2.2 — Short Run and Long Run

# 2.2 — Short Run and Long Run
## ECON 306 • Microeconomic Analysis • Spring 2021
### Ryan Safner<br> Assistant Professor of Economics <br> <a href="mailto:safner@hood.edu"><i class="fa fa-paper-plane fa-fw"></i>safner@hood.edu</a> <br> <a href="https://github.com/ryansafner/microS21"><i class="fa fa-github fa-fw"></i>ryansafner/microS21</a><br> <a href="https://microS21.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i>microS21.classes.ryansafner.com</a><br>

---

# Outline

### [Production in the Short Run](#5)
### [The Firm's Problem: Long Run](#16)
### [Isoquants and MRTS](#23)
### [Isocost Lines](#36)

---

# The “Runs” of Production

- “Time”-frame usefully divided between short vs. long run analysis

- .hi[Short run]: at least one factor of production is .hi-purple[fixed] (too costly to change)
`$$q=f(\bar{k},l)$$`
  - Assume **capital** is fixed (i.e. number of factories, storefronts, etc)
  - Short-run decisions only about using **labor**
    
]

]

---

# The “Runs” of Production

- “Time”-frame usefully divided between short vs. long run analysis

- .hi[Long run]: all factors of production are .hi-purple[variable] (can be changed)
`$$q=f(k,l)$$`

]

]

---

# Production in the Short Run

---

# Production in the Short Run: Example

.pull-left[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.smaller[
.green[**Example**:] Consider a firm with the production function
`$$q=k^{0.5}l^{0.5}$$`

- Suppose in the short run, the firm has 4 units of capital.
]
]

.smallest[
1. Derive the short run production function.
2. What is the total product (output) that can be made with 4 workers?
3. What is the total product (output) that can be made with 5 workers?

]
]
--

]

---

# 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="2.2-slides_files/figure-html/unnamed-chunk-2-1.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="2.2-slides_files/figure-html/unnamed-chunk-8-1.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="2.2-slides_files/figure-html/unnamed-chunk-10-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Production Functions and Marginal Product

- A quick trick to roughly<sup>.magenta[†]</sup> estimate `$MP_l$`

`$$MP_l \approx \frac{q_2-q_1}{l_2-l_1}$$`

| `$l$` | `$q$` | `$MP_l$` |
|-----|-----|:------:|
| `$0$` | `$0.00$` | `$-$` |
| `$1$` | `$2.00$` | `$2.00-0.00=2.00$` |
| `$2$` | `$2.83$` | `$2.83-2.00=0.83$` |
| `$3$` | `$3.46$` | `$3.46-2.83=0.63$` |

.footnote[<sup>.magenta[†]</sup> Note these are *approximate*. Technically, `\$MP_l\$` is defined via calculus as an *infinitesimal* change in `\$l\$`, whereas these are discrete changes.]

---

# 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
`$$AP_k = \frac{q}{k}$$`
]

]

---

# Production in the Short Run: Example II

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**:] Suppose a firm has the following production function:
`$$q=2k+l^2$$`

- Suppose in the short run, the firm has 10 units of capital.
]

1. Write an equation for the short run production function.
2. Calculate the total product(s), marginal product(s), and average product(s) for each of the first 5 workers.

---

# 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* `$k$`

]

---

# The Firm's Problem

.pull-left[
.smallest[
- Based on what we've discussed, we can fill in a constrained optimization model for the firm
    - But don't write this one down just yet!

- The **firm's problem** is:

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

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

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

- It's actually much easier to break this into **2 stages**. See today’s [class notes](/content/2.1-content) page for an example using only one stage.
]
]

---

# 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: 
]
]
.pull-right[

---

# 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:

2<sup>nd</sup> Stage: .hi[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
]
]

---

# Long Run Production

<table class="table" style="width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="2"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="6"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">k</div></th>
</tr>
  <tr>
   <th style="text-align:left;">  </th>
   <th style="text-align:left;">  </th>
   <th style="text-align:left;"> 0 </th>
   <th style="text-align:left;"> 1 </th>
   <th style="text-align:left;"> 2 </th>
   <th style="text-align:left;"> 3 </th>
   <th style="text-align:left;"> 4 </th>
   <th style="text-align:left;"> 5 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;font-weight: bold;vertical-align: middle !important;" rowspan="6"> l </td>
   <td style="text-align:left;font-weight: bold;"> 0 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 0.00 </td>
  </tr>
  <tr>
   
   <td style="text-align:left;font-weight: bold;"> 1 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 1.00 </td>
   <td style="text-align:left;"> 1.41 </td>
   <td style="text-align:left;"> 1.73 </td>
   <td style="text-align:left;"> 2.00 </td>
   <td style="text-align:left;"> 2.24 </td>
  </tr>
  <tr>
   
   <td style="text-align:left;font-weight: bold;"> 2 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 1.41 </td>
   <td style="text-align:left;"> 2.00 </td>
   <td style="text-align:left;"> 2.45 </td>
   <td style="text-align:left;"> 2.83 </td>
   <td style="text-align:left;"> 3.16 </td>
  </tr>
  <tr>
   
