1.3: Perfect Competition I: Short Run

# 1.3: Perfect Competition I: Short Run
## 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>

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# Profit Maximization in the Short Run

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# A *Competitive* Market

- We assume (for now) the firm is in a .hi[competitive] industry:

1. Firms' products are **perfect substitutes**

2. Firms are .hi-purple["price-takers"], no one firm can affect the *market price*

3. Market **entry and exit are free**.hi[†]
]

---

# Visualizing Total Profit As `$R(q)-C(q)$`

- `$\pi(q)=R(q)-C(q)$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Visualizing Total Profit As `$R(q)-C(q)$`

- `$\pi(q)=R(q)-C(q)$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" style="display: block; margin: auto;" />
]

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# Visualizing Total Profit As `$R(q)-C(q)$`

- `$\pi(q)=R(q)-C(q)$`

- Graph: find `$q^*$` to max `$\pi \implies q^*$`, max distance between `$R(q)$` and `$C(q)$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Visualizing Total Profit As `$R(q)-C(q)$`

- `$\pi(q)=R(q)-C(q)$`

- Graph: find `$q^*$` to max `$\pi \implies q^*$`, max distance between `$R(q)$` and `$C(q)$`

- Slopes must be equal:
`$$MR(q)=MC(q)$$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Visualizing Total Profit As `$R(q)-C(q)$`

- `$\pi(q)=R(q)-C(q)$`

- Graph: find `$q^*$` to max `$\pi \implies q^*$`, max distance between `$R(q)$` and `$C(q)$`

- Slopes must be equal:
`$$MR(q)=MC(q)$$`

- At `$q^*=5$`:
  - `$R(q)=50$`
  - `$C(q)=40$`
  - `$\pi(q)=10$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" style="display: block; margin: auto;" />
]

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# Visualizing Profit Per Unit As `$MR(q)$` and `$MC(q)$`

- At low output `$q<q^*$`, can increase `$\pi$` by producing *more*: `$MR(q)>MC(q)$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" style="display: block; margin: auto;" />
]

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# Visualizing Profit Per Unit As `$MR(q)$` and `$MC(q)$`

- At high output `$q>q^*$`, can increase `$\pi$` by producing *less*: `$MR(q)<MC(q)$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-7-1.png" width="504" style="display: block; margin: auto;" />
]

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# Visualizing Profit Per Unit As `$MR(q)$` and `$MC(q)$`

- `$\pi$` is *maximized* where `$MR(q)=MC(q)$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" style="display: block; margin: auto;" />
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# Comparative Statics

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# If Market Price Changes I

- Suppose the market price *increases*

- Firm--always setting `$MR=MC$`--will respond by *producing more*

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" style="display: block; margin: auto;" />
]

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# If Market Price Changes II

- Suppose the market price *decreases*

- Firm--always setting `$MR=MC$`--will respond by *producing less*

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" style="display: block; margin: auto;" />
]

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# If Market Price Changes III

- .hi-purple[The firm's marginal cost curve is its (inverse) supply curve].hi[†]
`$$Supply=MC(q)$$`
 - How it will supply the optimal amount of output in response to the market price

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" style="display: block; margin: auto;" />
]

# Calculating Profit

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# Calculating Average Profit as `$AR(q)-AC(q)$`

- Profit is 
`$$\pi(q)=R(q)-C(q)$$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" style="display: block; margin: auto;" />
]

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# Calculating Average Profit as `$AR(q)-AC(q)$`

- Profit is 
`$$\pi(q)=R(q)-C(q)$$`

- Profit per unit can be calculated as:
`$$\begin{align*}
\frac{\pi(q)}{q}&=AR(q)-AC(q)\\
&=p-AC(q)\\
\end{align*}$$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" style="display: block; margin: auto;" />
]

---
# Calculating Average Profit as `$AR(q)-AC(q)$`

- Profit is 
`$$\pi(q)=R(q)-C(q)$$`

- Profit per unit can be calculated as:
`$$\begin{align*}
\frac{\pi(q)}{q}&=AR(q)-AC(q)\\
&=p-AC(q)\\
\end{align*}$$`

- Multiply by `$q$` to get total profit:
`$$\pi(q)=q\left[p-AC(q) \right]$$`

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-14-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Calculating Average Profit as `$AR(q)-AC(q)$`

- At market price of `$p^*=$` $10

- At `$q^*=5$` (per unit):

- At `$q^*=5$` (totals):

]

```
## geom_segment: arrow = NULL, arrow.fill = NULL, lineend = butt, linejoin = round, na.rm = FALSE
## stat_identity: na.rm = FALSE
## position_identity
```
]

---

# Calculating Average Profit as `$AR(q)-AC(q)$`

- At market price of `$p^*=$` $10

- At `$q^*=5$` (per unit):
  - `$AR(5)=$` $10/unit

- At `$q^*=5$` (totals):
  - `$R(5)=$` $50
]

