2.3 — Cost Minimization

ECON 306 • Microeconomic Analysis • Spring 2022

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com

Recall: The Firm's Two Problems

1^st Stage: firm's profit maximization problem:

Choose: < output >
In order to maximize: < profits >

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

2^nd Stage: firm's cost minimization problem:

Choose: < inputs >
In order to minimize: < cost >
Subject to: < producing the optimal output >

Minimizing costs $⟺$ maximizing profits

Solving the Cost Minimization Problem

The Firm's Cost Minimization Problem

The firm's cost minimization problem is:

Choose: < inputs: $l, k$ >
In order to minimize: < total cost: $w l + r k$ >
Subject to: < producing the optimal output: $q^{*} = f (l, k)$ >

The Cost Minimization Problem: Tools

Our tools for firm's input choices:
Choice: combination of inputs $(l, k)$
Production function/isoquants: firm's technological constraints
- How the firm trades off between inputs
Isocost line: firm's total cost (for given output and input prices)
- How the market trades off between inputs

The Cost Minimization Problem: Verbally

The firms's cost minimization problem:

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

The Cost Minimization Problem: Math

$min_{l, k} w l + r k$ $s . t . q^{*} = f (l, k)$

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

The Firm's Least-Cost Input Combination: Graphically

Graphical solution: Lowest isocost line tangent to desired isoquant (A)

The Firm's Least-Cost Input Combination: Graphically

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 does not produce desired output
D is cheaper, does not produce desired output

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

$\begin{aligned} Isoquant curve slope & = Isocost line slope \end{aligned}$

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

$\begin{aligned} Isoquant curve slope & = Isocost line slope \\ M R T S_{l, k} & = \frac{w}{r} \\ \frac{M P_{l}}{M P_{k}} & = \frac{w}{r} \\ 0.5 & = 0.5 \end{aligned}$

Marginal benefit = Marginal cost
- Firm would exchange at same rate as market
No other combination of (l,k) exists at current prices & output that could produce $q^{⋆}$ at lower cost!

Two Equivalent Rules

Rule 1

$\frac{M P_{l}}{M P_{k}} = \frac{w}{r}$

Easier for calculation (slopes)

Two Equivalent Rules

Rule 1

$\frac{M P_{l}}{M P_{k}} = \frac{w}{r}$

Easier for calculation (slopes)

Rule 2

$\frac{M P_{l}}{w} = \frac{M P_{k}}{r}$

Easier for intuition (next slide)

The Equimarginal Rule Again I

$\frac{M P_{l}}{w} = \frac{M P_{k}}{r} = \dots = \frac{M P_{n}}{p_{n}}$

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 $M P_{l} > M P_{k})$
- But each option has a different cost, so we weight each option by its price, hence $\frac{M P_{n}}{p_{n}}$

The Equimarginal Rule Again II

Any optimum in economics: no better alternatives exist under current constraints
No possible change in your inputs to produce $q^{*}$ that would lower cost

The Firm's Least-Cost Input Combination: Example

Example:

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

The marginal products are:

$\begin{aligned} M P_{l} & = 2 k \\ M P_{k} & = 2 l \end{aligned}$

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

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

Returns to Scale

The returns to scale of production: change in output when all inputs are increased at the same rate (scale)

Returns to Scale

The returns to scale of production: change in output when all inputs are increased at the same rate (scale)
Constant returns to scale: output increases at same proportionate rate to inputs change
- e.g. double all inputs, output doubles

Returns to Scale

The returns to scale of production: change in output when all inputs are increased at the same rate (scale)
Constant returns to scale: output increases at same proportionate rate to inputs change
- e.g. double all inputs, output doubles
Increasing returns to scale: output increases more than proportionately to inputs change
- e.g. double all inputs, output more than doubles

Returns to Scale

The returns to scale of production: change in output when all inputs are increased at the same rate (scale)
Constant returns to scale: output increases at same proportionate rate to inputs change
- e.g. double all inputs, output doubles
Increasing returns to scale: output increases more than proportionately to inputs change
- e.g. double all inputs, output more than doubles
Decreasing returns to scale: output increases less than proportionately to inputs change
- e.g. double all inputs, output less than doubles

Returns to Scale: Example

Example: Do the following production functions exhibit constant returns to scale, increasing returns to scale, or decreasing returns to scale?

