The Employer’s Dilemma

class: center, middle, inverse, title-slide

# The Employer’s Dilemma
## EC 350: Labor Economics
### <a href="https://kyleraze.com">Kyle Raze</a>
### Winter 2022

---

# Agenda

1. Difference-in-differences, revisited
    - Card and Krueger (1994)
    - Difference-in-differences with regression
    - Parallel trends assumption 
    - Discuss Hoynes and Patel (2018)
2. The Employer's Dilemma
    - Labor demand
    - Profit maximization
    - Production technology
    - Short-run demand curve

---
class: inverse, middle

# Difference-in-differences, revisited

---
# Card and Krueger (1994), revisited

.pull-left[
### **Raw data comparison**

Difference-in-differences .mono[=] .pink[0.59] .mono[-] .purple[-2.16] .mono[=] 2.75

<table class="table" style="margin-left: auto; margin-right: auto;">
<caption>Average employment per store</caption>
 <thead>
  <tr>
   <th style="text-align:left;"> Group </th>
   <th style="text-align:center;"> After </th>
   <th style="text-align:center;"> Before </th>
   <th style="text-align:center;"> Difference </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;line-height: 110%;"> .pink[Treatment (NJ)] </td>
   <td style="text-align:center;line-height: 110%;"> 21.03 </td>
   <td style="text-align:center;line-height: 110%;"> 20.44 </td>
   <td style="text-align:center;line-height: 110%;"> 0.59 </td>
  </tr>
  <tr>
   <td style="text-align:left;line-height: 110%;"> .purple[Control (PA)] </td>
   <td style="text-align:center;line-height: 110%;"> 21.17 </td>
   <td style="text-align:center;line-height: 110%;"> 23.33 </td>
   <td style="text-align:center;line-height: 110%;"> -2.16 </td>
  </tr>
</tbody>
</table>
]
.pull-right[
<img src="08-Employers_files/figure-html/unnamed-chunk-2-1.svg" style="display: block; margin: auto;" />
]

---
# Card and Krueger (1994), revisited

.pull-left[
### **Raw data comparison**

Difference-in-differences .mono[=] .pink[0.59] .mono[-] .purple[-2.16] .mono[=] 2.75

`$$\begin{align} \text{Employment}_{it} = \alpha &+ \beta ~ \text{NJ}_i + \gamma ~ \text{After}_t \\ &+ \delta ~ \text{NJ}_i \times \text{After}_t + \varepsilon_{it} \end{align}$$`

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Parameter </th>
   <th style="text-align:center;"> (1) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;color: #272822 !important;line-height: 110%;color: #272822 !important;"> Intercept </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 23.33 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;color: #272822 !important;">  </td>
   <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (1.07) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #272822 !important;line-height: 110%;color: #272822 !important;"> NJ </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -2.89 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;color: #272822 !important;">  </td>
   <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (1.19) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #272822 !important;line-height: 110%;color: #272822 !important;"> After </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -2.16 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;color: #272822 !important;">  </td>
   <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (1.52) </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #272822 !important;line-height: 110%;color: #272822 !important;"> NJ × After </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2.75 </td>
  </tr>
  <tr>
   <td style="text-align:left;color: #272822 !important;color: #c2bebe !important;line-height: 110%;color: #272822 !important;">  </td>
   <td style="text-align:center;color: #272822 !important;color: #c2bebe !important;line-height: 110%;"> (1.69) </td>
  </tr>
</tbody>
</table>
]

---
# Card and Krueger (1994), revisited

### **Regression .mono[-->] raw data averages**

**Step 1:** Specify the regression model.

`$$\text{Employment}_{it} = \alpha + \beta ~ \text{NJ}_i + \gamma ~ \text{After}_t + \delta ~ \text{NJ}_i \times \text{After}_t + \varepsilon_{it}$$`

**Step 2:** Find the expected value of the model for each state and period.

- Average employment in **Pennsylvania** restaurants **before** New Jersey's wage increase 
--
<br> `$\quad = E\left[\text{Employment}_{it} ~|~ \text{NJ}_i = 0~\wedge~\text{After}_t = 0 \right]$`
--
<br> `$\quad = \alpha + \beta ~ (0) + \gamma ~ (0) + \delta ~ (0) \times (0) + (0)$`
--
<br> `$\quad = \alpha$`
--
<br> `$\quad = 23.33$`

---
# Card and Krueger (1994), revisited

### **Regression .mono[-->] raw data averages**

**Step 1:** Specify the regression model.

`$$\text{Employment}_{it} = \alpha + \beta ~ \text{NJ}_i + \gamma ~ \text{After}_t + \delta ~ \text{NJ}_i \times \text{After}_t + \varepsilon_{it}$$`

