2.1: Bertrand Competition

class: center, middle, inverse, title-slide

# 2.1: Bertrand Competition
## ECON 326 · Industrial Organization · Spring 2020
### Ryan Safner Assistant Professor of Economics <a href="mailto:safner@hood.edu"> safner@hood.edu</a> <a href="https://github.com/ryansafner/IOs20"> ryansafner/IOs20</a> <a href="https://IOs20.classes.ryansafner.com"> IOs20.classes.ryansafner.com</a>

---

# A More Rigorous Oligopoly/Cartel Problem

.smaller[
.content-box-green[
.hi-green[Example]: Suppose Squeaky Clean (Firm 1) and Biobase (Firm 2) are the only two producers of chlorine for swimming pools. The inverse market demand for chlorine is
`$$P=32-2Q$$`
where `$Q=q_{1}+q_{2}$` is measured in tons, and `$P$` is $/ton. Assume only a constant marginal cost of $16 for both firms
]

1. If the two firms collude and agree to act as a monopolist and evenly split the market, how much will each firm produce, what will be the market price, and how much profit will each firm earn?

2. Under this agreement, does either firm have an incentive to cheat (i.e. by producing an additional ton of chlorine)?
]

---

# Models of Oligopoly

.pull-left[
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**

]

.pull-right[
.center[
![:scale 100%](https://www.dropbox.com/s/tymdsxbqaq2scxq/chessknights.jpg?raw=1)
]
]

---

# Bertrand Competition

.left-column[
.center[
![:scale 80%](https://www.dropbox.com/s/difq3lgs9wwltru/Bertrand.jpg?raw=1)

.smaller[
Joseph Bertrand

1822-1890
]
]
]
.right-column[
- .hi["Bertrand competition"]: two (or more) firms compete on **price** to sell the **same good**

- Firms set their prices **simultaneously**

- Consumers are indifferent between the brands and .hi-purple[always buy from the seller with the lowest price]

]

---

# Bertrand Competition: Example

.pull-left[

- Suppose two firms, .hi-red[Walmart] and .hi-blue[Target] stock and sell identical HDTVs

- Costs each firm $200 to stock an HDTV

- Let `$Q$` be the *total* quantity purchased by consumers from the entire market (i.e. both firms)
  - `$Q=\color{red}{q_w}+\color{blue}{q_t}$`

- Denote .hi-red[Walmart]'s price as .hi-red[`\$p_w\$`] and .hi-blue[Target]'s price as .hi-blue[`\$p_t\$`]
]

.pull-right[
.center[
![](https://www.dropbox.com/s/tukgbue8pwu0x2x/hdtvs.jpg?raw=1)
]
]

---

# Bertrand Competition: Example

.pull-left[
- Demand for HDTV's at .hi-red[Walmart]:

]

---

# Bertrand Competition: Example

.pull-left[
- Demand for HDTV's at .hi-red[Walmart]:
 - `$Q$` if .red[`\$p_w\$`] `$<$` .blue[`\$p_t\$`]
]

---

# Bertrand Competition: Example

.pull-left[
- Demand for HDTV's at .hi-red[Walmart]:
 - `$Q$` if .red[`\$p_w\$`] `$<$` .blue[`\$p_t\$`]
 - `$\frac{Q}{2}$` if .red[`\$p_w\$`] `$=$` .blue[`\$p_t\$`]
]

---

# Bertrand Competition: Example

.pull-left[
- Demand for HDTV's at .hi-red[Walmart]:
 - `$Q$` if .red[`\$p_w\$`] `$<$` .blue[`\$p_t\$`]
 - `$\frac{Q}{2}$` if .red[`\$p_w\$`] `$=$` .blue[`\$p_t\$`]
 - `$0$` if .red[`\$p_w\$`] `$>$` .blue[`\$p_t\$`]
]

.pull-right[
<img src="2.1-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Bertrand Competition: Example

.pull-left[
- Demand for HDTV's at .hi-blue[Target]:
 - `$0$` if .red[`\$p_w\$`] `$<$` .blue[`\$p_t\$`]
 - `$\frac{Q}{2}$` if .red[`\$p_w\$`] `$=$` .blue[`\$p_t\$`]
 - `$Q$` if .red[`\$p_w\$`] `$>$` .blue[`\$p_t\$`]
]

---

# Bertrand Competition: Example

.pull-left[

- The only way to sell TVs is to match or beat your competitor's price

]

.pull-right[
.center[
![](https://www.dropbox.com/s/tukgbue8pwu0x2x/hdtvs.jpg?raw=1)
]
]

---

# Bertrand Competition: Example

.pull-left[

- The only way to sell TVs is to match or beat your competitor's price

- Suppose you are .hi-red[Walmart]

For a known .hi-blue[`\$p_t\$`], setting your price

`$$\color{red}{p_w}=\color{blue}{p_t}-\epsilon$$`

for any arbitrary `$\epsilon > 0$` captures you the entire market `$Q$`

- Same for .hi-blue[Target] for `$\color{red}{p_w}$`

]

.pull-right[
.center[
![](https://www.dropbox.com/s/tukgbue8pwu0x2x/hdtvs.jpg?raw=1)
]
]

---

# Bertrand Competition: Example

.pull-left[

- Won't charge `$p<MC$`, earn losses

- Firms continue undercutting one another until .hi-red[`\$p_w\$`] `$=$` .hi-blue[`\$p_t\$`] `$=MC$`

- .hi-purple[Nash Equilibrium]: `$$\big( \color{red}{p_w = MC}, \color{blue}{p_t=MC} \big)$$`
  - Firms earn no profits!

