2.4 — Costs of Production

# 2.4 — Costs of Production
## 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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# 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
]

]

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# A *Competitive* Market

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

1. Firms’ products are .hi-purple[perfect substitutes]

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

3. Market .hi-purple[entry and exit are free].magenta[†]
]

.footnote[.magenta[†] Remember this feature. It turns out to be the most important feature that distinguishes different types of industries!]

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# Profit

- Recall that profit is is:
`$$\pi=\underbrace{pq}_{revenues}-\underbrace{(wl+rk)}_{costs}$$`

- We’ll first take a closer look at costs today, then at revenues

- Next class we'll put them together to find `$q^*$` that maximizes `$\pi$` (the first stage problem)

]

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# Opportunity Costs in Production

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# Costs in Economics are Opportunity Costs

- Remember, .hi[economic costs] are different from 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%](https://www.dropbox.com/s/mb9yaadujqe38fj/disappear.jpg?raw=1)
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]

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# Costs in Economics are Opportunity Costs

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

- One of the most difficult concepts to think about!
]

![:scale 70%](https://www.dropbox.com/s/mb9yaadujqe38fj/disappear.jpg?raw=1)
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]

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# Costs in Economics 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
    - Resource's value in *alternative* uses

]

![:scale 70%](https://www.dropbox.com/s/mb9yaadujqe38fj/disappear.jpg?raw=1)
]
]

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# Costs in Economics are Opportunity Costs

- Because resources are scarce, and have rivalrous uses,

- 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 for *"pulling it out"* of an alternate use in the economy!

]

![:scale 70%](https://www.dropbox.com/s/mb9yaadujqe38fj/disappear.jpg?raw=1)
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# Opportunity Costs in Production

.content-box-green[
.green[**Examples**]:
.smallest[
- If you choose to 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
]
]
]

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# Opportunity Costs and Economic Profit

.pull-left[
| Item | Amount |
|----|----|
| Revenues | $600,000 |
| Supplies | ($20,000) |
| Electricity and Water | ($10,000) |
| Employee Salaries | ($300,000) |
| Craig' Salary | ($200,000) |

]

.pull-right[
.smaller[
- Craig could close his firm and rent out the building he owns for $50,000 per year.
- Instead of running his own business, Craig could work at a larger consulting firm and expect to earn $300,000 per year.
]
]
--

2. What is Craig's Consulting's accounting profit? economic profit?
]

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# Opportunity Cost is Hard for People

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# 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]

]

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# Sunk Costs: Examples

.pull-left[
.center[
![:scale 50%](https://www.dropbox.com/s/3167irxi1s2vnof/sharknado.jpg?raw=1)
]
]

.pull-right[
.center[
![](https://www.dropbox.com/s/8ozye0d9nustg7k/paperwritingfrustration.jpg?raw=1)
]
]

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# Sunks Costs: Examples

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# The Sunk Cost Fallacy

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# 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-purple["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 away from)?

]

![:scale 70%](https://www.dropbox.com/s/mb9yaadujqe38fj/disappear.jpg?raw=1)
]
]

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# The Accounting vs. Economic Point of View II

- **Social implications**: are consumers *best* off with you using scarce resources (with alternative uses!) to produce your current product?

- Remember: **this is an _economics_ course**, not a *business* course!
  - What might be good/bad for one business might have bad/good *consequences* for society!
    - e.g. monopoly vs. competition
]

![](https://www.dropbox.com/s/mb9yaadujqe38fj/disappear.jpg?raw=1)
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# Costs in the Short Run

- .hi[Total cost function, `\$C(q)\$`] relates output `$q$` to the total cost of production `$C$`

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

- 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

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# Fixed vs. Variable costs: Examples

**Fixed costs**: the aicraft

**Variable costs**: getting one more customer in a seat
]
]
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# Fixed vs. Variable costs: Examples

.pull-left[
.center[
![:scale 100%](https://www.dropbox.com/s/tyqf0tckydm01bs/carfactory.jpg?raw=1)
]
]

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

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

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# Fixed vs. Variable costs: Examples

**Fixed costs**: the retail space

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

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# 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

`$$C(q)=q^2+q+10$$`

]

1. Write a function for the fixed costs, `$f$`.

2. Write a function for the variable costs, `$VC(q)$`.

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# Cost Functions: Example, Visualized

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

]
]

<img src="2.4-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" />
]

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# 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:

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

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# Marginal Cost

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

`$$MC(q) = \frac{\Delta C(q)}{\Delta q} \approx \frac{C_2-C_1}{q_2-q_1}$$`

- 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"

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# The Importance of Marginal Cost

[Dazexiang Rebellion against the Qin Dynasty (209 B.C.)](https://en.wikipedia.org/wiki/Dazexiang_uprising)
]

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# Average and Marginal Costs: Example