**Step 2:** Find the expected value of the model for each state and period.

- Average employment in **Pennsylvania** restaurants **after** New Jersey's wage increase 
--
<br> `$\quad = E\left[\text{Employment}_{it} ~|~ \text{NJ}_i = 0~\wedge~\text{After}_t = 1 \right]$`
--
<br> `$\quad = \alpha + \beta ~ (0) + \gamma ~ (1) + \delta ~ (0) \times (1) + (0)$`
--
<br> `$\quad = \alpha + \gamma$`
--
<br> `$\quad = 23.33 - 2.16$`
--
<br> `$\quad = 21.17$`

---
# Card and Krueger (1994), revisited

### **Regression .mono[-->] raw data averages**

**Step 1:** Specify the regression model.

`$$\text{Employment}_{it} = \alpha + \beta ~ \text{NJ}_i + \gamma ~ \text{After}_t + \delta ~ \text{NJ}_i \times \text{After}_t + \varepsilon_{it}$$`

**Step 2:** Find the expected value of the model for each state and period.

- Average employment in **New Jersey** restaurants **before** New Jersey's wage increase 
--
<br> `$\quad = E\left[\text{Employment}_{it} ~|~ \text{NJ}_i = 1~\wedge~\text{After}_t = 0 \right]$`
--
<br> `$\quad = \alpha + \beta ~ (1) + \gamma ~ (0) + \delta ~ (1) \times (0) + (0)$`
--
<br> `$\quad = \alpha + \beta$`
--
<br> `$\quad = 23.33 - 2.89$`
--
<br> `$\quad = 20.44$`

---
# Card and Krueger (1994), revisited

### **Regression .mono[-->] raw data averages**

**Step 1:** Specify the regression model.

`$$\text{Employment}_{it} = \alpha + \beta ~ \text{NJ}_i + \gamma ~ \text{After}_t + \delta ~ \text{NJ}_i \times \text{After}_t + \varepsilon_{it}$$`

**Step 2:** Find the expected value of the model for each state and period.

- Average employment in **New Jersey** restaurants **after** New Jersey's wage increase 
--
<br> `$\quad = E\left[\text{Employment}_{it} ~|~ \text{NJ}_i = 1~\wedge~\text{After}_t = 1 \right]$`
--
<br> `$\quad = \alpha + \beta ~ (1) + \gamma ~ (1) + \delta ~ (1) \times (1) + (0)$`
--
<br> `$\quad = \alpha + \beta + \gamma + \delta$`
--
<br> `$\quad = 23.33 - 2.89 - 2.16 + 2.75$`
--
<br> `$\quad = 21.03$`

---
# Card and Krueger (1994), revisited

## **Parallel trends assumption**

> If New Jersey hadn't increased its minimum wage, New Jersey's fast-food employment would have continued on the same trend as fast-food employment in Pennsylvania.

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-5-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
**Variables**
- .hi[D] .mono[=] Minimum wage
- .hi[Y] .mono[=] Fast-food employment
- .hi[S] .mono[=] State (NJ or PA)
- .hi[T] .mono[=] Time (before or after)
- .hi[U] .mono[=] Unobserved changes within each state

A difference-in-differences comparison explicitly controls for .hi[S] and .hi[T].
]

---
# Card and Krueger (1994), revisited

## **Parallel trends assumption**

**Q:** Why is it important to articulate this assumption?

**A:** Allows us to **direct our skepticism** toward variables that confound the effects of treatment!

.pull-left[
.center[**Valid difference-in-differences**]
<img src="08-Employers_files/figure-html/unnamed-chunk-6-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
.center[**Invalid difference-in-differences**]
<img src="08-Employers_files/figure-html/unnamed-chunk-7-1.svg" style="display: block; margin: auto;" />
]

---
# Card and Krueger (1994), revisited

## **Parallel trends assumption**

Fundamentally **untestable**, but **falsifiable** with additional years of data.

**Best-case scenario?** Parallel trends before treatment .mono[-->] **fail to reject** parallel trends.

---
# Card and Krueger (1994), revisited

## **Parallel trends assumption**

Fundamentally **untestable**, but **falsifiable** with additional years of data.

