1.2: Technology and Cost

# 1.2: Technology and Cost
## ECON 326 · Industrial Organization · Spring 2020
### 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/IOs20"><i class="fa fa-github fa-fw"></i> ryansafner/IOs20</a><br> <a href="https://IOs20.classes.ryansafner.com"> <i class="fa fa-globe fa-fw"></i> IOs20.classes.ryansafner.com</a><br>

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# Models Refresher

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# The Two Major Models of Price Theory

## Optimization

- Agents have .hi[objectives] they value

- Agents face .hi[constraints]

- Make .hi[tradeoffs] to maximize objectives within constraints

## Equilibrium

- Agents .hi[compete] with others over **scarce** resources

- Agents .hi[adjust] behaviors based on prices

- .hi[Stable outcomes] when adjustments stop

]

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# Relationship Between Optimization & Equilibrium

.pull-left[
- If people can **learn** and **change** their behavior, they will always **switch** to a higher-valued option

- If there are no alternatives that are better, people are at an .hi[optimum]

- If everyone is at an optimum, the system is in .hi[equilibrium]
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![:scale 70%](https://www.dropbox.com/s/j7xenkgo3p71vke/equilibriumbalance.png?raw=1)
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# Constrained Optimization III

- All constrained optimization models have three moving parts:

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

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.pull-right[
.center[
![:scale 100%](https://www.dropbox.com/s/qvr240t5j6t3arm/optimize.jpeg?raw=1)
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# Constrained Optimization III

- All constrained optimization models have three moving parts:

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

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

]

.pull-right[
.center[
![:scale 100%](https://www.dropbox.com/s/qvr240t5j6t3arm/optimize.jpeg?raw=1)
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# Constrained Optimization III

- All constrained optimization models have three moving parts:

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

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

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

]

.pull-right[
.center[
![:scale 100%](https://www.dropbox.com/s/qvr240t5j6t3arm/optimize.jpeg?raw=1)
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# This Black Box We Call "Firms"

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

- Synonymous with .hi[production function]

- Fully replicable

- We'll explore (and explode) this much later
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.pull-right[
.center[
![:scale 70%](https://www.dropbox.com/s/zu2c5ijzqvz5sbh/inputoutput.png?raw=1)
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# What Do Firms Do? I

- Assume "firm" is agent to model:

- So what do firms do?

- How would we set up an optimization model:

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

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

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

]

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# What Do Firms Do? II

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.center[
![:scale 70%](https://www.dropbox.com/s/zu2c5ijzqvz5sbh/inputoutput.png?raw=1)
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# What Do Firms Do? II

- **Inputs**: `$x_1, x_2, \cdots, x_n$`
  - <span class="green">**Examples**: worker efforts, warehouse space, electricity, loans, gasoline, cardboard, fertilizer, computers, software programs, etc<span>
  
]

.pull-right[
.center[
![:scale 70%](https://www.dropbox.com/s/zu2c5ijzqvz5sbh/inputoutput.png?raw=1)
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# What Do Firms Do? II

- **Inputs**: `$x_1, x_2, \cdots, x_n$`
  - <span class="green">**Examples**: worker efforts, warehouse space, electricity, loans, oil, fertilizer, software programs, etc<span>
  
- **Output**: `$q$`
  - <span class="green">**Examples**: gasoline, cars, legal services, mobile apps, vegetables, consulting advice, financial reports, etc<span>

]

.pull-right[
.center[
![:scale 70%](https://www.dropbox.com/s/zu2c5ijzqvz5sbh/inputoutput.png?raw=1)
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# What Do Firms Do? III

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

]

.pull-right[
.center[
![:scale 70%](https://www.dropbox.com/s/zu2c5ijzqvz5sbh/inputoutput.png?raw=1)
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# Production Function as Recipe

![](https://www.dropbox.com/s/0820xwecp9au978/recipe1.PNG?raw=1)
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![](https://www.dropbox.com/s/7sbjwufc10xxalh/recipe2.PNG?raw=1)
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# Factors of Production I

`$$q=f(t,l,k,e,a)$$`

- Economists typically classify inputs in categories, known as the .hi[factors of production (FOP)] or "the .hi[factors]"

