2.3: Cost Minimization

# 2.3: Cost Minimization
## ECON 306 · Microeconomic Analysis · Fall 2019
### Ryan Safner Assistant Professor of Economics <a href="mailto:safner@hood.edu"> safner@hood.edu</a> <a href="https://github.com/ryansafner/microf19"> ryansafner/microf19</a> <a href="https://microF19.classes.ryansafner.com"> microF19.classes.ryansafner.com</a>

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# Recall: The Firm's Two Problems

- First Stage: the **firm's profit maximization problem**:

1. **Choose:** .blue[ < output >]

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

- Second Stage: the **firm's cost minimization problem**:

1. **Choose:** .blue[ < inputs >]

2. **In order to _minimize_:** .green[< cost >]

3. **Subject to:** .red[< producing the optimal output >]

- Minimizing costs `$\iff$` maximizing profits

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# Solving the Cost Minimization Problem

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# The Firm's Cost Minimization Problem

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

1. **Choose:** .blue[ < inputs: `\$l, k\$`>]

2. **In order to maximize:** .green[< total cost: `\$$wl+rk\$` >]

3. **Subject to:** .red[< producing the optimal output: `\$q^*=f(l,k)\$` >]

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# The Cost Minimization Problem: Tools

- .blue[Choice]: combination of inputs `$(l, k)$`

- .red[Production function/isoquants]: firm's technological constraints
    - How the *firm* trades off between inputs
- .green[Isocost line]: firm's total cost (for given output and input prices)
    - How the *market* trades off between inputs
- The **firms's cost minimization problem**: choose `$l$` and `$k$` to minimize total cost subject to producing the optimal amount of output
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# The Cost Minimization Problem: Math

`$$\min_{l,k} wl+rk$$`
`$$s.t. q^*=f(l,k)$$`

- This requires calculus to solve. We will look at **graphs** instead!
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# The Firm's Least-Cost Input Combination: Graphically

- .shout[Graphical solution]: Lowest isocost line *tangent* to desired isoquant (A)

- B produces same output as A, but higher cost

- C is same cost as A, but produces less than desired output

- D produces is cheaper, but produces less than desired output

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<img src="12-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" />
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# The Firm's Least-Cost Input Combination: Why A?

`$$\begin{align*}
\text{Isoquant curve slope} &= \text{Isocost line slope} \\\end{align*}$$`

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<img src="12-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" />
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# The Firm's Least-Cost Input Combination: Why A?

`$$\begin{align*}
\text{Isoquant curve slope} &= \text{Isocost line slope} \\
| MRTS_{l,k} | &= | \frac{w}{r} | \\
| \frac{MP_l}{MP_k} | &= | \frac{w}{r} | \\
| -0.5 | &= | -0.5 | \\\end{align*}$$`

- *Firm* would exchange at same rate as market

- **No other combination** of `$(l,k)$` **exists** at current prices & output that could lower cost to produce `$q^*$`!

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<img src="12-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" />
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# Two Equivalent Rules

## Rule 1

`$$\frac{MP_l}{MP_k}  =  \frac{w}{r}$$`

- Easier for solving math problems (slope of each curve)

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<img src="12-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" />
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# Two Equivalent Rules

## Rule 1

`$$\frac{MP_l}{MP_k}  =  \frac{w}{r}$$`

- Easier for solving math problems (slope of each curve)

## Rule 2

`$$\frac{MP_l}{w}  =  \frac{MP_k}{r}$$`

- Easier for intuition (next slide)

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<img src="12-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" />
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# The Equimarginal Rule Again I

`$$\frac{MP_l}{w}  =  \frac{MP_k}{r} = \cdots = \frac{MP_n}{p_n}$$`

- .shout[Equimarginal Rule]: the cost of production is minimized where the **marginal product per dollar spent** is **equalized** across all `$n$` possible inputs

- Firm will always choose an option that gives higher marginal product (e.g. if `$MP_l > MP_k)$`
    - But each option has a different cost, so we weight each option by its cost, hence `$\frac{MP_l}{w}$`
    
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# The Equimarginal Rule Again II

`$$\frac{MP_l}{w}  =  \frac{MP_k}{r} = \cdots = \frac{MP_n}{p_n}$$`

- Why is this the optimum?

