class: center, middle, inverse, title-slide # 2.3 — Cost Minimization ## ECON 306 · Microeconomic Analysis · Fall 2020 ### Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microF20
microF20.classes.ryansafner.com
--- # Recall: The Firm's Two Problems .pull-left[ .smallest[ - 1<sup>st</sup> Stage: .hi[firm's profit maximization problem]: 1. **Choose:** .hi-blue[ < output >] 2. **In order to maximize:** .hi-green[< profits >] - We'll cover this later...first we'll explore: - 2<sup>nd</sup> Stage: .hi[firm's cost minimization problem]: 1. **Choose:** .hi-blue[ < inputs >] 2. **In order to _minimize_:** .hi-green[< cost >] 3. **Subject to:** .hi-red[< producing the optimal output >] - Minimizing costs `\(\iff\)` maximizing profits ] ] .pull-right[ .center[ ![](https://www.dropbox.com/s/zwows56ydpqj9u6/management.jpg?raw=1) ] ] --- class: inverse, center, middle # Solving the Cost Minimization Problem --- # The Firm's Cost Minimization Problem .pull-left[ - 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)\\)` >] ] .pull-right[ .center[ ![](https://www.dropbox.com/s/nw0v6bsho0ab4zq/choices.jpg?raw=1) ] ] --- # The Cost Minimization Problem: Tools .pull-left[ - Our tools for firm's input choices: - .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 ] .pull-right[ .center[ ![](https://www.dropbox.com/s/nw0v6bsho0ab4zq/choices.jpg?raw=1) ] ] --- # The Cost Minimization Problem: Verbally .pull-left[ - The .hi[firms's cost minimization problem]: > choose a combination of `\(l\)` and `\(k\)` to minimize total cost that produces the optimal amount of output ] .pull-right[ .center[ ![](https://www.dropbox.com/s/nw0v6bsho0ab4zq/choices.jpg?raw=1) ] ] --- # The Cost Minimization Problem: Math .pull-left[ `$$\min_{l,k} wl+rk$$` `$$s.t. q^*=f(l,k)$$` - This requires calculus to solve. We will look at **graphs** instead! ] .pull-right[ .center[ ![](https://www.dropbox.com/s/nw0v6bsho0ab4zq/choices.jpg?raw=1) ] ] --- # The Firm's Least-Cost Input Combination: Graphically .pull-left[ - .hi[Graphical solution]: **Lowest** .hi-red[isocost line] **_tangent_ to desired** .hi-green[isoquant] (A) ] .pull-right[ <img src="2.3-slides_files/figure-html/unnamed-chunk-1-1.png" width="504" /> ] --- # The Firm's Least-Cost Input Combination: Graphically .pull-left[ - .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 produces less than desired output - D produces is cheaper, but produces less than desired output ] .pull-right[ <img src="2.3-slides_files/figure-html/unnamed-chunk-2-1.png" width="504" /> ] --- # The Firm's Least-Cost Input Combination: Why A? .pull-left[ `$$\begin{align*} \color{green}{\text{Isoquant curve slope}} &= \color{red}{\text{Isocost line slope}} \\ \end{align*}$$` ] .pull-right[ <img src="2.3-slides_files/figure-html/unnamed-chunk-3-1.png" width="504" /> ] --- # The Firm's Least-Cost Input Combination: Why A? .pull-left[ `$$\begin{align*} \color{green}{\text{Isoquant curve slope}} &= \color{red}{\text{Isocost line slope}} \\ \color{green}{| MRTS_{l,k} |} &= \color{red}{| \frac{w}{r} |} \\ \color{green}{| \frac{MP_l}{MP_k} |} &= \color{red}{| \frac{w}{r} |} \\ \color{green}{| -0.5 |} &= \color{red}{| -0.5 |} \\\end{align*}$$` - .hi-green[Firm] would exchange at same rate as .hi-red[market] - **No other combination** of `\((l,k)\)` **exists** at current prices & output that could lower cost to produce `\(q^*\)`! ] .pull-right[ <img src="2.3-slides_files/figure-html/unnamed-chunk-4-1.png" width="504" /> ] --- # Two Equivalent Rules .pull-left[ ## Rule 1 `$$\frac{MP_l}{MP_k} = \frac{w}{r}$$` - Easier for solving math problems ] .pull-right[ <img src="2.3-slides_files/figure-html/unnamed-chunk-5-1.png" width="504" /> ] --- # Two Equivalent Rules .pull-left[ ## Rule 1 `$$\frac{MP_l}{MP_k} = \frac{w}{r}$$` - Easier for solving math problems ## Rule 2 `$$\frac{MP_l}{w} = \frac{MP_k}{r}$$` - Easier for intuition (next slide) ] .pull-right[ <img src="2.3-slides_files/figure-html/unnamed-chunk-6-1.png" width="504" /> ] --- # 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 cost, hence `\(\frac{MP_n}{p_n}\)` --- # The Equimarginal Rule Again II `$$\frac{MP_l}{w} = \frac{MP_k}{r} = \cdots = \frac{MP_n}{p_n}$$` - Why is this the optimum? - .hi-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! - Should use more `\(l\)` and less `\(k\)`! - This will raise `\(MP_k\)` and lower `\(MP_l\)`! - Continue until cost-adjusted marginal products are equalized --- # The Equimarginal Rule Again III .pull-left[ - 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 ] .pull-right[ .center[ ![:scale 100%](https://www.dropbox.com/s/qvr240t5j6t3arm/optimize.jpeg?raw=1) ] ] --- # The Firm's Least-Cost Input Combination: Example .content-box-green[ .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: `$$\begin{align*} MP_l&=2k\\ MP_k&=2l\\\end{align*}$$` ] ] .smallest[ 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? ] --- class: inverse, center, middle # Returns to Scale --- # Returns to Scale - The .hi[returns to scale] of production refers to the change in output when all inputs are increased *at the same rate* -- - .hi-purple[Constant returns to scale]: output increases at same proportionate rate as inputs increase - e.g. if you double all inputs, output doubles -- - .hi-purple[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 -- - .hi-purple[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 --- # 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. `\(q=2l^{0.3}k^{0.3}\)` --- # 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! --- # 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 .hi-purple[output elasticity of capital] - A 1% increase in `\(k\)` leads to a `\(\alpha\)`% 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 `\(q\)` --- # Output-Expansion Paths & Cost Curves .center[ ![](https://www.dropbox.com/s/0lccry50i7x8shl/outputexpansion.png?raw=1) Goolsbee et. al (2011: 246) ] .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 ]