg Competition: Example

With $q_{c}^{*} = 45$ and $q_{p}^{*} = 22.5$ , this sets a market-clearing price of:

$\begin{aligned} P & = 5 - 0.05 (67.5) \\ P & = $ 1.625 \end{aligned}$

Coke's profit would be:

$\begin{aligned} π_{c} & = (1.625 - 0.50) 45 \\ π_{c} & = $ 50.625 \end{aligned}$

Pepsi's profit would be:

$\begin{aligned} π_{p} & = (1.625 - 0.50) 22.5 \\ π_{p} & = $ 25.3125 \end{aligned}$

Stackelberg-Nash Equilibrium, The Market

Cournot vs. Stackelberg Competition
  FirmCournot (p = $2.00)Stackelberg (p = $1.63)
outputprofitoutputprofit
Coke30.00$45.0045.00$50.63
Pepsi30.0$45.0022.50$25.31
INDUSTRY60.0$90.0067.50$75.94


  

Firm	Cournot (p = $2.00)	Stackelberg (p = $1.63)
Coke	30.00	$45.00	45.00	$50.63
Pepsi	30.0	$45.00	22.50	$25.31
INDUSTRY	60.0	$90.00	67.50	$75.94

Stackelberg and First-Mover Advantage

Stackelberg leader clearly has a first-mover advantage over the follower
- Leader: $q^{*} = 45$ , π = $50.63
- Follower: $q^{*} = 22.5$ , π = $25.31
If firms compete simultaneously (Cournot): $q^{*} = 30$ , π = $45.00 each
Leading $≻$ simultaneous $≻$ Following

Stackelberg and First-Mover Advantage

Stackelberg Nash equilibrium requires perfect information for both leader and follower
- Follower must be able to observe leader's output to choose its own
- Leader must believe follower will see leader's output and react optimally
Imperfect information reduces the game to (simultaneous) Cournot competition

Stackelberg and First-Mover Advantage

Again, leader cannot act like a monopolist
- A strategic game! Market output (that pushes down market price) is $Q = q_{c} + q_{p}$
Leader's choice of 45 is optimal only if follower responds with 22.5

Comparing All Oligopoly Models

Firm	Bertrand (p = $0.50)		Cournot (p = $2.00)		Stackelberg (p = $1.63)		Collusion (p = $2.75)
Firm	output	profit	output	profit	output	profit	output	profit
Coke	45.00	$0.00	30.00	$45.00	45.00	$50.63	22.50	$50.63
Pepsi	45.00	$0.00	30.00	$45.00	22.50	$25.31	22.50	$50.63
INDUSTRY	90.00	$0.00	60.00	$90.00	67.50	$75.94	45.00	$101.25

Output: $Q_{m} < Q_{c} < Q_{s} < Q_{b}$
Market price: $P_{b} < P_{s} < P_{c} < P_{m}$
Profit: $π_{b} = 0 < π_{s} < π_{c} < π_{m}$

Where subscript $m$ is monopoly (collusion), $c$ is Cournot, $s$ is Stackelberg, $b$ is Bertrand

3.2 — Stackelberg Competition

ECON 326 • Industrial Organization • Spring 2023

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/ioS23
ioS23.classes.ryansafner.com

Stackelberg Competition: Moblab

Each of you is one Airline competing against another in a duopoly
- Each pays same per-flight cost
- Market price determined by total number of flights in market
LeadAir first chooses its number of flights, publicly announced
FollowAir then chooses its number of flights

Stackelberg Competition

Henrich von Stackelberg

1905-1946

“Stackelberg competition”: Cournot-style competition, two (or more) firms compete on quantity to sell the same good
Again, firms’ joint output determines the market price faced by all firms
But firms set their quantities sequentially
- Leader produces first
- Follower produces second

Stackelberg Competition: Example

Return to Coke and Pepsi again, with a constant marginal cost of $0.50 and the (inverse) market demand:

$\begin{aligned} P & = 5 - 0.05 Q \\ Q & = q_{c} + q_{p} \end{aligned}$

Stackelberg Competition: Example

$\begin{aligned} q_{c}^{*} & = 45 - 0.5 q_{p} \\ q_{p}^{*} & = 45 - 0.5 q_{c} \end{aligned}$

Suppose now that Coke is the leader and produces $q_{c}$ first

Stackelberg Competition: Example

$\begin{aligned} q_{c}^{*} & = 45 - 0.5 q_{p} \\ q_{p}^{*} & = 45 - 0.5 q_{c} \end{aligned}$

Suppose now that Coke is the leader and produces $q_{c}$ first
Coke knows exactly how Pepsi will respond to its output: $q_{p}^{*} = 45 - 0.5 q_{c}$

Stackelberg Competition: Example

$\begin{aligned} q_{c}^{*} & = 45 - 0.5 q_{p} \\ q_{p}^{*} & = 45 - 0.5 q_{c} \end{aligned}$

Suppose now that Coke is the leader and produces $q_{c}$ first
Coke knows exactly how Pepsi will respond to its output: $q_{p}^{*} = 45 - 0.5 q_{c}$
Coke, as leader, in theory faces entire market demand
- But not rational to act like a monopolist!
- knows that Pepsi (the follower) will still produce afterwards, which pushes down market price for both firms!