**Worst-case scenario?** Differential trends before treatment .mono[-->] **reject** parallel trends.

---
# Hoynes and Patel (2018)

## Discussion

**Q.sub[1]:** What comparisons does the study make?

**Q.sub[2]:** What do those comparisons suggest about the effect of the Earned Income Tax Credit on poverty?

**Q.sub[3]:** What do we need to believe to interpret the results as causal?

**Q.sub[4]:** What are the policy implications?

---
class: inverse, middle

# The Employer's Dilemma

---
# The Employer's Dilemma

**Q:** How do employers make decisions about hiring (and firing) workers?

- What tradeoffs do employers face in competitive markets?

**Q:** Why should we care?

- **A:** Labor market outcomes ultimately depend on **interactions** between workers *and* employers!

Before modeling interactions between workers and employers, we will first develop a model of **labor demand**.

---
# Labor demand

**Q:** How is labor different from the goods and services that consumers demand?

**A:** Labor is a **derived demand**.

- Consumers don't demand labor itself, but rather the goods that labor produces.

**Q:** In what ways is labor different from other factors of production?

**A:** Many!

- **You can't own a worker!** Rather, you can only rent a worker's services.
- **Workers need motivation!** Office supplies don't get bored and browse Reddit, but people do.
- **Workers care about working conditions!** Most robots can handle a 95-degree warehouse, but many people would struggle.

---
# Profit maximization

## **Objective function**

We assume that the employer seeks to maximize profit: `$$\begin{align} \Pi &= \text{TR} - \text{TC} \\ &= pq - (wE + rK) \\ &= pq - wE - rK \end{align}$$`

- `$\Pi$` represents profit, measured in dollars.
- `$pq$` represents total revenue, where `$p$` is the output price and `$q$` is the quantity of output.
- `$wE$` represents the wage bill, where `$w$` is the market wage and `$E$` is the number of full-time equivalent workers.
- `$rK$` represents capital expenses, where `$r$` is the rental rate of capital and `$K$` is the amount of capital.

The employer will choose a **profit-maximizing** level of output to be produced by a **cost-minimizing** bundle of labor and capital.

---
# Profit maximization

## **Objective function**

We assume that the employer seeks to maximize profit: `$$\begin{align} \Pi &= pq - wE - rK \end{align}$$`

We will also assume that the employer is a **price-taker**.

- The **choices** of the employer **have no impact on market prices** for the output good, labor, or capital.
- In other words, the market for the output good is **perfectly competitive**, as are the markets for labor and capital.

---
# Production technology

## **Production function**

> A mathematical description of the relationship between inputs and output in an employer's production process.

> .center[Inputs .mono[-->] production technology .mono[-->] output]

We will make **three main assumptions** about the **production technology** used by the employer:

1. Labor and capital are the only inputs (*i.e.,* `$q = f(E, K)$`)

2. Workers are homogeneous

3. Marginal productivity eventually decreases

---
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-10-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Total product of labor]

> The amount of output from a given quantity of labor, *holding the amount of capital and other inputs constant*.

]

---
count: false
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-11-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Total product of labor]

> The amount of output from a given quantity of labor, *holding the amount of capital and other inputs constant*.

]

---
count: false
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-12-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Total product of labor]

> The amount of output from a given quantity of labor, *holding the amount of capital and other inputs constant*.

]

---
count: false
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-13-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Total product of labor]

> The amount of output from a given quantity of labor, *holding the amount of capital and other inputs constant*.

]

---
count: false
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-14-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Total product of labor]

> The amount of output from a given quantity of labor, *holding the amount of capital and other inputs constant*.

]

---
count: false
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-15-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Total product of labor]

> The amount of output from a given quantity of labor, *holding the amount of capital and other inputs constant*.

]

---
count: false
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-16-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Total product of labor]

> The amount of output from a given quantity of labor, *holding the amount of capital and other inputs constant*.

]

---
count: false
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-17-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Total product of labor]

> The amount of output from a given quantity of labor, *holding the amount of capital and other inputs constant*.

]

---
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-18-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Marginal product of labor]

> The change in output from a one-unit increase in labor, *holding the amount of capital and other inputs constant*.

`$$\text{MP}_E = \dfrac{\Delta q}{\Delta E}$$`

]

---
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-19-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Marginal product of labor]

> The change in output from a one-unit increase in labor, *holding the amount of capital and other inputs constant*.

`$$\text{MP}_E = \dfrac{\Delta q}{\Delta E}$$`

]

---
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-20-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Marginal product of capital]

> The change in output from a one-unit increase in capital, *holding the amount of labor and other inputs constant*.

`$$\text{MP}_K = \dfrac{\Delta q}{\Delta K}$$`
]

---
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-21-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple["Law" of diminishing returns].super[.hi-pink[<span>&#8224;</span>]]

.footnote[.super[.hi-pink[<span>&#8224;</span>]] Also known as "diminishing marginal productivity."]