   <td style="text-align:left;font-weight: bold;"> 3 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 1.73 </td>
   <td style="text-align:left;"> 2.45 </td>
   <td style="text-align:left;"> 3.00 </td>
   <td style="text-align:left;"> 3.46 </td>
   <td style="text-align:left;"> 3.87 </td>
  </tr>
  <tr>
   
   <td style="text-align:left;font-weight: bold;"> 4 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 2.00 </td>
   <td style="text-align:left;"> 2.83 </td>
   <td style="text-align:left;"> 3.46 </td>
   <td style="text-align:left;"> 4.00 </td>
   <td style="text-align:left;"> 4.47 </td>
  </tr>
  <tr>
   
   <td style="text-align:left;font-weight: bold;"> 5 </td>
   <td style="text-align:left;"> 0.00 </td>
   <td style="text-align:left;"> 2.24 </td>
   <td style="text-align:left;"> 3.16 </td>
   <td style="text-align:left;"> 3.87 </td>
   <td style="text-align:left;"> 4.47 </td>
   <td style="text-align:left;"> 5.00 </td>
  </tr>
</tbody>
</table>
]

- So how does the firm choose the *optimal* combination??
]
---

# Mapping Input-Combination Choices Graphically

3-D Production Function
<div id="htmlwidget-617b2c236e2a1e314dbe" style="width:504px;height:396px;" class="plotly html-widget"></div>
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]
]
.pull-right[
.center[
2-D Isoquant Contours

]
]

---

# Isoquants and MRTS

---

# Isoquant Curves

- We can draw an .hi-blue[isoquant] indicating all combinations of `$l$` and `$k$` that yield the same `$q$`

]

]

---

# Isoquant Curves

- We can draw an .hi-blue[isoquant] indicating all combinations of `$l$` and `$k$` that yield the same `$q$`

- Combinations *above* curve yield .hi-green[more output]; on a .hi-green[higher curve]
  - `$D > A = B = C$`

]

]

---

# Isoquant Curves

- We can draw an .hi-blue[isoquant] indicating all combinations of `$l$` and `$k$` that yield the same `$q$`

- Combinations *above* curve yield .hi-green[more output]; on a .hi-green[higher curve]
  - `$D > A = B = C$`

- Combinations *below* the curve yield .hi-red[less output]; on a .hi-red[lower curve]
  - `$E < A = B = C$`
]

]

---

# Marginal Rate of *Technical* Substitution I

.pull-left[
- If your firm uses fewer workers, how much more capital would it need to produce the same amount?

]

---

# Marginal Rate of *Technical* Substitution I

.pull-left[
- If your firm uses fewer workers, how much more capital would it need to produce the same amount?

- .hi[Marginal Rate of Technical Substitution (MRTS)]: rate at which firm trades off one input for another to *yield same output*

- Firm's .hi-purple[relative value] of using `$l$` in production based on its tech:

> “We could give up (MRTS) units of k to use 1 more unit of l to produce the same output.”

]

---

# Marginal Rate of *Technical* Substitution II

---

# Marginal Rate of *Technical* Substitution II

- MRTS is the slope of the isoquant
`$$MRTS_{l,k}=-\frac{\Delta k}{\Delta l} = \frac{rise}{run}$$`

- Amount of `$k$` given up for 1 more `$l$`

- Note: slope (MRTS) changes along the curve!

- .hi-purple[Law of diminishing returns]!

]
.pull-right[

]

---

# MRTS and Marginal Products

- Relationship between `$MP$` and `$MRTS$`:

`$$\begin{align*}
\underbrace{\frac{\Delta k}{\Delta l}}_{MRTS} &= -\frac{MP_l}{MP_k}\\ \end{align*}$$`

- See proof in [today's class notes](/content/2.2-content)

- Sound familiar? 🧐

]
.pull-right[

]

---

# Special Case I: Perfect Substitutes

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Consider Bank Tellers `\$(l)\$` and ATMs `\$(k)\$` 
]
- Suppose 1 ATM can do the work of 2 bank tellers

- .hi[Perfect substitutes]: inputs that can be substituted at same fixed rate and yield same output

- `$MRTS_{l,k}=-0.5$` (a constant!)

]

.pull-right[
<img src="2.2-slides_files/figure-html/unnamed-chunk-20-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Special Case II: Perfect Complements

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Consider buses `\$(k)\$` and bus drivers `\$(l)\$`
]
- Must combine together in fixed proportions (1:1)

- .hi[Perfect complements]: inputs must be used together in same fixed proportion to produce output

- `$MRTS_{l,k}$`: ?