$q = 4 l + 2 k$
$q = 2 l k$
$q = 2 l^{0.3} k^{0.3}$

Returns to Scale: Cobb-Douglas

One reason Cobb-Douglas functions are great: easy to determine returns to scale:
$q = A k^{α} l^{β}$
$α + β = 1$ : constant returns to scale
$α + β > 1$ : increasing returns to scale
$α + β < 1$ : decreasing returns to scale
Note this trick only works for Cobb-Douglas functions!

Cobb-Douglas: Constant Returns Case

A common case of Cobb-Douglas is often written as:
$q = A k^{α} l^{1 - α}$ (i.e., the exponents sum to 1, constant returns)
$α$ is the output elasticity of capital
- A 1% increase in $k$ leads to an $α$ % increase in $q$
$1 - α$ is the output elasticity of labor
- A 1% increase in $l$ leads to a $(1 - α)$ % increase in $q$

Output-Expansion Paths & Cost Curves

Goolsbee et. al (2011: 246)

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

2.3 — Cost Minimization

ECON 306 • Microeconomic Analysis • Spring 2022

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com

Recall: The Firm's Two Problems

1^st Stage: firm's profit maximization problem:

Choose: < output >
In order to maximize: < profits >

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

2^nd Stage: firm's cost minimization problem:

Choose: < inputs >
In order to minimize: < cost >
Subject to: < producing the optimal output >

Minimizing costs $⟺$ maximizing profits

Solving the Cost Minimization Problem

The Firm's Cost Minimization Problem

The firm's cost minimization problem is:

Choose: < inputs: $l, k$ >
In order to minimize: < total cost: $w l + r k$ >
Subject to: < producing the optimal output: $q^{*} = f (l, k)$ >

The Cost Minimization Problem: Tools

Our tools for firm's input choices:
Choice: combination of inputs $(l, k)$
Production function/isoquants: firm's technological constraints
- How the firm trades off between inputs
Isocost line: firm's total cost (for given output and input prices)
- How the market trades off between inputs

The Cost Minimization Problem: Verbally

The firms's cost minimization problem:

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

The Cost Minimization Problem: Math

$min_{l, k} w l + r k$ $s . t . q^{*} = f (l, k)$

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

The Firm's Least-Cost Input Combination: Graphically

Graphical solution: Lowest isocost line tangent to desired isoquant (A)

The Firm's Least-Cost Input Combination: Graphically

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 does not produce desired output
D is cheaper, does not produce desired output

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

$\begin{aligned} Isoquant curve slope & = Isocost line slope \end{aligned}$

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

$\begin{aligned} Isoquant curve slope & = Isocost line slope \\ M R T S_{l, k} & = \frac{w}{r} \\ \frac{M P_{l}}{M P_{k}} & = \frac{w}{r} \\ 0.5 & = 0.5 \end{aligned}$

Marginal benefit = Marginal cost
- Firm would exchange at same rate as market
No other combination of (l,k) exists at current prices & output that could produce $q^{⋆}$ at lower cost!

Two Equivalent Rules

Rule 1

$\frac{M P_{l}}{M P_{k}} = \frac{w}{r}$

Easier for calculation (slopes)

Two Equivalent Rules

Rule 1

$\frac{M P_{l}}{M P_{k}} = \frac{w}{r}$

Easier for calculation (slopes)

Rule 2

$\frac{M P_{l}}{w} = \frac{M P_{k}}{r}$

Easier for intuition (next slide)

The Equimarginal Rule Again I

$\frac{M P_{l}}{w} = \frac{M P_{k}}{r} = \dots = \frac{M P_{n}}{p_{n}}$

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 $M P_{l} > M P_{k})$
- But each option has a different cost, so we weight each option by its price, hence $\frac{M P_{n}}{p_{n}}$

The Equimarginal Rule Again II

Any optimum in economics: no better alternatives exist under current constraints
No possible change in your inputs to produce $q^{*}$ that would lower cost

The Firm's Least-Cost Input Combination: Example

Example:

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

The marginal products are:

$\begin{aligned} M P_{l} & = 2 k \\ M P_{k} & = 2 l \end{aligned}$

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

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

Returns to Scale

The returns to scale of production: change in output when all inputs are increased at the same rate (scale)