> For a fixed amount of capital, the **marginal product** of labor **eventually declines** as employment increases.

Early gains from specialization give way to crowded capital inputs.

]

---
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-22-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Average product of labor]

> The amount of output produced by the typical worker, *holding the amount of capital and other inputs constant*.

`$$\text{AP}_E = \dfrac{q}{E}$$`

]

---
# Production technology

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-23-1.svg" style="display: block; margin: auto;" />
]
.pull-right[
## .hi-purple[Average product of labor]

The marginal product curve intersects the average product curve where average product is maximized.

- When the average product curve is **increasing**, marginal product **is greater than** average product.
- When the average product curve is **decreasing**, marginal product **is less than** average product.

]

---
# Production technology

**Q:** What is the marginal product of each worker?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> Marginal product (MP) </th>
   <th style="text-align:center;"> Average product (AP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
</tbody>
</table>

---
count: false
# Production technology

**Q:** What is the marginal product of each worker?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> Marginal product (MP) </th>
   <th style="text-align:center;"> Average product (AP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
</tbody>
</table>

---
count: false
# Production technology

**Q:** What is the marginal product of each worker?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> Marginal product (MP) </th>
   <th style="text-align:center;"> Average product (AP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
</tbody>
</table>

---
count: false
# Production technology

**Q:** What is the marginal product of each worker?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> Marginal product (MP) </th>
   <th style="text-align:center;"> Average product (AP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
</tbody>
</table>

---
count: false
# Production technology

**Q:** What is the marginal product of each worker?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> Marginal product (MP) </th>
   <th style="text-align:center;"> Average product (AP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
</tbody>
</table>

---
# Production technology

**Q:** What is the average product for each level of employment?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> Marginal product (MP) </th>
   <th style="text-align:center;"> Average product (AP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
</tbody>
</table>

---
count: false
# Production technology

**Q:** What is the average product for each level of employment?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> Marginal product (MP) </th>
   <th style="text-align:center;"> Average product (AP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 900 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 700 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 500 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
  </tr>
</tbody>
</table>

---
# Valuing production

## **Marginal revenue product of labor**

> The change in total revenue from a one-unit increase in labor, *holding capital and other inputs constant*.

`$$\begin{align} \text{MRP}_E &= \dfrac{\Delta\text{TR}}{\Delta q} \times \dfrac{\Delta q}{\Delta E} \\ &= \text{MR} \times \text{MP}_E \end{align}$$`

In a **perfectly competitive market** for the output good, price does not depend on a firm's level of output (*i.e.,* `$\text{MR} = p$`).

- **Implication?** Marginal revenue product is the same as the **value of marginal product of labor**:

`$$\text{VMP}_E = \text{MRP}_E = p \times \text{MP}_E$$`

---
# Valuing production

## **Value of average product of labor**

> The amount of revenue the typical worker produces for the firm.

`$$\text{VAP}_E = p \times \text{AP}_E$$`

---
# Valuing production

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-31-1.svg" style="display: block; margin: auto;" />
]
.pull-right[

The relationship between the value of average product and the value of marginal product is the same as the relationship between average product and marginal product.

**The difference?** The average product and marginal product curves are now "scaled up" by the price of the output good.

- Vertical axis is now in dollars instead of units of output.
- In this example, `$p$` .mono[=] $150.

]

---
# Valuing production

**Q:** If the price of the output good is $2, what is the marginal revenue product of each worker?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> MP </th>
   <th style="text-align:center;"> AP </th>
   <th style="text-align:center;"> Marginal revenue product (MRP) </th>
   <th style="text-align:center;"> Value of MP (VAP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 900 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 700 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 500 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
</tbody>
</table>

---
count: false
# Valuing production

**Q:** If the price of the output good is $2, what is the marginal revenue product of each worker?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> MP </th>
   <th style="text-align:center;"> AP </th>
   <th style="text-align:center;"> Marginal revenue product (MRP) </th>
   <th style="text-align:center;"> Value of MP (VAP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $2000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 900 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 700 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 500 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -$400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
</tbody>
</table>

---
# Valuing production

**Q:** If the price of the output good is $2, what is the value of average product?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> MP </th>
   <th style="text-align:center;"> AP </th>
   <th style="text-align:center;"> Marginal revenue product (MRP) </th>
   <th style="text-align:center;"> Value of MP (VAP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $2000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 900 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 700 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 500 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -$400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;">  </td>
  </tr>
</tbody>
</table>

---
count: false
# Valuing production

**Q:** If the price of the output good is $2, what is the value of average product?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;"> MP </th>
   <th style="text-align:center;"> AP </th>
   <th style="text-align:center;"> Marginal revenue product (MRP) </th>
   <th style="text-align:center;"> Value of MP (VAP) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $2000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $2000 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 900 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1800 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1600 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 700 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1400 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1200 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 500 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1000 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -$400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $800 </td>
  </tr>
</tbody>
</table>

---
# Short run *vs.* long run

## **Short run**

> The time span over which a business can adjust some inputs (*e.g.,* labor), but cannot adjust others (*e.g.,* capital).