.content-box-green[
.green[**Example**]: suppose firm could get a higher marginal product per $1 spent on `$l$` than for `$k$` (i.e. "more bang for your buck"!)

- Not minimizing costs!
- Use more `$l$` and less `$k$`!
    - This will raise `$MP_k$` and lower `$MP_l$`!

- Continues until cost-adjusted marginal products are equalized

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# The Equimarginal Rule Again III

- Any .shout[optimum] in economics: no better alternatives exist under current constraints

- No possible change in your inputs to produce `$q^*$` that would lower cost

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.pull-right[
.center[
![:scale 100%](https://www.dropbox.com/s/qvr240t5j6t3arm/optimize.jpeg?raw=1)
]
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# The Firm's Least-Cost Input Combination: Example

Your firm can use labor l and capital k to produce output according to the production function:
`$$q=2lk$$`

The marginal products are:

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

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You want to produce 100 units, the price of labor is $10, and the price of capital is $5.

1. What is the least-cost combination of labor and capital that produces 100 units of output?
2. How much does this combination cost?

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# Returns to Scale

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# Returns to Scale

- The .shout[returns to scale] of production refers to the change in output when all inputs are increased *at the same rate*

`$$q=f(k,l) \rightarrow q=f(ck,cl) \text{ for some }c$$`

- .whisper[Constant returns to scale]: output increases at the same proportionate rate as inputs increase
    - e.g. if you double all inputs, output doubles
    `$$f(2k,2l)=2f(k,l)$$`

- .whisper[Increasing returns to scale]: output increases *more than* proportionately to the change in inputs
 - e.g. if you double all inputs, output *more than* doubles
 `$$f(ck,cl)<cf(k,l)$$`

- .whisper[Decreasing returns to scale]: output increases *less than* proportionately to the change in inputs
    - e.g. if you double all inputs, output *less than* doubles
    `$$f(ck,cl)>cf(k,l)$$`

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# Returns to Scale: Example

.content-box-green[
.green[**Example**:] Does each of the following production functions exhibit constant returns to scale, increasing returns to scale, or decreasing returns to scale?
]
1. `$q=4l+2k$`

2. `$q=2lk$`

3. `$q=2l^{0.3}k^{0.3}$`
 
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# Returns to Scale: Cobb-Douglas

- One reason we often use Cobb-Douglas production functions is to easily determine returns to scale:     
`$$q=Ak^\alpha l^\beta$$`

- `$\alpha + \beta = 1$`: constant returns to scale
- `$\alpha + \beta >1$`: increasing returns to scale
- `$\alpha + \beta <1$`: decreasing returns to scale
- Note this trick *only* works for Cobb-Douglas functions!

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# Cobb-Douglas: Constant Returns Case

- In the constant returns to scale case (most common), Cobb-Douglas is often written as:     
`$$q=Ak^\alpha l^{1-\alpha}$$`

- `$\alpha$` is the .whisper[output elasticity of capital]
    - A 1% increase in `$k$` leads to a `$\alpha$`% increase in `$q$`

- `$1-\alpha$` is the .whisper[output elasticity of labor]
    - A 1% increase in `$l$` leads to a `$(1-\alpha)$`% increase in `$q$`

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# Output-Expansion Paths & Cost Curves

Goolsbee et. al (2011: 246)
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- **Output Expansion Path**: curve illustrating the changes in the optimal mix of inputs and the total cost to produce an increasing amount of output

- **Total Cost curve**: curve showing the total cost of producing different amounts of output (next class)

- See next class' notes page to see how we go from our least-cost combinations over a range of outputs to derive a total cost function