Stackelberg Competition as Sequential Game

This is a sequential game, so we should solve this via backward induction
Though Pepsi will move second (last), it will be responding to Coke's output
So Coke must know how Pepsi will react in order to choose its optimal output

Stackelberg Competition: ExampleSubstitute follower's reaction function into (inverse) market demand function faced by leader

  

Stackelberg Competition: Example

Substitute follower's reaction function into (inverse) market demand function faced by leader

$\begin{aligned} P & = 5 - 0.05 q_{c} - 0.05 p_{p} \\ P & = 5 - 0.05 q_{c} - 0.05 (45 - 0.5 q_{c}) \\ P & = 2.75 - 0.025 q_{c} \end{aligned}$

Stackelberg Competition: Example

Substitute follower's reaction function into (inverse) market demand function faced by leader

$\begin{aligned} P & = 5 - 0.05 q_{c} - 0.05 p_{p} \\ P & = 5 - 0.05 q_{c} - 0.05 (45 - 0.5 q_{c}) \\ P & = 2.75 - 0.025 q_{c} \end{aligned}$

Now find $M R (q)$ for Coke from this by doubling the slope:

$M R_{c} = 2.75 - 0.05 q_{c}$

Stackelberg Competition: Example

Now Coke can find its optimal quantity:

$\begin{aligned} M R_{c} & = M C \\ 2.75 - 0.05 q_{c} & = 0.50 \\ 45 & = q_{c}^{*} \end{aligned}$

Stackelberg Competition: Example

Now Coke can find its optimal quantity:

$\begin{aligned} M R_{c} & = M C \\ 2.75 - 0.05 q_{c} & = 0.50 \\ 45 & = q_{c}^{*} \end{aligned}$

Pepsi will optimally respond by producing:

$\begin{aligned} q_{p}^{*} & = 45 - 0.5 q_{c} \\ q_{p}^{*} & = 45 - 0.5 (45) \\ q_{p}^{*} & = 22.5 \end{aligned}$

Stackelberg Competition: Example

Stackelberg Nash Equilibrium: $(q_{c}^{*} = 45, q_{p}^{*} = 22.5)$

Stackelberg Competition: Example

With $q_{c}^{*} = 45$ and $q_{p}^{*} = 22.5$ , this sets a market-clearing price of:

$\begin{aligned} P & = 5 - 0.05 (67.5) \\ P & = $ 1.625 \end{aligned}$

Stackelberg Competition: Example

With $q_{c}^{*} = 45$ and $q_{p}^{*} = 22.5$ , this sets a market-clearing price of:

$\begin{aligned} P & = 5 - 0.05 (67.5) \\ P & = $ 1.625 \end{aligned}$

Coke's profit would be:

$\begin{aligned} π_{c} & = (1.625 - 0.50) 45 \\ π_{c} & = $ 50.625 \end{aligned}$

Stackelberg Competition: Example

With $q_{c}^{*} = 45$ and $q_{p}^{*} = 22.5$ , this sets a market-clearing price of:

$\begin{aligned} P & = 5 - 0.05 (67.5) \\ P & = $ 1.625 \end{aligned}$

Coke's profit would be:

$\begin{aligned} π_{c} & = (1.625 - 0.50) 45 \\ π_{c} & = $ 50.625 \end{aligned}$

Pepsi's profit would be:

$\begin{aligned} π_{p} & = (1.625 - 0.50) 22.5 \\ π_{p} & = $ 25.3125 \end{aligned}$

Stackelberg-Nash Equilibrium, The Market

Cournot vs. Stackelberg Competition

Stackelberg and First-Mover Advantage

Stackelberg leader clearly has a first-mover advantage over the follower
- Leader: $q^{*} = 45$ , π = $50.63
- Follower: $q^{*} = 22.5$ , π = $25.31
If firms compete simultaneously (Cournot): $q^{*} = 30$ , π = $45.00 each
Leading $≻$ simultaneous $≻$ Following

Stackelberg and First-Mover Advantage

Stackelberg Nash equilibrium requires perfect information for both leader and follower
- Follower must be able to observe leader's output to choose its own
- Leader must believe follower will see leader's output and react optimally
Imperfect information reduces the game to (simultaneous) Cournot competition

Stackelberg and First-Mover Advantage

Again, leader cannot act like a monopolist
- A strategic game! Market output (that pushes down market price) is $Q = q_{c} + q_{p}$
Leader's choice of 45 is optimal only if follower responds with 22.5

Comparing All Oligopoly Models

Output: $Q_{m} < Q_{c} < Q_{s} < Q_{b}$
Market price: $P_{b} < P_{s} < P_{c} < P_{m}$
Profit: $π_{b} = 0 < π_{s} < π_{c} < π_{m}$

Where subscript $m$ is monopoly (collusion), $c$ is Cournot, $s$ is Stackelberg, $b$ is Bertrand

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3.2 — Stackelberg Competition

ECON 326 • Industrial Organization • Spring 2023

Ryan Safner Associate Professor of Economics safner@hood.edu ryansafner/ioS23 ioS23.classes.ryansafner.com

Stackelberg Competition: Moblab

Stackelberg Competition: Moblab

Stackelberg Competition

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition as Sequential Game

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg-Nash Equilibrium, The Market

Cournot vs. Stackelberg Competition

Stackelberg and First-Mover Advantage

Stackelberg and First-Mover Advantage

Stackelberg and First-Mover Advantage

Comparing All Oligopoly Models

Stackelberg Competition: Moblab

Help

3.2 — Stackelberg Competition

3.2 — Stackelberg Competition

ECON 326 • Industrial Organization • Spring 2023

Ryan Safner Associate Professor of Economics safner@hood.edu ryansafner/ioS23 ioS23.classes.ryansafner.com

Stackelberg Competition: Moblab

Stackelberg Competition: Moblab

Stackelberg Competition

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition as Sequential Game

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg Competition: Example

Stackelberg-Nash Equilibrium, The Market

Cournot vs. Stackelberg Competition

Stackelberg and First-Mover Advantage

Stackelberg and First-Mover Advantage

Stackelberg and First-Mover Advantage

Comparing All Oligopoly Models

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/ioS23
ioS23.classes.ryansafner.com

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/ioS23
ioS23.classes.ryansafner.com