<img src="1.3-slides_files/figure-html/unnamed-chunk-16-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Calculating Average Profit as `$AR(q)-AC(q)$`

- At market price of `$p^*=$` $10

- At `$q^*=5$` (per unit):
  - `$AR(5)=$` $10/unit
  - `$AC(5)=$` $7/unit

- At `$q^*=5$` (totals):
  - `$R(5)=$` $50
  - `$C(5)=$` $35
]

<img src="1.3-slides_files/figure-html/unnamed-chunk-17-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Calculating Average Profit as `$AR(q)-AC(q)$`

- At market price of `$p^*=$` $10

- At `$q^*=5$` (per unit):
  - `$AR(5)=$` $10/unit
  - `$AC(5)=$` $7/unit
  - `$A\pi(5)=$` $3/unit
  
- At `$q^*=5$` (totals):
  - `$R(5)=$` $50
  - `$C(5)=$` $35
  - `$\pi(5)=$` $15
]

<img src="1.3-slides_files/figure-html/unnamed-chunk-18-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Calculating Average Profit as `$AR(q)-AC(q)$`

- At market price of `$p^*=$` $2

- At `$q^*=1$` (per unit):

- At `$q^*=1$` (totals):

]

<img src="1.3-slides_files/figure-html/unnamed-chunk-19-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Calculating Average Profit as `$AR(q)-AC(q)$`

- At market price of `$p^*=$` $2

- At `$q^*=1$` (per unit):
  - `$AR(1)=$` $2/unit

- At `$q^*=5$` (totals):
  - `$R(1)=$` $2
]

<img src="1.3-slides_files/figure-html/unnamed-chunk-20-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Calculating Average Profit as `$AR(q)-AC(q)$`

- At market price of `$p^*=$` $2

- At `$q^*=5$` (per unit):
  - `$AR(1)=$` $2/unit
  - `$AC(1)=$` $7/unit

- At `$q^*=5$` (totals):
  - `$R(1)=$` $2
  - `$C(5)=$` $35
]

<img src="1.3-slides_files/figure-html/unnamed-chunk-21-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Calculating Average Profit as `$AR(q)-AC(q)$`

- At market price of `$p^*=$` $2

- At `$q^*=5$` (per unit):
  - `$AR(1)=$` $2/unit
  - `$AC(1)=$` $10/unit
  - `$A\pi(1)=$` -$8/unit
  
- At `$q^*=5$` (totals):
  - `$R(1)=$` $2
  - `$C(1)=$` $10
  - `$\pi(1)=$` -$8
]

<img src="1.3-slides_files/figure-html/unnamed-chunk-22-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Short-Run Shut-Down Decisions

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# Short-Run Shut-Down Decisions

- What if a firm's profits at `$q^*$` are *negative* (i.e. it earns **losses**)?

- Should it produce at all?

]

.pull-right[
.center[
![:scale 90%](https://www.dropbox.com/s/9ljdldig9i71e2r/emptywarehouse.jpg?raw=1)
]
]

---

# Short-Run Shut-Down Decisions

- Suppose firm chooses to produce **nothing** `$(q=0)$`:

- If it has fixed costs, its profits are:

`$$\begin{align*}
\pi(q)&=pq-C(q)\\
\end{align*}$$`

]

.pull-right[
.center[
![:scale 90%](https://www.dropbox.com/s/9ljdldig9i71e2r/emptywarehouse.jpg?raw=1)
]
]

---

# Short-Run Shut-Down Decisions

- Suppose firm chooses to produce **nothing** `$(q=0)$`:

- If it has fixed costs, its profits are:

`$$\begin{align*}
\pi(q)&=pq-C(q)\\
\pi(q)&=pq-f-VC(q)\\
\end{align*}$$`

]

.pull-right[
.center[
![:scale 90%](https://www.dropbox.com/s/9ljdldig9i71e2r/emptywarehouse.jpg?raw=1)
]
]

---

# Short-Run Shut-Down Decisions

- Suppose firm chooses to produce **nothing** `$(q=0)$`:

- If it has fixed costs, its profits are:

`$$\begin{align*}
\pi(q)&=pq-C(q)\\
\pi(q)&=pq-f-VC(q)\\
\pi(0)&=-f\\
\end{align*}$$`

]

.pull-right[
.center[
![:scale 90%](https://www.dropbox.com/s/9ljdldig9i71e2r/emptywarehouse.jpg?raw=1)
]
]

---

# Short-Run Shut-Down Decisions

- A firm should choose to produce nothing `$(q=0)$` only when:

`$$\begin{align*}
\pi \text{ from producing} &< \pi \text{ from not producing}\\
\end{align*}$$`

---

# Short-Run Shut-Down Decisions

- A firm should choose to produce nothing `$(q=0)$` only when:

