1.2 — Technology and Cost

# 1.2 — Technology and Cost

## ECON 326 • Industrial Organization • Spring 2023

### Ryan Safner Associate Professor of Economics <a href="mailto:safner@hood.edu">safner@hood.edu</a> <a href="https://github.com/ryansafner/ioS23">ryansafner/ioS23</a> <a href="https://ioS23.classes.ryansafner.com"> ioS23.classes.ryansafner.com</a> 
---

# Outline

### [Short Run Production Concepts](#27)
### [Costs in the Short Run](#37)
### [Costs in the Long Run](#65)
### [Revenues](#75)

---

# This Black Box We Call "Firms"

- .hi[Firm] is a mere .hi-purple[production process]:
  - a bundle of technology, physical assets, and individuals

- Synonymous with .hi[production function]

- Fully replicable

- We'll explore (and explode) this much later
]

---

# What Do Firms Do? I

- We'll assume “the firm” is the agent to model:

- So what do firms do?

- How would we set up an optimization model:

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

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

3. **Subject to:** .hi-red[< some constraints >]

]

---

# What Do Firms Do? II

---

# What Do Firms Do? II

---
# What Do Firms Do? II

- **Inputs**: `$x_1, x_2, \cdots, x_n$`
 - **Examples**: worker efforts, warehouse space, electricity, loans, oil, cardboard, fertilizer, computers, software programs, etc
 
- **Output**: `$q$`
 - **Examples**: gas, cars, legal services, mobile apps, vegetables, consulting advice, financial reports, etc
]
]

---

# What Do Firms Do? III

.pull-left[
- .hi[Technology] or a .hi[production function]: rate at which firm can convert specified **inputs** `$(x_1, x_2, \cdots, x_n)$` into **output** `$(q)$`
`$$q=f(x_1, x_2, \cdots, x_n)$$`

]

---

# Production Function as Recipe

![](../images/recipe1.PNG)
]
]

![](../images/recipe2.PNG)
]

]

---

# Factors of Production I

`$$q=A \,f(t,l,k)$$`

.pull-left[
.smaller[
- Economists typically classify inputs, called the .hi[“factors of production” (FOP)]:

<table>
 <thead>
 <tr>
 <th style="text-align:left;"> Factor </th>
 <th style="text-align:left;"> Owned By </th>
 <th style="text-align:left;"> Earns </th>
 </tr>
 </thead>
<tbody>
 <tr>
 <td style="text-align:left;"> Land (t) </td>
 <td style="text-align:left;"> Landowners </td>
 <td style="text-align:left;"> Rent </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Labor (l) </td>
 <td style="text-align:left;"> Laborers </td>
 <td style="text-align:left;"> Wages </td>
 </tr>
 <tr>
 <td style="text-align:left;"> Capital (k) </td>
 <td style="text-align:left;"> Capitalists </td>
 <td style="text-align:left;"> Interest </td>
 </tr>
</tbody>
</table>
]
.smallest[
- `$A$`: .b["total factor productivity"] (ideas/knowledge/institutions)

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

]

---

# Factors of Production II

`$$q=f(l,k)$$`

- We will assume just two inputs: labor `$l$` and capital `$k$`

]

]

---

# What Does a Firm Maximize?

- We assume firms .hi-purple[maximize profit `\$(\pi)\$`]

- Not true for all firms
 - **Examples**: non-profits, charities, civic associations, government agencies, criminal organizations, etc

- Even profit-seeking firms may also want to maximize *additional* things
 - **Examples**: goodwill, sustainability, social responsibility, etc

]

---

# Profits Have a Bad Rap These Days

---

# What is Profit?

- In economics, .hi-purple[profit] is simply **benefits minus (opportunity) costs**

]

---

# What is Profit?

- In economics, .hi-purple[profit] is simply **benefits minus (opportunity) costs**

- Suppose firm sells **output** `$q$` at price `$p$`

]

---

# What is Profit?

- In economics, .hi-purple[profit] is simply **benefits minus (opportunity) costs**

