2.2 — Production Technology

ECON 306 • Microeconomic Analysis • Spring 2023

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microS23
microS23.classes.ryansafner.com

Outline

The “Runs” of Production

“Time”-frame usefully divided between short vs. long run analysis
Short run: at least one factor of production is fixed (too costly to change) $q = f (\bar{k}, l)$
- Assume capital is fixed (i.e. number of factories, storefronts, etc)
- Short-run decisions only about using labor

The “Runs” of Production

“Time”-frame usefully divided between short vs. long run analysis
Long run: all factors of production are variable (can be changed) $q = f (k, l)$

Production in the Short Run

Production in the Short Run: Example

Example: Consider a firm with the production function $q = k^{0.5} l^{0.5}$

Suppose in the short run, the firm has 4 units of capital.

Derive the short run production function.
What is the total product (output) that can be made with 4 workers?
What is the total product (output) that can be made with 5 workers?

Production in the Short Run: Example

Example: Consider a firm with the production function $q = k^{0.5} l^{0.5}$

Suppose in the short run, the firm has 4 units of capital.

Derive the short run production function.
What is the total product (output) that can be made with 4 workers?
What is the total product (output) that can be made with 5 workers?

Marginal Products

The marginal product of an input is the additional output produced by one more unit of that input (holding all other inputs constant)
Like marginal utility
Similar to marginal utilities, I will give you the marginal product equations

Marginal Product of Labor

Marginal product of labor $(M P_{l})$ : additional output produced by adding one more unit of labor (holding $k$ constant) $M P_{l} = \frac{Δ q}{Δ l}$
$M P_{l}$ is slope of $T P$ at each value of $l$ !
- Note: via calculus: $\frac{\partial q}{\partial l}$

Marginal Product of Capital

Marginal product of capital $(M P_{k})$ : additional output produced by adding one more unit of capital (holding $l$ constant) $M P_{k} = \frac{Δ q}{Δ k}$
$M P_{k}$ is slope of $T P$ at each value of $k$ !
- Note: via calculus: $\frac{\partial q}{\partial k}$
Note we don't consider capital in the short run!

Diminishing Returns

Law of Diminishing Returns: adding more of one factor of production holding all others constant will result in successively lower increases in output
In order to increase output, firm will need to increase all factors!

Diminishing Returns

Law of Diminishing Returns: adding more of one factor of production holding all others constant will result in successively lower increases in output
In order to increase output, firm will need to increase all factors!

Average Product of Labor (and Capital)

Average product of labor $(A P_{l})$ : total output per worker $A P_{l} = \frac{q}{l}$
A measure of labor productivity
Average product of capital $(A P_{k})$ : total output per unit of capital $A P_{k} = \frac{q}{k}$

The Firm's Problem: Long Run

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 $k$

The Firm's Problem

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:

Choose: < inputs and output >
In order to maximize: < profits >
Subject to: < technology >

It's actually much easier to break this into 2 stages. See today’s class notes page for an example using only one stage.

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:

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

Long Run Production

Example: $q = \sqrt{l k}$

Many input-combinations yield the same output!
So how does the firm choose the optimal combination?

Mapping Input-Combination Choices Graphically

3-D Production Function

2-D Isoquant Contours

Isoquants and MRTS

Isoquant Curves

We can draw an isoquant indicating all combinations of $l$ and $k$ that yield the same $q$

Isoquant Curves

We can draw an isoquant indicating all combinations of $l$ and $k$ that yield the same $q$
Combinations above curve yield more output; on a higher curve
- $D > A = B = C$

Isoquant Curves

We can draw an isoquant indicating all combinations of $l$ and $k$ that yield the same $q$
Combinations above curve yield more output; on a higher curve
- $D > A = B = C$
Combinations below the curve yield less output; on a lower curve
- $E < A = B = C$

Marginal Rate of Technical Substitution I

If your firm uses fewer workers, how much more capital would it need to produce the same amount?

