2.3 — Cost Minimization

# 2.3 — Cost Minimization

## ECON 306 • Microeconomic Analysis • Spring 2023

### Ryan Safner Associate Professor of Economics <a href="mailto:safner@hood.edu">safner@hood.edu</a> <a href="https://github.com/ryansafner/microS23">ryansafner/microS23</a> <a href="https://microS23.classes.ryansafner.com"> microS23.classes.ryansafner.com</a>

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# Recall: The Firm's Two Problems

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

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

- We'll cover this later...first we'll explore:

2nd Stage: .hi[firm's cost minimization problem]:

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

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

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

- Minimizing costs `$\iff$` maximizing profits
]
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# 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 minimize:** .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

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

- .hi-red[Production function/isoquants]: firm's technological constraints
    - How the *firm* trades off between inputs
- .hi-green[Isocost line]: firm's total cost (for given output and input prices)
    - How the *market* trades off between inputs
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# The Cost Minimization Problem: Verbally

> choose a combination of `$l$` and `$k$` to minimize total cost that produces the optimal amount of output

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# The Cost Minimization Problem: Math

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

- This requires calculus to solve. We will look at **graphs** instead!
]

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# The Firm's Least-Cost Input Combination: Graphically

- .hi[Graphical solution]: **Lowest** .hi-red[isocost line] **_tangent_ to desired** .hi-green[isoquant] (A)

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<img src="2.3-slides_files/figure-html/soln-1.png" width="504" style="display: block; margin: auto;" />
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# The Firm's Least-Cost Input Combination: Graphically

- .hi[Graphical solution]: **Lowest** .hi-red[isocost line] **_tangent_ to desired** .hi-green[isoquant] (A)

- B produces same output as A, but higher cost

- C is same cost as A, but does not produce desired output

- D is cheaper, does not produce desired output

]

<img src="2.3-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" style="display: block; margin: auto;" />
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# The Firm's Least-Cost Input Combination: Why A?

.pull-left[
.smallest[
`$$\begin{align*}
\color{green}{\text{Isoquant curve slope}} &= \color{red}{\text{Isocost line slope}} \\ \end{align*}$$`
]
]

<img src="2.3-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" style="display: block; margin: auto;" />
]

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# The Firm's Least-Cost Input Combination: Why A?

.pull-left[
.smallest[
`$$\begin{align*}
\color{green}{\text{Isoquant curve slope}} &= \color{red}{\text{Isocost line slope}} \\
\color{green}{MRTS_{l,k}} &= \color{red}{\frac{w}{r}} \\
\color{green}{\frac{MP_l}{MP_k}} &= \color{red}{\frac{w}{r}} \\
\color{green}{0.5} &= \color{red}{0.5 } \\\end{align*}$$`

- .hi-green[Marginal benefit] = .hi-red[Marginal cost]
  - .hi-green[Firm] would exchange at same rate as .hi-red[market]

- .hi-purple[No other combination of (l,k) exists at current prices & output that could produce `\$q^\star\$` at lower cost!]
]
]

<img src="2.3-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" style="display: block; margin: auto;" />
]

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# Two Equivalent Rules

## Rule 1

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

- Easier for calculation (slopes)

]

<img src="2.3-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" style="display: block; margin: auto;" />
]

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# Two Equivalent Rules

## Rule 1

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

- Easier for calculation (slopes)

## Rule 2

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

- Easier for intuition (next slide)

]

<img src="2.3-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" style="display: block; margin: auto;" />
]

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# The Equimarginal Rule Again I

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

- .hi[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 price, hence `$\frac{MP_n}{p_n}$`
    
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# The Equimarginal Rule Again II

- Any .hi[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

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.smallest[
.green[**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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]

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

.pull-left[
.quitesmall[
- The .hi[returns to scale] of production: change in output when **all** inputs are increased .hi-turquoise[at the same rate] (scale)
]
]

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# Returns to Scale

.pull-left[
.quitesmall[
- The .hi[returns to scale] of production: change in output when **all** inputs are increased .hi-turquoise[at the same rate] (scale)

- .hi-purple[Constant returns to scale]: output increases at .hi-turquoise[same proportionate rate] to inputs change
    - e.g. double all inputs, output doubles
]
]

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# Returns to Scale

.pull-left[
.quitesmall[
- The .hi[returns to scale] of production: change in output when **all** inputs are increased .hi-turquoise[at the same rate] (scale)

- .hi-purple[Constant returns to scale]: output increases at .hi-turquoise[same proportionate rate] to inputs change
    - e.g. double all inputs, output doubles

- .hi-purple[Increasing returns to scale]: output increases .hi-turquoise[more than proportionately] to inputs change
    - e.g. double all inputs, output *more than* doubles
]
]

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# Returns to Scale

.pull-left[
.quitesmall[
- The .hi[returns to scale] of production: change in output when **all** inputs are increased .hi-turquoise[at the same rate] (scale)

- .hi-purple[Constant returns to scale]: output increases at .hi-turquoise[same proportionate rate] to inputs change
    - e.g. double all inputs, output doubles

- .hi-purple[Increasing returns to scale]: output increases .hi-turquoise[more than proportionately] to inputs change.magenta[†]
 - e.g. double all inputs, output *more than* doubles

- .hi-purple[Decreasing returns to scale]: output increases .hi-turquoise[less than proportionately] to inputs change
    - e.g. double all inputs, output *less than* doubles
]
]

.footnote[.magenta[†] See my new newsletter [Increasing Returns](https://increasingreturns.substack.com) for more on the importance of this idea
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# Returns to Scale: Example

.bg-washed-green.b--dark-green.ba.bw2.br3.shadow-5.ph4.mt5[
.green[**Example**:] Do 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 Cobb-Douglas functions are great: easy to 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

.pull-left[
.smallest[
- A common case of Cobb-Douglas is often written as:     
`$$q=Ak^\alpha l^{1-\alpha}$$`
(i.e., the exponents sum to 1, constant returns)

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

- `$1-\alpha$` is the .hi-purple[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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.smallest[
- **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

]