- Suppose firm sells **output** `$q$` at price `$p$`

- It can buy each **input** `$x_i$` at an associated price `$p_i$`, i.e.
    - labor `$l$` at wage rate `$w$`
    - capital `$k$` at rental rate `$r$`

]

---

# What is Profit?

- In economics, .hi-purple[profit] is simply **benefits minus (opportunity) costs**

- Suppose firm sells **output** `$q$` at price `$p$`

- It can buy each **input** `$x_i$` at an associated price `$p_i$`, i.e.
    - labor `$l$` at wage rate `$w$`
    - capital `$k$` at rental rate `$r$`

- The profit of selling `$q$` units and using inputs `$l,k$` is:

]

---

# Who Gets the Profits? I

]

---

# Reminder from Macroeconomics: “The Circular Flow”

---

# Who Gets the Profits? I

- .hi-purple[The firm's costs are all of the factor-owner's incomes!]
    - Landowners, laborers, creditors are all paid rent, wages, and interest, respectively

]

---

# Who Gets the Profits? I

- Profits are the .hi-purple[residual value] leftover after paying all factors

- Profits are income for the .hi[residual claimant(s)] of the production process (i.e. **owner(s)** of a firm):
    - Entrepreneurs
    - Shareholders

]

---

# Who Gets the Profits? II

- Residual claimants have incentives to maximize firm's profits, as this *maximizes their own income*

- Entrepreneurs and shareholders are the only participants in production that are *not* guaranteed an income!
    - Starting and owning a firm is inherently **risky**!

]

# People Overestimate Profits

.source[Source: [American Enterprise Institute](https://www.aei.org/carpe-diem/the-public-thinks-the-average-company-makes-a-36-profit-margin-which-is-about-5x-too-high-part-ii/)]

---

# Profits and Entrepreneurship: A Preview

- In markets, production must face the .hi[profit test]:
 - Is consumer's willingness to pay `$>$` opportunity cost of inputs?

- Profits are an indication that **value is being created for society**

- Losses are an indication that **value is being destroyed for society**

- Survival in markets *requires* firms continually create value & earn profits
]

---

# The Firm's Optimization Problem I

- So what do firms do?

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

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

3. **Subject to:** .hi-red[< technology >]

- We've so far assumed they maximize profits and they are limited by their technology

]

---

# The Firm's Optimization Problem II

- Prices?
 - Depends on the market the firm is operating in!
 - Study of industrial organization
 
- Essential question: .hi-turquoise[how competitive is a market?] This will influence what firms (can) do
]

---

# Short-Run Production Concepts

---

# Marginal Products

.pull-left[
- The .hi[marginal product] of an input is the *additional* output produced by *one more unit* of that input (*holding all other inputs constant*)

- Like marginal utility

- Similar to marginal utilities, I will give you the marginal product equations

]

.pull-right[
<img src="1.2-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Marginal Product of Labor

- .hi[Marginal product of labor `\$(MP_l)\$`]: additional output produced by adding one more unit of labor (holding `$k$` constant)
`$$MP_l = \frac{\Delta q}{\Delta l}$$`

- `$MP_l$` is slope of `$TP$` at each value of `$l$`! 
  - Note: via calculus: `$\frac{\partial q}{\partial l}$`
]

]

---

# Marginal Product of Capital

- .hi[Marginal product of capital `\$(MP_k)\$`]: additional output produced by adding one more unit of capital (holding `$l$` constant)
`$$MP_k = \frac{\Delta q}{\Delta k}$$`

- `$MP_k$` is slope of `$TP$` at each value of `$k$`! 
  - Note: via calculus: `$\frac{\partial q}{\partial k}$`

- Note we don't consider capital in the short run!

]

]

---

# Diminishing Returns

- .hi-purple[Law of Diminishing Returns]: adding more of one factor of production **holding all others constant** will result in successively lower increases in output

- In order to increase output, firm will need to increase *all* factors!