.smaller[
.green[**Example**]: A small farm grows strawberries on 5 acres of land that it rents for $200 a week. The farm can hire workers at a wage of $250/week for each worker. The table below shows how the output of strawberries (in truckloads) varies with the number of workers hired:]

.pull-left[
.smaller[
| Output | Labor |
|--------|------:|
| 0 | 0 | 
| 1 | 1 | 
| 2 | 3 |
| 3 | 7 |
| 4 | 12 |
| 5 | 18 |
]
]
.pull-right[

1. If labor is the only variable cost, calculate the `$MC(q)$` and `$AC(q)$` for each of the first 5 truckloads.
]
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# Average and Marginal Costs: Visualized

.quitesmall[
| `$q$` | `$C(q)$` | `$MC(q)$` | `$AFC(q)$` | `$AVC(q)$` | `$AC(q)$` |
|----:|----:|--------:|-------:|
| `$0$` | `$10$` | `$-$` | `$-$` | `$-$` | `$-$` |
| `$1$` | `$12$` | `$2$` | `$10.00$` | `$2$` | `$12.00$` |
| `$2$` | `$16$` | `$4$` | `$5.00$` | `$3$` | `$8.00$` |
| `$3$` | `$22$` | `$6$` | `$3.33$` | `$4$` | `$7.30$` |
| `$4$` | `$30$` | `$8$` | `$2.50$` | `$5$` | `$7.50$` |
| `$5$` | `$40$` | `$10$` | `$2.00$` | `$6$` | `$8.00$` |
| `$6$` | `$52$` | `$12$` | `$1.67$` | `$7$` | `$8.70$` |
| `$7$` | `$66$` | `$14$` | `$1.43$` | `$8$` | `$9.40$` |
| `$8$` | `$82$` | `$16$` | `$1.25$` | `$9$` | `$10.25$` |
| `$9$` | `$100$` | `$18$` | `$1.11$` | `$10$` | `$11.10$` |
| `$10$` | `$120$` | `$20$` | `$1.00$` | `$11$` | `$12.00$` |

]
]

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# Relationship Between Marginal and Average

- marginal `$>$` average, average `$\uparrow$`
]
]

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# Relationship Between Marginal and Average

- marginal `$>$` average, average `$\uparrow$`

- marginal `$<$` average, average `$\downarrow$`
]
]

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# Relationship Between Marginal and Average

- marginal `$>$` average, average `$\uparrow$`

- marginal `$<$` average, average `$\downarrow$`

- When marginal `$=$` average, average is **maximized/minimized**
- .hi-purple[When `\$MC=AC\$`, `\$AC\$` is at a *minimum*]
- .hi-purple[When `\$MC=AVC\$`, `\$AVC\$` is at a *minimum*]
 
- Economic importance (later): Break-even price and shut-down price
]
]
.pull-right[
<img src="2.4-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" />
]

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# Short Run Costs: Example

.content-box-green[
.green[**Example**:] Suppose a firm's cost structure is described by: 
`$$\begin{align*}
C(q)&=15q^2+8q+45\\
MC(q)&=30q+8\\ \end{align*}$$`
]

1. Write expressions for the firm's **fixed costs**, **variable costs**, **average fixed costs**, **average variable costs**, and **average (total) costs**.

2. Find the minimum average (total) cost.

3. Find the minimum average variable cost.

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# Costs: Example: Visualized

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# 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 `$k$`)

]

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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

]

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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

- .purple[Long run average cost (LRAC)] curve "envelopes" the lowest (optimal) parts of all the SRAC curves!

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

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# Long Run Costs & Scale Economies I

- .hi[Economies of scale]: costs fall with output
  - `$AFC > AVC(q)$`

- .hi[Diseconomies of scale]: costs rise with output
 - `$AFC < AVC(q)$`
 
- .hi[Constant economies of scale]: costs don't change with output
 - Firm at minimum average cost

]

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# Long Run Costs & Scale Economies I

- .hi-purple[Returns to Scale] (last class): a **technological** relationship between inputs & output

- .hi[Economies of Scale] (this class): an **economic** relationship between output and average costs
]

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# Long Run Costs & Scale Economies II

- .hi[Minimum Efficient Scale]: `$q$` with the lowest `$AC(q)$`

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

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

<img src="2.4-slides_files/figure-html/unnamed-chunk-9-1.png" width="504" />
]

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# Long Run Costs and Scale Economies: Example

`$$\begin{align*}
LRC(q)&= 32000q-250q^2+q^3\\
LRMC(q)&=32000-500q+3q^2\\ \end{align*}$$`
]

1. At what levels of output will the firm face economies of scale and diseconomies of scale? (Hint: This firm has a `$U$`-shaped LRAC.)

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# Long Run Costs and Scale Economies: Example