Marginal Rate of Technical Substitution I

If your firm uses fewer workers, how much more capital would it need to produce the same amount?
Marginal Rate of Technical Substitution (MRTS): rate at which firm trades off one input for another to yield same output
Firm's relative value of using $l$ in production based on its tech:

“We could give up (MRTS) units of k to use 1 more unit of l to produce the same output.”

Marginal Rate of Technical Substitution II

MRTS is the slope of the isoquant $M R T S_{l, k} = - \frac{Δ k}{Δ l} = \frac{r i s e}{r u n}$
Amount of $k$ given up for 1 more $l$
Note: slope (MRTS) changes along the curve!
Law of diminishing returns!

MRTS and Marginal Products

Relationship between $M P$ and $M R T S$ :

$\begin{aligned} \underset{M R T S}{\underset{⏟}{\frac{Δ k}{Δ l}}} & = - \frac{M P_{l}}{M P_{k}} \end{aligned}$

See proof in today's class notes
Sound familiar? 🧐

Special Case I: Perfect Substitutes

Example: Consider Bank Tellers $(l)$ and ATMs $(k)$

Suppose 1 ATM can do the work of 2 bank tellers
Perfect substitutes: inputs that can be substituted at same fixed rate and yield same output
$M R T S_{l, k} = - 0.5$ (a constant!)

Special Case II: Perfect Complements

Example: Consider buses $(k)$ and bus drivers $(l)$

Must combine together in fixed proportions (1:1)
Perfect complements: inputs must be used together in same fixed proportion to produce output
$M R T S_{l, k}$ : ?

Common Case: Cobb-Douglas Production Functions

Again: very common functional form in economics is Cobb-Douglas

$q = A k^{a} l^{b}$

Where $a, b > 0$
- often $a + b = 1$
$A$ is total factor productivity

Practice

Example: Suppose a firm has the following production function: $q = 2 l k$

Where its marginal products are:

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

Put $l$ on the horizontal axis and $k$ on the vertical axis. Write an equation for $M R T S_{l, k}$ .
Would input combinations of $(1, 4)$ and $(2, 2)$ be on the same isoquant?
Sketch a graph of the isoquant from part 2.

Isocost Lines

If your firm can choose among many input combinations to produce $q$ , which combinations are optimal?
Those combination that are cheapest
Denote prices of each input as:
- $w$ : price of labor (wage)
- $r$ : price of capital
Let $C$ be total cost of using inputs $(l, k)$ at market prices $(w, r)$ to produce $q$ units of output:

$C (w, r, q) = w l + r k$

The Isocost Line, Graphically

$w l + r k = C$

The Isocost Line, Graphically

$w l + r k = C$

Solve for $k$ to graph

$k = \frac{C}{r} - \frac{w}{r} l$

The Isocost Line, Graphically

$w l + r k = C$

Solve for $k$ to graph

$k = \frac{C}{r} - \frac{w}{r} l$

Vertical-intercept: $\frac{C}{r}$
Horizontal-intercept: $\frac{C}{w}$

The Isocost Line, Graphically

$w l + r k = C$

Solve for $k$ to graph

$k = \frac{C}{r} - \frac{w}{r} l$

Vertical-intercept: $\frac{C}{r}$
Horizontal-intercept: $\frac{C}{w}$
slope: $- \frac{w}{r}$

The Isocost Line: Example

Example: Suppose your firm has a purchasing budget of $50. Market wages are $5/worker-hour and the mark rental rate of capital is $10/machine-hour. Let $l$ be on the horizontal axis and $k$ be on the vertical axis.

Write an equation for the isocost line (in graphable form).
Graph the isocost line.

