2.4 — Stackelberg Competition

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# 2.4 — Stackelberg Competition
## ECON 316 • Game Theory • Fall 2021
### Ryan Safner Assistant Professor of Economics <a href="mailto:safner@hood.edu">safner@hood.edu</a> <a href="https://github.com/ryansafner/gameF21">ryansafner/gameF21</a> <a href="https://gameF21.classes.ryansafner.com"> gameF21.classes.ryansafner.com</a>

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# Models of Oligopoly

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Three canonical models of Oligopoly

1. .hi[Bertrand competition]
  - Firms **simultaneously** compete on **price**
2. .hi[Cournot competition]
  - Firms **simultaneously** compete on **quantity**
3. .hi[Stackelberg competition]
  - Firms **sequentially** compete on **quantity**

]

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.center[
![:scale 100%](../images/chessknights.jpg)
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# Stackelberg Competition

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.center[
![:scale 70%](../images/stackelberg.jpeg)

.smaller[
Henrich von Stackelberg

1905-1946
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]
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- .hi["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**
  - .hi-purple[Leader] produces first
  - .hi-purple[Follower] produces second

]

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# Stackelberg Competition: Example

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.hi-green[Example]: Return to .hi-red[Saudi Arabia `\$(sa)\$`] and .hi-blue[Iran `\$(i)\$`], again with the market (inverse) demand curve:
`$$\begin{align*}
P&=200-3Q\\
Q&=\color{red}{q_{sa}}+\color{blue}{q_i}\\
\end{align*}$$`

]

- We solved for Saudi Arabia and Iran's .hi-purple[reaction functions] in **Cournot competition** last class:
`$$\begin{align*}
\color{red}{q_{sa}^*}&\color{red}{=30-0.5}\color{blue}{q_i}\\
\color{blue}{q_i^*}&\color{blue}{=30-0.5}\color{red}{q_{sa}}\\
\end{align*}$$`

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# Stackelberg Competition: Example

`$$\begin{align*}
\color{red}{q_{sa}^*}&\color{red}{=30-0.5}\color{blue}{q_i}\\
\color{blue}{q_i^*}&\color{blue}{=30-0.5}\color{red}{q_{sa}}\\
\end{align*}$$`

- Suppose .hi-red[Saudi Arabia] is the .hi-purple[Stackelberg leader] and produces `$q_{sa}$` **first**

--
- Saudi Arabia knows exactly how Iran will respond to its output

`$$\color{blue}{q_i^*}\color{blue}{=30-0.5}\color{red}{q_{sa}}$$`

- .hi-red[Saudi Arabia], as leader, essentially faces **entire market demand**
  - But **can't** act like a pure monopolist!
  - knows that .hi-blue[follower] will still produce afterwards, which pushes down market price for both firms!

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# Stackelberg Competition: Example

- Substitute .hi-blue[follower]'s reaction function into (inverse) market demand function faced by .hi-red[leader]

`$$\begin{align*}
P&=200-3q_{sa}-3(30-0.5q_{sa})\\
P&=110-1.5q_{sa}\\
\end{align*}$$`

- Now find `$MR(q)$` for .hi-red[Saudi Arabia] from this by doubling the slope:

`$$MR_{Leader}=110-3q_{sa}$$`

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# Stackelberg Competition: Example

- Now .hi-red[Saudi Arabia] can find its optimal quantity:

`$$\begin{align*}
MR_{Leader}&=MC\\
110-3q_{sa}&=20\\
30&=q_{sa}^*\\
\end{align*}$$`

- .hi-blue[Iran] will optimally respond by producing:

`$$\begin{align*}
q_i^*&=30-0.5q_{sa}\\
q_i^*&=30-0.5(30)\\
q_i^*&=15\\
\end{align*}$$`

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# Stackelberg Equilibrium, Graphically

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<img src="2.4-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" style="display: block; margin: auto;" />
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- **Stackelberg Nash Equilibrium**: `$$\big( \color{red}{q^*_{sa}=30}, \color{blue}{q^*_{i}=15} \big)$$`

]

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# Stackelberg Competition: Example

- With `$\color{red}{q^*_{sa}=30}$` and `$\color{blue}{q^*_i=15}$`, this sets a market-clearing price of:

`$$\begin{align*}
P&=200-3(45)\\
P&=65\\
\end{align*}$$`

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- .hi-red[Saudi Arabia's] profit would be:

`$$\begin{align*}
\pi_{sa}&=30(65-20)\\
\pi_{sa}&=\$1,350\\
\end{align*}$$`

]

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- .hi-blue[Iran's] profit would be:

`$$\begin{align*}
\pi_{i}&=15(65-20)\\
\pi_{i}&=\$675\\
\end{align*}$$`

]

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# Stackelberg Equilibrium, The Market

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# Cournot vs. Stackelberg Competition

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![:scale 80%](../images/cournot-stackelberg-compare.png)

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- **Leader** Saudi Arabia `$\uparrow$` its output and `$\uparrow$` profits

- **Follower** Iran forced to `$\downarrow$` its output and accept `$\downarrow$` profits

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# Stackelberg and First-Mover Advantage

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<img src="2.4-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" style="display: block; margin: auto;" />

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- Stackelberg **leader** clearly has a .hi-purple[first-mover advantage] over the **follower**
  - **Leader**: `$q^*=30$`, `$\pi=1,350$`
  - **Follower**: `$q^*=15$`, `$\pi=675$`

- If firms compete **simultaneously** (.hi[Cournot]): `$q^*=20$`, `$\pi=1,200$` each

- Leading `$\succ$` simultaneous `$\succ$` Following
]

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# Stackelberg and First-Mover Advantage

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<img src="2.4-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" style="display: block; margin: auto;" />

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- Stackelberg Nash equilibrium requires .hi-purple[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

- .hi-purple[Imperfect information] reduces the game to (simultaneous) .hi[Cournot competition]
]

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# Stackelberg and First-Mover Advantage

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<img src="2.4-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" style="display: block; margin: auto;" />
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- Again, leader *cannot* act like a monopolist 
  - A strategic game! Market output (that pushes down market price) is `$Q=q_{sa}+q_{i}$`
    
- Leader's choice of 30 is optimal **only if** follower responds with 15
]

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# Comparing All Oligopoly Models

.center[
![](../images/oligopoly-comparison.png)
]

- Output: `$Q_m < Q_c < Q_s < Q_b$`
- Market price: `$P_b < P_s < P_c < P_m$`
- Profit: `$\pi_b=0 < \pi_s < \pi_c < \pi_m$`

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

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# Stackelberg Competition: Moblab

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![:scale 80%](../images/stackelbergmoblab.png)

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# Stackelberg Competition: Moblab

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.center[
![](../images/stackelbergmoblab.png)
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- 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

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