]

<img src="1.2-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Diminishing Returns

- .hi-purple[Law of Diminishing Returns]: adding more of one factor of production **holding all others constant** will result in successively lower increases in output

- In order to increase output, firm will need to increase *all* factors!

]

<img src="1.2-slides_files/figure-html/unnamed-chunk-11-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Average Product of Labor (and Capital)

- .hi[Average product of labor `\$(AP_l)\$`]: total output per worker
`$$AP_l = \frac{q}{l}$$`

- A measure of *labor productivity*

- .hi[Average product of capital `\$(AP_k)\$`]: total output per unit of capital
`$$AP_k = \frac{q}{k}$$`
]

]

---

# The Firm's Problem: Long Run

---

# The Long Run

- In the long run, *all* factors of production are .hi-purple[variable]

`$$q=f(k,l)$$`

- Can build more factories, open more storefronts, rent more space, invest in machines, etc.

- So the firm can choose both `$l$` *and* `$k$`

]

---

# Production Costs are Opportunity Costs

- Remember, .hi[economic costs] are broader than the common conception of “cost”
  - .hi-purple[Accounting cost]: monetary cost
  - .hi[Economic cost]: value of next best alternative use of resources given up (i.e. .hi[opportunity cost])

]

![:scale 70%](../images/disappear.jpg)
]
]

---

# Production Costs are Opportunity Costs

- This leads to the difference between:
  - .hi-purple[Accounting profit]: revenues minus accounting costs
  - .hi[Economic profit]: revenues minus accounting + .ul[opportunity] costs

- A really difficult concept to think about!
]

![:scale 70%](../images/disappear.jpg)
]
]

---

# Production Costs are Opportunity Costs

- Another helpful perspective:
  - .hi-purple[Accounting cost]: what you **historically** paid for a resource
  - .hi[Economic cost]: what you can **currently** get in the market for a selling a resource (it’s value in *alternative* uses)

]

![:scale 70%](../images/disappear.jpg)
]
]

---

# A Reminder: It's Demand all the Way Down!

- .hi-red[Supply] is actually .hi-blue[Demand] in disguise!

- An .hi[(opportunity) cost] to buy (scarce) inputs for production because **other people** .hi-blue[demand] those same inputs to consume or produce **other valuable things**!
  - Price necessary to **pull them out of other valuable productive uses** in the economy!
]

---

# Production Costs are Opportunity Costs

- Because resources are scarce, and have rivalrous uses, .hi-turquoise[how do we know we are using resources efficiently??]

- In functioning markets, .hi-purple[the market price measures the opportunity cost of using a resource for an alternative use]

- Firms not only pay for direct use of a resource, but also indirectly compensate society for *“pulling the resource out”* of alternate uses in the economy!

]

![:scale 70%](../images/disappear.jpg)
]
]

---

# Production Costs are Opportunity Costs

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Examples**]:
.smallest[
- If you start a business, you may give up your salary at your current job
- If you invest in a factory, you give up other investment opportunities
- If you use an office building you own, you cannot rent it to other people
- If you hire a skilled worker, you must pay them a high enough salary to deter them from working for other firms
]
]
]

---

# Opportunity Costs vs. Sunk Costs

- Opportunity cost is a *forward-looking* concept

- Choices made in the *past* with *non-recoverable* costs are called .hi[sunk costs]

- Sunk costs *should not* enter into future decisions

- Many people have difficulty letting go of unchangeable past decisions: .hi-purple[sunk cost fallacy]

]

---

# Common Sunk Costs in Business

- Licensing fees, long-term lease contracts

- Specific capital (with no alternative use): uniforms, menus, signs

- Research & Development spending

- Advertising spending

]

# The Accounting vs. Economic Point of View I

- Helpful to consider two points of view:

1. .hi-purple[“Accounting point of view”]: are you taking in more cash than you are spending?

2. .hi[“Economic point of view”]: is your product you making the *best social* use of your resources
  - i.e. are there higher-valued uses of your resources you are keeping them out of?