Interpreting the Isocost Line

Points on the line are same total cost
- A: $$ 5 (0 l) + $ 10 (5 k) = $ 50$
- B: $$ 5 (10 l) + $ 10 (0 k) = $ 50$
- C: $$ 5 (2 l) + $ 10 (4 k) = $ 50$
- D: $$ 5 (6 l) + $ 10 (2 k) = $ 50$

Interpreting the Isocost Line

Points on the line are same total cost
- A: $$ 5 (0 l) + $ 10 (5 k) = $ 50$
- B: $$ 5 (10 l) + $ 10 (0 k) = $ 50$
- C: $$ 5 (2 l) + $ 10 (4 k) = $ 50$
- D: $$ 5 (6 l) + $ 10 (2 k) = $ 50$
Points beneath the line are cheaper (but may produce less)
- E: $$ 5 (3 l) + $ 10 (2 k) = $ 35$

Interpreting the Isocost Line

Points on the line are same total cost
- A: $$ 5 (0 l) + $ 10 (5 k) = $ 50$
- B: $$ 5 (10 l) + $ 10 (0 k) = $ 50$
- C: $$ 5 (2 l) + $ 10 (4 k) = $ 50$
- D: $$ 5 (6 l) + $ 10 (2 k) = $ 50$
Points beneath the line are cheaper (but may produce less)
- E: $$ 5 (3 l) + $ 10 (2 k) = $ 35$
Points above the line are more expensive (and may produce more)
- F: $$ 5 (6 l) + $ 10 (4 k) = $ 70$

Interpretting the Slope

Slope: tradeoff between $l$ and $k$ at market prices
- Market “exchange rate” between $l$ and $k$
Relative price of $l$ or the opportunity cost of $l$ :

Hiring 1 more unit of $l$ requires giving up $(\frac{w}{r})$ units of $k$

Changes in Relative Factor Prices I

Changes in relative factor prices: rotate the line

Example: An increase in the price of $l$

Slope changes: $- \frac{w^{'}}{r}$

Changes in Relative Factor Prices II

Changes in relative factor prices: rotate the line

Example: An increase in the price of $k$

Slope changes: $- \frac{w}{r^{'}}$

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2.2 — Production Technology

ECON 306 • Microeconomic Analysis • Spring 2023

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microS23
microS23.classes.ryansafner.com

Outline

Production in the Short Run

The Firm's Problem: Long Run

Isoquants and MRTS

Isocost Lines

The “Runs” of Production

“Time”-frame usefully divided between short vs. long run analysis
Short run: at least one factor of production is fixed (too costly to change) $q = f (\bar{k}, l)$
- Assume capital is fixed (i.e. number of factories, storefronts, etc)
- Short-run decisions only about using labor

The “Runs” of Production

“Time”-frame usefully divided between short vs. long run analysis
Long run: all factors of production are variable (can be changed) $q = f (k, l)$

Production in the Short Run

Production in the Short Run: Example

Example: Consider a firm with the production function $q = k^{0.5} l^{0.5}$

Suppose in the short run, the firm has 4 units of capital.

Derive the short run production function.
What is the total product (output) that can be made with 4 workers?
What is the total product (output) that can be made with 5 workers?

Production in the Short Run: Example

Example: Consider a firm with the production function $q = k^{0.5} l^{0.5}$

Suppose in the short run, the firm has 4 units of capital.

Derive the short run production function.
What is the total product (output) that can be made with 4 workers?
What is the total product (output) that can be made with 5 workers?

Marginal Products

The marginal product of an input is the additional output produced by one more unit of that input (holding all other inputs constant)
Like marginal utility
Similar to marginal utilities, I will give you the marginal product equations

Marginal Product of Labor

Marginal product of labor $(M P_{l})$ : additional output produced by adding one more unit of labor (holding $k$ constant) $M P_{l} = \frac{Δ q}{Δ l}$
$M P_{l}$ is slope of $T P$ at each value of $l$ !
- Note: via calculus: $\frac{\partial q}{\partial l}$

Marginal Product of Capital

Marginal product of capital $(M P_{k})$ : additional output produced by adding one more unit of capital (holding $l$ constant) $M P_{k} = \frac{Δ q}{Δ k}$
$M P_{k}$ is slope of $T P$ at each value of $k$ !
- Note: via calculus: $\frac{\partial q}{\partial k}$
Note we don't consider capital in the short run!

Diminishing Returns

Law of Diminishing Returns: adding more of one factor of production holding all others constant will result in successively lower increases in output
In order to increase output, firm will need to increase all factors!