]

.pull-right[
<img src="2.2-slides_files/figure-html/unnamed-chunk-21-1.png" width="504" style="display: block; margin: auto;" />
]

---

# Common Case: Cobb-Douglas Production Functions

- Again: very common functional form in economics is .hi[Cobb-Douglas]

`$$q=A \, k^al^b$$`

- Where `$a, b >0$`
  - often `$a+b=1$`

- `$A$` is total factor productivity
]

.pull-right[
<img src="2.2-slides_files/figure-html/unnamed-chunk-22-1.png" width="504" style="display: block; margin: auto;" />

]

---

# Practice

.smallest[
.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Suppose a firm has the following production function:
`$$q=2lk$$`

Where its marginal products are:

`$$\begin{align*}
MP_l&=2k\\
MP_k&=2l\\
\end{align*}$$`

1. Put `$l$` on the horizontal axis and `$k$` on the vertical axis. Write an equation for `$MRTS_{l,k}$`.

2. Would input combinations of `$(1, 4)$` and `$(2, 2)$` be on the same isoquant?

3. Sketch a graph of the isoquant from part 2. 
]
]

---

# Isocost Lines

---

# Isocost Lines

.pull-left[
.smallest[
- If your firm can choose among *many* input combinations to produce `$q$`, which combinations are optimal?

- Those combination that are .hi-purple[cheapest]

- Denote prices of each input as:
    - `$w$`: price of labor (wage)
    - `$r$`: price of capital

- Let `$C$` be .hi[total cost] of using inputs `$(l,k)$` at market prices `$(w,r)$` to produce `$q$` units of output:

`$$C(w,r,q)=wl+rk$$`
]
]

---

# The Isocost Line, Graphically

`$$wl+rk=C$$`

]

---

# The Isocost Line, Graphically

`$$wl+rk=C$$`

- Solve for `$k$` to graph

`$$k=\frac{C}{r}-\frac{w}{r}l$$`

]

<img src="2.2-slides_files/figure-html/BC-plot0-1.png" width="432" style="display: block; margin: auto;" />
]

---

# The Isocost Line, Graphically

`$$wl+rk=C$$`

- Solve for `$k$` to graph

`$$k=\frac{C}{r}-\frac{w}{r}l$$`

- Vertical-intercept: `$\frac{C}{r}$`
- Horizontal-intercept: `$\frac{C}{w}$`
]

<img src="2.2-slides_files/figure-html/BC-plot1-1.png" width="432" style="display: block; margin: auto;" />
]

---

# The Isocost Line, Graphically

`$$wl+rk=C$$`

- Solve for `$k$` to graph

`$$k=\frac{C}{r}-\frac{w}{r}l$$`

- Vertical-intercept: `$\frac{C}{r}$`
- Horizontal-intercept: `$\frac{C}{w}$`
- slope: `$-\frac{w}{r}$`

]

.pull-right[
<img src="2.2-slides_files/figure-html/BC-plot2-1.png" width="432" style="display: block; margin: auto;" />
]

---

# The Isocost Line: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**]: Suppose your firm has a purchasing budget of $50. Market wages are $5/worker-hour and the mark rental rate of capital is $10/machine-hour. Let `$l$` be on the horizontal axis and `$k$` be on the vertical axis.

1. Write an equation for the isocost line (in graphable form).

2. Graph the isocost line.
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# Interpreting Isocost Line

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- Points .blue[on] the line are same total cost
  - A: \$$5(0l)+$10(5k) = $50\$ 
  - B: \$$5(10l)+$10(0k) = $50\$
  - C: \$$5(2l)+$10(4k) = $50\$ 
  - D: \$$5(6l)+$10(2k) = $50\$
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<img src="2.2-slides_files/figure-html/BC2-plot1-1.png" width="432" style="display: block; margin: auto;" />
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# Interpreting Isocost Line

- Points .green[beneath] the line are .green[cheaper] (but may produce less)
  - E: \$$5(3l)+$10(2k) = $35\$

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<img src="2.2-slides_files/figure-html/BC2-plot2-1.png" width="432" style="display: block; margin: auto;" />
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# Interpreting the Isocost Line

- Points .green[beneath] the line are .green[cheaper] (but may produce less)
  - E: \$$5(3l)+$10(2k) = $35\$

- Points .red[above] the line are .red[more expensive] (and may produce more)
  - F: \$$5(6l)+$10(4k) = $70\$ 
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<img src="2.2-slides_files/figure-html/BC2-plot3-1.png" width="432" style="display: block; margin: auto;" />
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# Interpretting the Slope

- **Slope**: .hi-purple[tradeoff] between `$l$` and `$k$` at market prices
  - Market .hi-purple[“exchange rate”] between `$l$` and `$k$`

- .hi-purple[*Relative* price] of `$l$` or the .hi-purple[opportunity cost] of `$l$`:

> Hiring 1 more unit of `$l$` requires giving up `$\left(\frac{w}{r}\right)$` units of `$k$`

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<img src="2.2-slides_files/figure-html/BC-slope-plot-1.png" width="432" style="display: block; margin: auto;" />
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# Changes in Relative Factor Prices I

- Slope changes: `$-\frac{w'}{r}$`
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<img src="2.2-slides_files/figure-html/BC-x-change-plot-1.png" width="432" style="display: block; margin: auto;" />
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# Changes in Relative Factor Prices II

- Slope changes: `$-\frac{w}{r'}$`
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<img src="2.2-slides_files/figure-html/BC-y-change-plot-1.png" width="432" style="display: block; margin: auto;" />
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