]

![:scale 70%](../images/disappear.jpg)
]
]

---
# The Accounting vs. Economic Point of View II

.pull-left[
.smaller[
- .hi-turquoise[Implications for society]: are consumers *best* off with you using scarce resources (with alternative uses!) to produce your current product?

- Remember: .hi-turquoise[this is an .ul[*economics*] course, not a *business* course]!
  - .hi-purple[Economists are pro-market, *not* pro-business!]
  - What might be good/bad for **one** business might have bad/good *consequences* for society!
]
]

![:scale 70%](../images/disappear.jpg)
]
]

---

---

# Costs in the Short Run
 
- .hi[Total cost function, `\$C(q)\$`] relates output `$q$` to the total cost of production `$C$`.magenta[†]

`$$C(q)=f+VC(q)$$`

- Two kinds of short run costs:

**1.** .hi[Fixed costs, `\$f\$`] are costs that do not vary with output
  - Only true in the short run! (Consider this the cost of maintaining your capital)

**2.** .hi[Variable costs, `\$VC(q)\$`] are costs that vary with output (notice the variable in them!)
  - Typically, the more production of `$q$`, the higher the cost
  - e.g. firm is hiring *additional* labor

.source[.magenta[†] Assuming that (i) firms are always choosing input combinations that minimize total cost and (ii) input prices are constant. See more in [today’s appendix](/resources/appendices/2.4-appendix.html).]
---

# Fixed vs. Variable costs: Examples

**Fixed costs**: the aircraft, regulatory approval

**Variable costs**: providing one more flight
]
]
---

# Fixed vs. Variable costs: Examples

.pull-right[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[.hi-green[Example]: Car Factory

**Fixed costs**: the factory, machines in the factory

**Variable costs**: producing one more car
]
]

---

# Fixed vs. Variable costs: Examples

.pull-right[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[.hi-green[Example]: Starbucks

**Fixed costs**: the retail space, espresso machines

**Variable costs**: selling one more cup of coffee
]
]

---

# Fixed vs. Sunk costs

- .hi-purple[Sunk costs] are a *type* of .hi[fixed cost] that are *not* avoidable or recoverable

- Many .hi[fixed costs] can be avoided or changed in the long run

- Common .hi[fixed], but *not* .hi-purple[sunk], costs:
  - rent for office space, durable equipment, operating permits (that are renewed)

- When deciding to *stay* in business, .hi[fixed costs] matter, .hi-purple[sunk costs] do not!
]
]

# Cost Functions: Example, Visualized

.tiny[
| `$q$` | `$f$` | `$VC(q)$` | `$C(q)$` |
|----:|----:|--------:|-------:|
| `$0$` | `$10$` | `$0$` | `$10$` |
| `$1$` | `$10$` | `$2$` | `$12$` |
| `$2$` | `$10$` | `$6$` | `$16$` |
| `$3$` | `$10$` | `$12$` | `$22$` |
| `$4$` | `$10$` | `$20$` | `$30$` |
| `$5$` | `$10$` | `$30$` | `$40$` |
| `$6$` | `$10$` | `$42$` | `$52$` |
| `$7$` | `$10$` | `$56$` | `$66$` |
| `$8$` | `$10$` | `$72$` | `$82$` |
| `$9$` | `$10$` | `$90$` | `$100$` |
| `$10$` | `$10$` | `$110$` | `$120$` |

]
]

<img src="1.2-slides_files/figure-html/unnamed-chunk-14-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Average Costs

- .hi[Average Fixed Cost]: fixed cost per unit of output:

`$$AFC(q)=\frac{f}{q}$$`

- .hi[Average Variable Cost]: variable cost per unit of output:

`$$AVC(q)=\frac{VC(q)}{q}$$`

- .hi[Average (Total) Cost]: (total) cost per unit of output:

`$$AC(q)=\frac{C(q)}{q}$$`

---

# Marginal Cost

- .hi[Marginal Cost] is the change in total cost for each additional unit of output produced:

`$$MC(q) = \frac{\Delta C(q)}{\Delta q}$$`

- Calculus: first derivative of the cost function

- .hi-purple[Marginal cost is the *primary* cost that matters in making decisions]
  - All other costs are driven by marginal costs
  - This is the main cost that firms can “see”

---

# Average and Marginal Costs: Visualized

.center[
<img src="1.2-slides_files/figure-html/unnamed-chunk-15-1.png" width="1008" style="display: block; margin: auto;" />
]

---

# Relationship Between Marginal and Average

- If .red[marginal] `$<$` .orange[average], then .orange[average] `$\downarrow$`

]
]

.pull-right[
<img src="1.2-slides_files/figure-html/unnamed-chunk-16-1.png" width="504" style="display: block; margin: auto;" />
]

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# Relationship Between Marginal and Average

- If .red[marginal] `$<$` .orange[average], then .orange[average] `$\downarrow$`

- If .red[marginal] `$>$` .orange[average], then .orange[average] `$\uparrow$`
]
]

.pull-right[
<img src="1.2-slides_files/figure-html/unnamed-chunk-17-1.png" width="504" style="display: block; margin: auto;" />
]

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# Relationship Between Marginal and Average

- If .red[marginal] `$<$` .orange[average], then .orange[average] `$\downarrow$`

- If .red[marginal] `$>$` .orange[average], then .orange[average] `$\uparrow$`

- When .red[marginal] `$=$` .orange[average], .orange[average] is **maximized/minimized**

]
]

.pull-right[
<img src="1.2-slides_files/figure-html/unnamed-chunk-18-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Relationship Between Marginal and Average

- If .red[marginal] `$<$` .orange[average], then .orange[average] `$\downarrow$`

- If .red[marginal] `$>$` .orange[average], then .orange[average] `$\uparrow$`

- When .red[marginal] `$=$` .orange[average], .orange[average] is **maximized/minimized**
 - .hi-purple[When MC(q)=AC(q), AC(q) is at a *minimum*] (break-even price)
 - .hi-purple[When MC(q)=AVC(q), AVC(q) is at a *minimum*] (shut-down price)
 
]
]
.pull-right[
<img src="1.2-slides_files/figure-html/unnamed-chunk-19-1.png" width="504" style="display: block; margin: auto;" />
]

---

---

# Costs in the Long Run

- .hi[Long run]: firm can change all factors of production & vary scale of production

- .hi-purple[Long run average cost, LRAC(q)]: cost per unit of output when the firm can change *both* `$l$` and `$k$` to make more `$q$`

- .hi-purple[Long run marginal cost, LRMC(q)]: change in long run total cost as the firm produce an additional unit of `$q$` (by changing *both* `$l$` and/or `$k$`)

]

---

# Average Cost in the Long Run

- .hi[Long run]: firm can choose `$k$` (factories, locations, etc)

- Separate short run average cost (SRAC) curves for each amount of `$k$` potentially chosen

]

.pull-right[
<img src="1.2-slides_files/figure-html/unnamed-chunk-20-1.png" width="504" style="display: block; margin: auto;" />
]

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# Average Cost in the Long Run

- .hi[Long run]: firm can choose `$k$` (factories, locations, etc)

- Separate short run average cost (SRAC) curves for each amount of `$k$` potentially chosen

- .hi-purple[Long run average cost (LRAC)] curve “envelopes” the lowest (optimal) regions of all the SRAC curves!

.smallest[
> “Subject to producing the optimal amount of output, choose l and k to minimize cost”
]
]

.pull-right[
<img src="1.2-slides_files/figure-html/unnamed-chunk-21-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Long Run Costs & Scale Economies I

.pull-right[
.quitesmall[
- Further important properties about costs based on .hi[scale economies] of production: change in **average costs** when output is increased (scaled)

- .hi-purple[Economies of scale]: average costs .hi-turquoise[fall] with more output
  - High fixed costs `$AFC > AVC(q)$` low variable costs

- .hi-purple[Diseconomies of scale]: average costs .hi-turquoise[rise] with more output
 - Low fixed costs `$AFC < AVC(q)$` high variable costs
 