Diminishing Returns

Law of Diminishing Returns: adding more of one factor of production holding all others constant will result in successively lower increases in output
In order to increase output, firm will need to increase all factors!

Average Product of Labor (and Capital)

Average product of labor $(A P_{l})$ : total output per worker $A P_{l} = \frac{q}{l}$
A measure of labor productivity
Average product of capital $(A P_{k})$ : total output per unit of capital $A P_{k} = \frac{q}{k}$

The Firm's Problem: Long Run

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 $k$

The Firm's Problem

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:

Choose: < inputs and output >
In order to maximize: < profits >
Subject to: < technology >

It's actually much easier to break this into 2 stages. See today’s class notes page for an example using only one stage.

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:

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

Long Run Production

Example: $q = \sqrt{l k}$

Many input-combinations yield the same output!
So how does the firm choose the optimal combination?

Mapping Input-Combination Choices Graphically

3-D Production Function

2-D Isoquant Contours

Isoquants and MRTS

Isoquant Curves

We can draw an isoquant indicating all combinations of $l$ and $k$ that yield the same $q$

Isoquant Curves

We can draw an isoquant indicating all combinations of $l$ and $k$ that yield the same $q$
Combinations above curve yield more output; on a higher curve
- $D > A = B = C$

Isoquant Curves

We can draw an isoquant indicating all combinations of $l$ and $k$ that yield the same $q$
Combinations above curve yield more output; on a higher curve
- $D > A = B = C$
Combinations below the curve yield less output; on a lower curve
- $E < A = B = C$

Marginal Rate of Technical Substitution I

If your firm uses fewer workers, how much more capital would it need to produce the same amount?

Marginal Rate of Technical Substitution I

If your firm uses fewer workers, how much more capital would it need to produce the same amount?
Marginal Rate of Technical Substitution (MRTS): rate at which firm trades off one input for another to yield same output
Firm's relative value of using $l$ in production based on its tech:

“We could give up (MRTS) units of k to use 1 more unit of l to produce the same output.”

Marginal Rate of Technical Substitution II

MRTS is the slope of the isoquant $M R T S_{l, k} = - \frac{Δ k}{Δ l} = \frac{r i s e}{r u n}$
Amount of $k$ given up for 1 more $l$
Note: slope (MRTS) changes along the curve!
Law of diminishing returns!

MRTS and Marginal Products

Relationship between $M P$ and $M R T S$ :

$\begin{aligned} \underset{M R T S}{\underset{⏟}{\frac{Δ k}{Δ l}}} & = - \frac{M P_{l}}{M P_{k}} \end{aligned}$

See proof in today's class notes
Sound familiar? 🧐

Special Case I: Perfect Substitutes

Example: Consider Bank Tellers $(l)$ and ATMs $(k)$

Suppose 1 ATM can do the work of 2 bank tellers
Perfect substitutes: inputs that can be substituted at same fixed rate and yield same output
$M R T S_{l, k} = - 0.5$ (a constant!)

Special Case II: Perfect Complements

Example: Consider buses $(k)$ and bus drivers $(l)$

Must combine together in fixed proportions (1:1)
Perfect complements: inputs must be used together in same fixed proportion to produce output
$M R T S_{l, k}$ : ?

Common Case: Cobb-Douglas Production Functions

Again: very common functional form in economics is Cobb-Douglas

$q = A k^{a} l^{b}$

Where $a, b > 0$
- often $a + b = 1$
$A$ is total factor productivity

Practice

Example: Suppose a firm has the following production function: $q = 2 l k$

Where its marginal products are:

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

Put $l$ on the horizontal axis and $k$ on the vertical axis. Write an equation for $M R T S_{l, k}$ .
Would input combinations of $(1, 4)$ and $(2, 2)$ be on the same isoquant?
Sketch a graph of the isoquant from part 2.