In the short run, we will assume that the level of employment **E** can vary, but capital **K** is fixed at an initial level **K.sub[0]**.

- **Example:** A shop foreman can hire or fire workers or adjust hours, but they are unable to expand the factory by adding assembly lines, heavy machinery, or a new building.

---
# Short run *vs.* long run

## **Long run**

> The time span over which a business can adjust all inputs.

In the long-run, we will assume that both the level of employment **E** and capital **K** can vary.

- **Example:** An office manager can hire or fire workers, adjust hours, buy or sell desks and computers, or lease new office space.

---
class: clear-slide

**Q:** If the price of the output good is $2 and the market wage is $500 per week, how many workers should the employer hire?

<table>
 <thead>
  <tr>
   <th style="text-align:center;"> Workers (E) </th>
   <th style="text-align:center;"> Output (q) </th>
   <th style="text-align:center;">     MP     </th>
   <th style="text-align:center;">     AP     </th>
   <th style="text-align:center;">     MRP     </th>
   <th style="text-align:center;">     VAP     </th>
   <th style="text-align:center;"> Wage (w) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> — </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $500 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $2000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $2000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $500 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 900 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $500 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $500 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 4 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 700 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $500 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 5 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 600 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $500 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 6 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 3000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 500 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $0 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $1000 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $500 </td>
  </tr>
  <tr>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 7 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 2800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -200 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> 400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> -$400 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $800 </td>
   <td style="text-align:center;color: #272822 !important;line-height: 110%;"> $500 </td>
  </tr>
</tbody>
</table>

The employer should **think at the margin** and keep hiring as long as **MRP** `$\geq$` **w**.

**A:** The employer should hire 4 workers.

---
# Hiring in the short run

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-37-1.svg" style="display: block; margin: auto;" />
]
.pull-right[

## **Profit maximization**

An employer maximizes profit by hiring `$E^*$` workers where `$w = \text{MRP}_E$` and `$\text{MRP}_E$` is decreasing.

]

---
# Hiring in the short run

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-38-1.svg" style="display: block; margin: auto;" />
]
.pull-right[

## **Profit maximization**

**Q:** Why wouldn't an employer stop hiring while marginal revenue product is increasing?

]

---
count: false
# Hiring in the short run

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-39-1.svg" style="display: block; margin: auto;" />
]
.pull-right[

## **Profit maximization**

**Q:** Why wouldn't an employer stop hiring while marginal revenue product is increasing?

**A:** Because the employer would be "leaving money on the table."

- The employer could increase profit at the margin by hiring an additional worker.

]

---
# Hiring in the short run

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-40-1.svg" style="display: block; margin: auto;" />
]
.pull-right[

## **Profit maximization**

**Q:** What happens when marginal revenue product exceeds the value of average product?

]

---
count: false
# Hiring in the short run

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-41-1.svg" style="display: block; margin: auto;" />
]
.pull-right[

## **Profit maximization**

**Q:** What happens when marginal revenue product exceeds the value of average product?

**A:** The employer will shut down the business.

- Any wage that intersects MRP in this region will exceed VAP .mono[-->] business would operate at a loss!

]

---
# Hiring in the short run

.pull-left[
<img src="08-Employers_files/figure-html/unnamed-chunk-42-1.svg" style="display: block; margin: auto;" />
]
.pull-right[

## .hi-purple[Labor demand]

The portion of the MRP curve below the VAP curve traces out the .hi-purple[short-run labor demand] curve.

- Describes how an employer adjusts employment as the market wage changes, holding other inputs constant.
- **Downward sloping:** An employer wants to reduce staffing as the wage increases, *all else equal*.

]

---
# Housekeeping

**~~In-class~~ online midterm exam** on Monday, February 7th.

- Covers material from weeks 1 through 5.
- I will post a midterm review guide and some practice problems.

**Problem Set 2** due by Saturday, February 5th by 11:59pm.

- Covers material from weeks 4 and 5.