- .hi-purple[Constant economies of scale]: average costs .hi-turquoise[don’t change] with more output
 - Firm at minimum average cost (optimal plant size), called .hi[minimum efficient scale (MES)]
]
]

---

# Long Run Costs & Scale Economies II

- .hi[Minimum Efficient Scale]: `$q$` with the lowest `$AC(q)$`
  - “optimal firm size”

]

<img src="1.2-slides_files/figure-html/unnamed-chunk-22-1.png" width="504" style="display: block; margin: auto;" />
]

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# Long Run Costs & Scale Economies II

- .hi[Minimum Efficient Scale]: `$q$` with the lowest `$AC(q)$`
  - “optimal firm size”

- .hi-green[Economies of Scale]: `$\uparrow q$`, `$\downarrow AC(q)$`

]

<img src="1.2-slides_files/figure-html/unnamed-chunk-23-1.png" width="504" style="display: block; margin: auto;" />
]

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# Long Run Costs & Scale Economies II

- .hi[Minimum Efficient Scale]: `$q$` with the lowest `$AC(q)$`
  - “optimal firm size”

- .hi-green[Economies of Scale]: `$\uparrow q$`, `$\downarrow AC(q)$`

- .hi-red[Diseconomies of Scale]: `$\uparrow q$`, `$\uparrow AC(q)$`

]

<img src="1.2-slides_files/figure-html/unnamed-chunk-24-1.png" width="504" style="display: block; margin: auto;" />
]
---

# Long Run Costs & Scale Economies III

- Can measure economies of scale `$(S)$`:

`$$S(q) = \frac{AC(q)}{MC(q)}$$`
 - `$S>1$`: economies of scale at `$q$`
 - `$S<1$`: diseconomies of scale at `$q$`
 - `$S=1$`: minimum efficient scale at `$q$`
]

<img src="1.2-slides_files/figure-html/unnamed-chunk-25-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Economies of Scope

- We often assume .hi[single-product plants/firms], but in reality most firms/plants are .hi[multi-product]

- .hi[Economies of Scope]: cost of producing multiple products (e.g. `$q_1$` and `$q_2$`) in a single plant exceeds costs of producing a single product in each plant

`$$C(q_1,q_2) < C(q_1,0)+C(0,q_2)$$`
]

---

# Revenues

---

# Revenues for Firms in *Competitive* Industries I

.pull-left[
<img src="1.2-slides_files/figure-html/unnamed-chunk-26-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[
<img src="1.2-slides_files/figure-html/industry-graph-1.png" width="504" style="display: block; margin: auto;" />
]

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# Revenues for Firms in *Competitive* Industries I

.pull-left[
<img src="1.2-slides_files/figure-html/unnamed-chunk-27-1.png" width="504" style="display: block; margin: auto;" />
]

.pull-right[
<img src="1.2-slides_files/figure-html/unnamed-chunk-28-1.png" width="504" style="display: block; margin: auto;" />
]

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

- Where did the .red[supply curve] come from? You’ll know today

---

# Revenues for Firms in *Competitive* Industries II

.pull-left[
<img src="1.2-slides_files/figure-html/unnamed-chunk-29-1.png" width="504" style="display: block; margin: auto;" />
]

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

---

# Average and Marginal Revenues

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

- .hi[Marginal Revenue]: change in revenues for each additional unit of output sold:
`$$\color{#e64173}{MR(q) = \frac{\Delta R(q)}{\Delta q}}$$`
  - Calculus: first derivative of the revenues function
  - .hi-purple[For a .ul[*competitive*] firm (only), MR(q) = p, i.e. the price!]

---

# Average and Marginal Revenues: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: A firm sells bushels of wheat in a very competitive market. The current market price is .blue[$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

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

]
]

<img src="1.2-slides_files/figure-html/unnamed-chunk-30-1.png" width="504" style="display: block; margin: auto;" />
]

---

# 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="1.2-slides_files/figure-html/unnamed-chunk-31-1.png" width="504" style="display: block; margin: auto;" />
]