3.2 — Stackelberg Competition

class: title-slide

# 3.2 — Stackelberg Competition

## ECON 326 • Industrial Organization • Spring 2023

### Ryan Safner<br> Associate Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/ioS23"><i class="fa fa-github fa-fw"></i>ryansafner/ioS23</a><br> <a href="https://ioS23.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>ioS23.classes.ryansafner.com</a><br>

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# Stackelberg Competition: Moblab

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

]

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# Stackelberg Competition: Moblab

.pull-left[
- 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

]

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

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# Stackelberg Competition

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

.smallest[
Henrich von Stackelberg

1905-1946
]
]
]
.right-column[
- .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

.pull-left[

- Return to .hi-red[Coke] and .hi-blue[Pepsi] again, with a constant marginal cost of $0.50 and the (inverse) market demand:

`$$\begin{align*}
P&=5-0.05Q\\
Q&=\color{red}{q_{c}}+\color{blue}{q_p}\\
\end{align*}$$`

]

.pull-right[
.center[
![](../images/cokepepsi.jpg)
]
]

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# Stackelberg Competition: Example

`$$\begin{align*}
\color{red}{q_{c}^*}&=45-0.5\color{blue}{q_{p}}\\
\color{blue}{q_{p}^*}&=45-0.5\color{red}{q_{c}}\\
\\ \end{align*}$$`

- Suppose now that .hi-red[Coke] is the .hi-purple[leader] and produces `$\color{red}{q_c}$` **first**

- .hi-red[Coke] knows exactly how .hi-blue[Pepsi] will respond to its output:
`$$\color{blue}{q_{p}^*}=45-0.5\color{red}{q_{c}}$$`

- .hi-red[Coke], as leader, in theory faces **entire market demand**
  - But **not rational** to act like a monopolist!
  - knows that .hi-blue[Pepsi] (the .hi-purple[follower]) will still produce afterwards, which pushes down market price for both firms!

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# Stackelberg Competition as Sequential Game

.pull-left[

- This is a sequential game, so we should solve this via .hi-purple[backward induction]

- Though .hi-blue[Pepsi] will move second (last), it will be responding to .hi-red[Coke]'s output

- So .hi-red[Coke] must know how .hi-blue[Pepsi] will react in order to choose its optimal output
]

.pull-right[
.center[
![:scale 75%](../images/stackelberg_game.png)
]
]

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# Stackelberg Competition: Example

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

`$$\begin{align*}
P&=5-0.05\color{red}{q_c}-0.05\color{blue}{p_p}\\
P&=5-0.05\color{red}{q_{c}}-0.05\color{blue}{(}45-0.5\color{red}{q_{c}}\color{blue}{)}\\
P&=2.75-0.025\color{red}{q_{c}}\\
\end{align*}$$`

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

`$$MR_{c}=2.75-0.05q_{c}$$`

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# Stackelberg Competition: Example

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

`$$\begin{align*}
MR_{c}&=MC\\
2.75-0.05q_{c}&=0.50\\
45&=q_{c}^*\\
\end{align*}$$`
--

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

`$$\begin{align*}
q_p^*&=45-0.5q_{c}\\
q_p^*&=45-0.5(45)\\
q_p^*&=22.5\\
\end{align*}$$`

---

# Stackelberg Competition: Example

.pull-left[
<img src="3.2-slides_files/figure-html/unnamed-chunk-34-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[
- **Stackelberg Nash Equilibrium**: `$$\big( \color{red}{q^*_{c}=45}, \color{blue}{q^*_{p}=22.5} \big)$$`

]

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# Stackelberg Competition: Example

- With `$\color{red}{q^*_{c}=45}$` and `$\color{blue}{q^*_p=22.5}$`, this sets a market-clearing price of:

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

.pull-left[
- .hi-red[Coke's] profit would be:

`$$\begin{align*}
\pi_{c}&=(1.625-0.50)45\\
\pi_{c}&=\$50.625\\
\end{align*}$$`

]

.pull-right[
- .hi-blue[Pepsi's] profit would be:

`$$\begin{align*}
\pi_{p}&=(1.625-0.50)22.5\\
\pi_{p}&=\$25.3125\\
\end{align*}$$`

]

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# Stackelberg-Nash Equilibrium, The Market

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# Cournot vs. Stackelberg Competition

<div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-3c068f32eab275168532" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-3c068f32eab275168532">{"x":{"filter":"none","vertical":false,"class":"display","data":[["Coke","Pepsi","INDUSTRY"],["30.00","30.0","60.0"],["$45.00","$45.00","$90.00"],["45.00","22.50","67.50"],["$50.63","$25.31","$75.94"]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th rowspan=\"2\">Firm<\/th>\n      <th colspan=\"2\">Cournot (p = $2.00)<\/th>\n      <th colspan=\"2\">Stackelberg (p = $1.63)<\/th>\n    <\/tr>\n    <tr>\n      <th>output<\/th>\n      <th>profit<\/th>\n      <th>output<\/th>\n      <th>profit<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

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# Stackelberg and First-Mover Advantage

.pull-left[

- Stackelberg **leader** clearly has a .hi-purple[first-mover advantage] over the **follower**
  - **Leader**: `$q^*=45$`, π = $50.63
  - **Follower**: `$q^*=22.5$`, π = $25.31

- If firms compete **simultaneously** (.hi[Cournot]): `$q^*=30$`, π = $45.00 each

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

]

.pull-right[
<img src="3.2-slides_files/figure-html/stackrf-1.png" width="504" style="display: block; margin: auto;" />
]

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# Stackelberg and First-Mover Advantage

.pull-left[

- 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]

]

.pull-right[
<img src="3.2-slides_files/figure-html/unnamed-chunk-37-1.png" width="504" style="display: block; margin: auto;" />
]

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# Stackelberg and First-Mover Advantage

.pull-left[

- 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

]

.pull-right[
<img src="3.2-slides_files/figure-html/unnamed-chunk-38-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Comparing All Oligopoly Models

<div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-6ea30860779d71098f5e" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-6ea30860779d71098f5e">{"x":{"filter":"none","vertical":false,"class":"display","data":[["Coke","Pepsi","INDUSTRY"],["45.00","45.00","90.00"],["$0.00","$0.00","$0.00"],["30.00","30.00","60.00"],["$45.00","$45.00","$90.00"],["45.00","22.50","67.50"],["$50.63","$25.31","$75.94"],["22.50","22.50","45.00"],["$50.63","$50.63","$101.25"]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th rowspan=\"2\">Firm<\/th>\n      <th colspan=\"2\">Bertrand (p = $0.50)<\/th>\n      <th colspan=\"2\">Cournot (p = $2.00)<\/th>\n      <th colspan=\"2\">Stackelberg (p = $1.63)<\/th>\n      <th colspan=\"2\">Collusion (p = $2.75)<\/th>\n    <\/tr>\n    <tr>\n      <th>output<\/th>\n      <th>profit<\/th>\n      <th>output<\/th>\n      <th>profit<\/th>\n      <th>output<\/th>\n      <th>profit<\/th>\n      <th>output<\/th>\n      <th>profit<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"dom":"t","columnDefs":[],"order":[],"autoWidth":false,"orderClasses":false}},"evals":[],"jsHooks":[]}</script>

.smaller[
- 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
]