Returns to Scale

The returns to scale of production: change in output when all inputs are increased at the same rate (scale)
Constant returns to scale: output increases at same proportionate rate to inputs change
- e.g. double all inputs, output doubles

Returns to Scale

The returns to scale of production: change in output when all inputs are increased at the same rate (scale)
Constant returns to scale: output increases at same proportionate rate to inputs change
- e.g. double all inputs, output doubles
Increasing returns to scale: output increases more than proportionately to inputs change
- e.g. double all inputs, output more than doubles

Returns to Scale

The returns to scale of production: change in output when all inputs are increased at the same rate (scale)
Constant returns to scale: output increases at same proportionate rate to inputs change
- e.g. double all inputs, output doubles
Increasing returns to scale: output increases more than proportionately to inputs change
- e.g. double all inputs, output more than doubles
Decreasing returns to scale: output increases less than proportionately to inputs change
- e.g. double all inputs, output less than doubles

Returns to Scale: Example

Example: Do the following production functions exhibit constant returns to scale, increasing returns to scale, or decreasing returns to scale?

$q = 4 l + 2 k$
$q = 2 l k$
$q = 2 l^{0.3} k^{0.3}$

Returns to Scale: Cobb-Douglas

One reason Cobb-Douglas functions are great: easy to determine returns to scale:
$q = A k^{α} l^{β}$
$α + β = 1$ : constant returns to scale
$α + β > 1$ : increasing returns to scale
$α + β < 1$ : decreasing returns to scale
Note this trick only works for Cobb-Douglas functions!

Cobb-Douglas: Constant Returns Case

A common case of Cobb-Douglas is often written as:
$q = A k^{α} l^{1 - α}$ (i.e., the exponents sum to 1, constant returns)
$α$ is the output elasticity of capital
- A 1% increase in $k$ leads to an $α$ % increase in $q$
$1 - α$ is the output elasticity of labor
- A 1% increase in $l$ leads to a $(1 - α)$ % increase in $q$

Output-Expansion Paths & Cost Curves

Goolsbee et. al (2011: 246)

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

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2.3 — Cost Minimization

ECON 306 • Microeconomic Analysis • Spring 2022

Ryan Safner Assistant Professor of Economics safner@hood.edu ryansafner/microS22 microS22.classes.ryansafner.com

Recall: The Firm's Two Problems

Solving the Cost Minimization Problem

The Firm's Cost Minimization Problem

The Cost Minimization Problem: Tools

The Cost Minimization Problem: Verbally

The Cost Minimization Problem: Math

The Firm's Least-Cost Input Combination: Graphically

The Firm's Least-Cost Input Combination: Graphically

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

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

Two Equivalent Rules

Rule 1

Two Equivalent Rules

Rule 1

Rule 2

The Equimarginal Rule Again I

The Equimarginal Rule Again II

The Firm's Least-Cost Input Combination: Example

Returns to Scale

Returns to Scale

Returns to Scale

Returns to Scale

Returns to Scale

Returns to Scale: Example

Returns to Scale: Cobb-Douglas

Cobb-Douglas: Constant Returns Case

Output-Expansion Paths & Cost Curves

Recall: The Firm's Two Problems

Help

2.3 — Cost Minimization

2.3 — Cost Minimization

ECON 306 • Microeconomic Analysis • Spring 2022

Ryan Safner Assistant Professor of Economics safner@hood.edu ryansafner/microS22 microS22.classes.ryansafner.com

Recall: The Firm's Two Problems

Solving the Cost Minimization Problem

The Firm's Cost Minimization Problem

The Cost Minimization Problem: Tools

The Cost Minimization Problem: Verbally

The Cost Minimization Problem: Math

The Firm's Least-Cost Input Combination: Graphically

The Firm's Least-Cost Input Combination: Graphically

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

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

Two Equivalent Rules

Rule 1

Two Equivalent Rules

Rule 1

Rule 2

The Equimarginal Rule Again I

The Equimarginal Rule Again II

The Firm's Least-Cost Input Combination: Example

Returns to Scale

Returns to Scale

Returns to Scale

Returns to Scale

Returns to Scale

Returns to Scale: Example

Returns to Scale: Cobb-Douglas

Cobb-Douglas: Constant Returns Case

Output-Expansion Paths & Cost Curves

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com

Ryan Safner
Assistant Professor of Economics
safner@hood.edu
ryansafner/microS22
microS22.classes.ryansafner.com