]

.pull-right[
.center[
![](https://www.dropbox.com/s/tukgbue8pwu0x2x/hdtvs.jpg?raw=1)
]
]

---

# Bertrand Paradox

.pull-left[

- .hi[Bertrand Paradox]: competitive outcome can be acheived with just 2 firms!
  - `$p=MC$`, `$\pi=0$`

]

.pull-right[
.center[
![](https://www.dropbox.com/s/tukgbue8pwu0x2x/hdtvs.jpg?raw=1)
]
]

---

# Walmart's Reaction Curve

.pull-left[
<img src="2.1-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[

We can graph .hi-red[Walmart]'s .hi-purple[reaction curve] to .hi-blue[Target]'s price

]

---

# Walmart's Reaction Curve

.pull-left[
<img src="2.1-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[

We can graph .hi-red[Walmart]'s .hi-purple[reaction curve] to .hi-blue[Target]'s price

- e.g. if .hi-blue[Target] sets a price of .hi-blue[$500], .hi-red[Walmart]'s best response is .hi-red[`\$$500-\epsilon\$`]

]

---

# Walmart's Reaction Curve

.pull-left[
<img src="2.1-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[

We can graph .hi-red[Walmart]'s .hi-purple[reaction curve] to .hi-blue[Target]'s price

]

---

# Walmart's Reaction Curve

.pull-left[
<img src="2.1-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[

We can graph .hi-red[Walmart]'s .hi-purple[reaction curve] to .hi-blue[Target]'s price

- e.g. if .hi-blue[Target] sets a price of .hi-blue[$500], .hi-red[Walmart]'s best response is .hi-red[`\$$500-\epsilon\$`]
- e.g. if .hi-blue[Target] sets a price of .hi-blue[$300], .hi-red[Walmart]'s best response is .hi-red[`\$$300-\epsilon\$`]
- e.g. if .hi-blue[Target] sets a price of .hi-blue[$200], (MC) .hi-red[Walmart]'s best response is .hi-red[`\$$200\$`] (MC)

]

---

# Target's Reaction Curve

.pull-left[
<img src="2.1-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[

We can graph .hi-blue[Target]'s .hi-purple[reaction curve] to .hi-red[Walmart]'s price

]
---

# Target's Reaction Curve

.pull-left[
<img src="2.1-slides_files/figure-html/unnamed-chunk-7-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[

We can graph .hi-blue[Target]'s .hi-purple[reaction curve] to .hi-red[Walmart]'s price

- e.g. if .hi-red[Walmart] sets a price of .hi-red[$500], .hi-blue[Target]'s best response is .hi-blue[`\$$500-\epsilon\$`]

]

---

# Target's Reaction Curve

.pull-left[
<img src="2.1-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[

We can graph .hi-blue[Target]'s .hi-purple[reaction curve] to .hi-red[Walmart]'s price

]

---

# Target's Reaction Curve

.pull-left[
<img src="2.1-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[

We can graph .hi-blue[Target]'s .hi-purple[reaction curve] to .hi-red[Walmart]'s price

- e.g. if .hi-red[Walmart] sets a price of .hi-red[$500], .hi-blue[Target]'s best response is .hi-blue[`\$$500-\epsilon\$`]
- e.g. if .hi-red[Walmart] sets a price of .hi-red[$300], .hi-blue[Target]'s best response is .hi-blue[`\$$300-\epsilon\$`]
- e.g. if .hi-red[Walmart] sets a price of .hi-red[$200] (MC),  .hi-blue[Target]'s best response is .hi-blue[`\$$200\$`] (MC)

]

---

# Nash Equilibrium with Reaction Curves

.pull-left[
<img src="2.1-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[

Combine both curves on the same graph

- .hi-purple[Nash Equilibrium]: `$$\big( \color{red}{p_w = MC}, \color{blue}{p_t=MC} \big)$$`
    - Where both reaction curves intersect

- No longer an incentive to undercut or change price

]

---

# Bertrand Competition: Moblab

.center[
![:scale 80%](https://www.dropbox.com/s/kzv1hjt3cpf3emy/Bertrandmoblab.png?raw=1)
]

---

# Bertrand Competition: Moblab

.pull-left[
- Each of you selling identical Economics course notes

- You will be put into a market with one other player

- Each term, both of you simultaneously choose your price

- Firm(s) choosing the lowest price get **all** the customers

]

.pull-right[
.center[
![](https://www.dropbox.com/s/kzv1hjt3cpf3emy/Bertrandmoblab.png?raw=1)
]

]

---
# Bertrand Competition: Moblab

.pull-left[

- The lowest price `$p_L$` determines the market demand
`$$q=3600-200p_L$$`

- Both firms have $2 cost per unit sold

- `$p=10$` maximizes total market profits

]

.pull-right[
.center[
![](https://www.dropbox.com/s/kzv1hjt3cpf3emy/Bertrandmoblab.png?raw=1)
]

]

---

# Bertrand Competition: Moblab

.pull-left[
`$$q=3600-200p_L$$`

.hi-green[Example:]
- Suppose Firm 1 sets `$p=9$` and Firm 2 sets `$p=10$`

- Firm 2 sells 0, makes $0

- Firm 1 sells `$q=3,600-200(9)=1,800$` and earns `$1,800(9-2)=12,600$` profit

]

.pull-right[
.center[
![](https://www.dropbox.com/s/kzv1hjt3cpf3emy/Bertrandmoblab.png?raw=1)
]

]