2.5 — Short Run Profit Maximization

# 2.5 — Short Run Profit Maximization
## ECON 306 · Microeconomic Analysis · Fall 2020
### Ryan Safner Assistant Professor of Economics <a href="mailto:safner@hood.edu">safner@hood.edu</a> <a href="https://github.com/ryansafner/microF20">ryansafner/microF20</a> <a href="https://microF20.classes.ryansafner.com"> microF20.classes.ryansafner.com</a>

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# Revenues

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# Revenues for Firms in *Competitive* Industries I

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# Revenues for Firms in *Competitive* Industries I

- .blue[Demand for a firm's product] is **perfectly elastic** at the market price

- Where did the supply curve come from? You'll see

---

# Revenues for Firms in *Competitive* Industries II

- .hi[Total Revenue] `$R(q)=pq$`
]

---

# Average and Marginal Revenues

- .hi[Average Revenue]: revenue per unit of output
`$$AR(q)=\frac{R}{q}$$`
  - Is *always* equal to the price! Why?

- .hi[Marginal Revenue]: change in revenues for each additional unit of output sold:
`$$MR(q) = \frac{\Delta R(q)}{\Delta q} \approx \frac{R_2-R_1}{q_2-q_1}$$`
  - Calculus: first derivative of the revenues function
  - For a *competitive* firm, always equal to the price!

---

# Average and Marginal Revenues: Example

.content-box-green[
.green[**Example**]: A firm sells bushels of wheat in a very competitive market. The current market price is $10/bushel.
]

For the 1st bushel sold:

- What is the total revenue?

- What is the average revenue?

]

- What is the total revenue?

- What is the average revenue?

- What is the marginal revenue?
]

---

# Total Revenue, Example: Visualized

| `$q$` | `$R(q)$` |
|----:|-------:|
| `$0$` | `$0$` |
| `$1$` | `$10$` |
| `$2$` | `$20$` |
| `$3$` | `$30$` |
| `$4$` | `$40$` |
| `$5$` | `$50$` |
| `$6$` | `$60$` |
| `$7$` | `$70$` |
| `$8$` | `$80$` |
| `$9$` | `$90$` |
| `$10$` | `$100$` |

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" />
]

---

# Average and Marginal Revenue, Example: Visualized

| `$q$` | `$R(q)$` | `$AR(q)$` | `$MR(q)$` |
|----:|-------:|--------:|--------:|
| `$0$` | `$0$` | `$-$` | `$-$` |
| `$1$` | `$10$` | `$10$` | `$10$` |
| `$2$` | `$20$` | `$10$` | `$10$` |
| `$3$` | `$30$` | `$10$` | `$10$` |
| `$4$` | `$40$` | `$10$` | `$10$` |
| `$5$` | `$50$` | `$10$` | `$10$` |
| `$6$` | `$60$` | `$10$` | `$10$` |
| `$7$` | `$70$` | `$10$` | `$10$` |
| `$8$` | `$80$` | `$10$` | `$10$` |
| `$9$` | `$90$` | `$10$` | `$10$` |
| `$10$` | `$100$` | `$10$` | `$10$` |

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-7-1.png" width="504" />
]

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# Recall: The Firm's Two Problems

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

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

- We'll cover this later...first we'll explore:

- 2nd Stage: .hi-purple[firm's cost minimization problem]:

1. **Choose:** .hi-blue[ < inputs >]

2. **In order to _minimize_:** .hi-green[< cost >]

3. **Subject to:** .hi-red[< producing the optimal output >]

- Minimizing costs `$\iff$` maximizing profits
]

]

---

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

- `$\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}$`

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-8-1.png" width="504" />
]

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

- `$\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}$`

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" />
]

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

- `$\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}$`

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

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" />
]

---

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

- `$\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}$`

- 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="2.5-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" />
]

---

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

- `$\color{green}{\pi(q)}=\color{blue}{R(q)}-\color{red}{C(q)}$`

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

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

.smallest[
- At `$q^*=5$`:
  - `$\color{blue}{R(q)=50}$`
  - `$\color{red}{C(q)=40}$`
  - `$\color{green}{\pi(q)=10}$`
]
]

