2.5: Short Run Profit Maximization

# 2.5: Short Run Profit Maximization
## ECON 306 · Microeconomic Analysis · Fall 2019
### Ryan Safner Assistant Professor of Economics <a href="mailto:safner@hood.edu"> safner@hood.edu</a> <a href="https://github.com/ryansafner/microf19"> ryansafner/microf19</a> <a href="https://microF19.classes.ryansafner.com"> microF19.classes.ryansafner.com</a>

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# Recall: The Firm's Two Problems

- First Stage: the **firm's profit maximization problem**:

1. **Choose:** .blue[ < output >]

2. **In order to maximize:** .green[< profits >]

- All that remains: how to find `$q^*$`
]

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

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

]

<img src="14-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" />
]

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

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

]

<img src="14-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" />
]

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

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

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

]

<img src="14-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" />
]

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

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

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

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

]

<img src="14-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" />
]

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

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

- Graph: find `$q^*$` to max `$\pi \implies q^*$` where 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="14-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" />
]

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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="14-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" />
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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="14-slides_files/figure-html/unnamed-chunk-7-1.png" width="504" />
]

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

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

]

<img src="14-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" />
]

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# Comparative Statics

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# If Market Price Changes I

- Suppose the market price *increases*

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

]

<img src="14-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" />
]

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# If Market Price Changes II

- Suppose the market price *decreases*

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

]

<img src="14-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" />
]

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# If Market Price Changes II

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

- There is an exception to this! We will see shortly!

]

<img src="14-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" />
]

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# Calculating Profit

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

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

]

<img src="14-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" />
]

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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="14-slides_files/figure-html/unnamed-chunk-13-1.png" width="504" />
]

---
# 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="14-slides_files/figure-html/unnamed-chunk-14-1.png" width="504" />
]

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# 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="14-slides_files/figure-html/unnamed-chunk-16-1.png" width="504" />
]

---

# 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="14-slides_files/figure-html/unnamed-chunk-17-1.png" width="504" />
]

---

# 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="14-slides_files/figure-html/unnamed-chunk-18-1.png" width="504" />
]

---

# 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="14-slides_files/figure-html/unnamed-chunk-19-1.png" width="504" />
]

---

# 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="14-slides_files/figure-html/unnamed-chunk-20-1.png" width="504" />
]

---

# 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="14-slides_files/figure-html/unnamed-chunk-21-1.png" width="504" />
]

---

# 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="14-slides_files/figure-html/unnamed-chunk-22-1.png" width="504" />
]

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# 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:

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

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

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# The Firm's Short Run Supply Decision

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# The Firm's Short Run Supply Decision

---

# The Firm's Short Run Supply Decision

---

# The Firm's Short Run Supply Decision

---

# The Firm's Short Run Supply Decision

---

# The Firm's Short Run Supply Decision

.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-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

.content-box-green[
.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.

4. Write Bob's short-run inverse supply function.