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Factor </th>
   <th style="text-align:left;"> Owned By </th>
   <th style="text-align:left;"> Earns </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Land (t) </td>
   <td style="text-align:left;"> Landowners </td>
   <td style="text-align:left;"> Rent </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Labor (l) </td>
   <td style="text-align:left;"> Laborers </td>
   <td style="text-align:left;"> Wages </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Capital (k) </td>
   <td style="text-align:left;"> Capitalists </td>
   <td style="text-align:left;"> Interest </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Entrepreneurship (e) </td>
   <td style="text-align:left;"> Entrepreneurs </td>
   <td style="text-align:left;"> Profit </td>
  </tr>
</tbody>
</table>

- `$a$`: "total factor productivity" (ideas/knowledge/institutions)

]

]

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# Factors of Production II

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

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

]

]

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# What Does a Firm Maximize?

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

- Not true for all firms
    - <span class="green">**Examples**: non-profits, charities, civic associations, government agencies, criminal organizations, etc</span>

- Even profit-seeking firms may also want to maximize additional things
    - <span class="green">**Examples**: goodwill, sustainability, social responsibility, etc </span>

]

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# What is Profit?

- In economics, profit is simply **benefits minus (opportunity) costs**

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# What is Profit?

- In economics, profit is simply **benefits minus (opportunity) costs**

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

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# What is Profit?

- In economics, profit is simply **benefits minus (opportunity) costs**

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

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

]

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# What is Profit?

- In economics, profit is simply **benefits minus (opportunity) costs**

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

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

- The profit of selling `$q$` units and using inputs `$l,k$` is:
`$$\pi=\underbrace{pq}_{revenues}-\underbrace{(wl+rk)}_{costs}$$`

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# Who Gets the Profits? I

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

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

]

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# Who Gets the Profits? II

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

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# Who Gets the Profits? III

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

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

]

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# The Firm's Optimization Problem I

- So what do firms do?

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

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

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

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

]

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# The Firm's Optimization Problem II

- Prices?
    - Depends on the market the firm is operating in!
    - Study of <span class="shout">industrial organization</span>
    
- Essential question: how competitive is a market? This will influence what firms (can) do
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# The "Runs" of Production

- The "time"-frame of production can be usefully divided between short vs. long run analysis

- .hi[Short run]: at least one factor of production is *fix*ed* (too costly to change)
`$$q=f(\bar{k},l)$$`
    - We assume **capital** is fixed (i.e. number of factories, storefronts, etc)
    - Only decisions about how to use **labor**
    
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.pull-right[
.center[
![:scale 60%](https://www.dropbox.com/s/csmheqe2xzp1dst/timehourglass.jpg?raw=1)
]

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# The "Runs" of Production

- The "time"-frame of production can be usefully divided between short vs. long run analysis

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

]

.pull-right[
.center[
![:scale 60%](https://www.dropbox.com/s/csmheqe2xzp1dst/timehourglass.jpg?raw=1)
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# Marginal Products

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

- Like marginal utility

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.pull-right[
<img src="1.2-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" style="display: block; margin: auto;" />
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# 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 the slope of `$TP$` at every value of `$l$`!

- Note: via calculus: `$\frac{\partial q}{\partial l}$`
]

]

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# 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 the slope of `$TP$` at every value of `$k$`!

- Note: via calculus: `$\frac{\partial q}{\partial k}$`

]

]

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# Diminishing Returns

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

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

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

- .hi[Law of Diminishing Returns]: adding more of one factor of production *holding others constant* will result in successively lower increases in output<sup>.red[1]</sup>

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

]

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

- .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
`$$AP_k = \frac{q}{k}$$`
]

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# The Long Run

- In the long run, *all* factors of production are **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* `$k$`

]

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# The Firm's Problem

.pull-left[
- 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-purple[ < inputs and output >]

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

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

]

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# Costs in Economics are Opportunity Costs

- Costs in economics are different from common conception of "cost"
    - **Accounting cost**: monetary cost
    - **Economic cost**: value of next best use of resources given up (opportunity cost)

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