`$$\begin{align*}
\pi \text{ from producing} &< \pi \text{ from not producing}\\
\pi(q) &< -f \\
\end{align*}$$`

---

# Short-Run Shut-Down Decisions

- A firm should choose to produce nothing `$(q=0)$` only when:

`$$\begin{align*}
\pi \text{ from producing} &< \pi \text{ from not producing}\\
\pi(q) &< -f \\
pq-VC(q)-f &<-f\\
\end{align*}$$`

---

# Short-Run Shut-Down Decisions

- A firm should choose to produce nothing `$(q=0)$` only when:

`$$\begin{align*}
\pi \text{ from producing} &< \pi \text{ from not producing}\\
\pi(q) &< -f \\
pq-VC(q)-f &<-f\\
pq-VC(q) &< 0\\
\end{align*}$$`

---

# Short-Run Shut-Down Decisions

- A firm should choose to produce nothing `$(q=0)$` only when:

`$$\begin{align*}
\pi \text{ from producing} &< \pi \text{ from not producing}\\
\pi(q) &< -f \\
pq-VC(q)-f &<-f\\
pq-VC(q) &< 0\\
pq &< VC(q)\\
\end{align*}$$`

---

# Short-Run Shut-Down Decisions

- A firm should choose to produce nothing `$(q=0)$` only when:

`$$\begin{align*}
\pi \text{ from producing} &< \pi \text{ from not producing}\\
\pi(q) &< -f \\
pq-VC(q)-f &<-f\\
pq-VC(q) &< 0\\
pq &< VC(q)\\
\mathbf{p} &< \mathbf{AVC(q)}\\
\end{align*}$$`

---

# Short-Run Shut-Down Decisions

.pull-left[
- .hi[Shut down price]: firm will shut down production *in the short run* when `$p<AVC(q)$`
]

---

# The Firm's Short Run Supply Decision

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# The Firm's Short Run Supply Decision

.pull-left[
<img src="1.3-slides_files/figure-html/unnamed-chunk-23-1.png" width="504" style="display: block; margin: auto;" />
]

---

# The Firm's Short Run Supply Decision

.pull-left[
<img src="1.3-slides_files/figure-html/unnamed-chunk-24-1.png" width="504" style="display: block; margin: auto;" />
]

---

# The Firm's Short Run Supply Decision

.pull-left[
<img src="1.3-slides_files/figure-html/unnamed-chunk-25-1.png" width="504" style="display: block; margin: auto;" />
]

---

# The Firm's Short Run Supply Decision

.pull-left[
<img src="1.3-slides_files/figure-html/unnamed-chunk-26-1.png" width="504" style="display: block; margin: auto;" />
]

---

# The Firm's Short Run Supply Decision

.pull-left[
<img src="1.3-slides_files/figure-html/unnamed-chunk-27-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[
.center[
Firm's short run (inverse) supply:
]
`$$\begin{cases}
p=MC(q) & \text{if } p \geq AVC \\
q=0 & \text{If } p < AVC\\ \end{cases}$$`
]

---

# The Firm's Short Run Supply Decision

.pull-left[
<img src="1.3-slides_files/figure-html/unnamed-chunk-28-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[
.center[
Firm's short run (inverse) supply:
]
`$$\begin{cases}
p=MC(q) & \text{if } p \geq AVC \\
q=0 & \text{If } p < AVC\\ \end{cases}$$`
]

---

# Summary:

**1. Choose `$q^*$` such that `$MR(q)=MC(q)$`**

**2. Profit `$\pi=q[p-AC(q)]$`**

**3. Shut down if `$p<AVC(q)$`**

`$$\begin{cases}
p=MC(q) & \text{if } p \geq AVC\\
q=0 & \text{If } p < AVC\\ \end{cases}$$`

---

# Choosing the Profit-Maximizing Output `$q^*$`: Example I

.green[**Example**]: Bob's barbershop gives haircuts in a very competitive market, where barbers cannot differentiate their haircuts. The current market price of a haircut is $15. Bob's daily short run costs are given by:

`$$\begin{align*}
C(q) &= 0.5q^2\\
MC(q) &=q\\
\end{align*}$$`

1. How many haircuts per day would maximize Bob's profits?

2. How much profit will Bob earn per day?

3. Find Bob's shut down price.

---

# Choosing the Profit-Maximizing Output `$q^*$`: Example II

.green[**Example**]:
A firm has short-run costs given by:
`$$\begin{aligned}
C(q)&=q^2+1\\
MC(q)&=2q\\ \end{aligned}$$`

.smallest[
1. Write equations for fixed costs, variable costs, average fixed costs, average variable costs, and for average (total) costs.

2. Suppose the firm is in a competitive market, and the current market price is $4, how many units of output maximize profits?

3. How much profit will this firm earn?

4. At what market price would the firm break even `$(\pi=0)$`?

5. Below what market price would the firm shut down in the short run if it were earning losses?
]