<img src="2.5-slides_files/figure-html/unnamed-chunk-12-1.png" width="504" />
]

---

# 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="2.5-slides_files/figure-html/unnamed-chunk-13-1.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="2.5-slides_files/figure-html/unnamed-chunk-14-1.png" width="504" />
]

---

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

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

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-15-1.png" width="504" />
]

---

# 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="2.5-slides_files/figure-html/unnamed-chunk-16-1.png" width="504" />
]

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# If Market Price Changes II

- Suppose the market price *decreases*

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

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-17-1.png" width="504" />
]

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# If Market Price Changes II

- .hi-purple[The firm's marginal cost curve is its (inverse) supply curve].magenta[†]
`$$Supply=MC(q)$$`
 - 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="2.5-slides_files/figure-html/unnamed-chunk-18-1.png" width="504" />
]

---

# Calculating Profit

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

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

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-19-1.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:
`$$\begin{align*}
\frac{\pi(q)}{q}&=\color{blue}{AR(q)}-\color{orange}{AC(q)}\\
&=\color{blue}{p}-\color{orange}{AC(q)}\\
\end{align*}$$`

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-20-1.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:
`$$\begin{align*}
\frac{\pi(q)}{q}&=\color{blue}{AR(q)}-\color{orange}{AC(q)}\\
&=\color{blue}{p}-\color{orange}{AC(q)}\\
\end{align*}$$`

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

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-21-1.png" width="504" />
]

---

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

- At market price of .blue[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 .blue[p* = $10]

- At q* = 5 (per unit):
  - .blue[AR(5) = $10/unit]

- At q* = 5 (totals):
  - .blue[R(5) = $50]
]

<img src="2.5-slides_files/figure-html/unnamed-chunk-23-1.png" width="504" />
]

---

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

- At market price of .blue[p* = $10]

- At q* = 5 (per unit):
  - .blue[AR(5) = $10/unit]
  - .orange[AC(5) = $7/unit]

- At q* = 5 (totals):
  - .blue[R(5) = $50]
  - .red[C(5) = $35]
]

<img src="2.5-slides_files/figure-html/unnamed-chunk-24-1.png" width="504" />
]

---

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

- At market price of .blue[p* = $10]

- At q* = 5 (per unit):
  - .blue[AR(5) = $10/unit]
  - .orange[AC(5) = $7/unit]
  - .green[A`\$\pi\$`(5) = $3/unit]

- At q* = 5 (totals):
  - .blue[R(5) = $50]
  - .red[C(5) = $35]
  - .green[`\$\pi\$` = $15]
]

<img src="2.5-slides_files/figure-html/unnamed-chunk-25-1.png" width="504" />
]

---

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

- At market price of .blue[p* = $2]

- At q* = 1 (per unit):

- At q* = 1 (totals):

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-26-1.png" width="504" />
]

---

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

- At market price of .blue[p* = $2]

- At q* = 1 (per unit):
  - .blue[AR(1) = $2/unit]

- At q* = 1 (totals):
  - .blue[R(1) = $2]
]

<img src="2.5-slides_files/figure-html/unnamed-chunk-27-1.png" width="504" />
]

---

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

- At market price of .blue[p* = $2]

- At q* = 1 (per unit):
  - .blue[AR(1) = $2/unit]
  - .orange[AC(1) = $10/unit]

- At q* = 1 (totals):
  - .blue[R(1) = $2]
  - .red[C(1) = $10]

]

<img src="2.5-slides_files/figure-html/unnamed-chunk-28-1.png" width="504" />
]

---

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

- At market price of .blue[p* = $2]

- At q* = 1 (per unit):
  - .blue[AR(1) = $2/unit]
  - .orange[AC(1) = $10/unit]
  - .green[A`\$\pi\$`(1) = -$8/unit]

- At q* = 1 (totals):
  - .blue[R(1) = $2]
  - .red[C(1) = $10]
  - .green[`\$\pi\$`(1) = -$8]
]

<img src="2.5-slides_files/figure-html/unnamed-chunk-29-1.png" width="504" />
]

---

# 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** `$(f>0)$`, 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** `$(f>0)$`, 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** `$(f>0)$`, 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

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

---

# 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

---

# 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.