Isocost Lines

If your firm can choose among many input combinations to produce $q$ , which combinations are optimal?
Those combination that are cheapest
Denote prices of each input as:
- $w$ : price of labor (wage)
- $r$ : price of capital
Let $C$ be total cost of using inputs $(l, k)$ at market prices $(w, r)$ to produce $q$ units of output:

$C (w, r, q) = w l + r k$

The Isocost Line, Graphically

$w l + r k = C$

The Isocost Line, Graphically

$w l + r k = C$

Solve for $k$ to graph

$k = \frac{C}{r} - \frac{w}{r} l$

The Isocost Line, Graphically

$w l + r k = C$

Solve for $k$ to graph

$k = \frac{C}{r} - \frac{w}{r} l$

Vertical-intercept: $\frac{C}{r}$
Horizontal-intercept: $\frac{C}{w}$

The Isocost Line, Graphically

$w l + r k = C$

Solve for $k$ to graph

$k = \frac{C}{r} - \frac{w}{r} l$

Vertical-intercept: $\frac{C}{r}$
Horizontal-intercept: $\frac{C}{w}$
slope: $- \frac{w}{r}$

The Isocost Line: Example

Write an equation for the isocost line (in graphable form).
Graph the isocost line.

Interpreting the Isocost Line

Points on the line are same total cost
- A: $$ 5 (0 l) + $ 10 (5 k) = $ 50$
- B: $$ 5 (10 l) + $ 10 (0 k) = $ 50$
- C: $$ 5 (2 l) + $ 10 (4 k) = $ 50$
- D: $$ 5 (6 l) + $ 10 (2 k) = $ 50$

Interpreting the Isocost Line

Points on the line are same total cost
- A: $$ 5 (0 l) + $ 10 (5 k) = $ 50$
- B: $$ 5 (10 l) + $ 10 (0 k) = $ 50$
- C: $$ 5 (2 l) + $ 10 (4 k) = $ 50$
- D: $$ 5 (6 l) + $ 10 (2 k) = $ 50$
Points beneath the line are cheaper (but may produce less)
- E: $$ 5 (3 l) + $ 10 (2 k) = $ 35$

Interpreting the Isocost Line

Points on the line are same total cost
- A: $$ 5 (0 l) + $ 10 (5 k) = $ 50$
- B: $$ 5 (10 l) + $ 10 (0 k) = $ 50$
- C: $$ 5 (2 l) + $ 10 (4 k) = $ 50$
- D: $$ 5 (6 l) + $ 10 (2 k) = $ 50$
Points beneath the line are cheaper (but may produce less)
- E: $$ 5 (3 l) + $ 10 (2 k) = $ 35$
Points above the line are more expensive (and may produce more)
- F: $$ 5 (6 l) + $ 10 (4 k) = $ 70$

Interpretting the Slope

Slope: tradeoff between $l$ and $k$ at market prices
- Market “exchange rate” between $l$ and $k$
Relative price of $l$ or the opportunity cost of $l$ :

Hiring 1 more unit of $l$ requires giving up $(\frac{w}{r})$ units of $k$

Changes in Relative Factor Prices I

Changes in relative factor prices: rotate the line

Example: An increase in the price of $l$

Slope changes: $- \frac{w^{'}}{r}$

Changes in Relative Factor Prices II

Changes in relative factor prices: rotate the line

Example: An increase in the price of $k$

Slope changes: $- \frac{w}{r^{'}}$

2.2 — Production Technology

ECON 306 • Microeconomic Analysis • Spring 2023

Ryan Safner Associate Professor of Economics safner@hood.edu ryansafner/microS23 microS23.classes.ryansafner.com

Outline

The “Runs” of Production

The “Runs” of Production

Production in the Short Run

Production in the Short Run: Example

Production in the Short Run: Example

Marginal Products

Marginal Product of Labor

Marginal Product of Capital

Diminishing Returns

Diminishing Returns

Average Product of Labor (and Capital)

The Firm's Problem: Long Run

The Long Run

The Firm's Problem

The Firm's Two Problems

The Firm's Two Problems

Long Run Production

Mapping Input-Combination Choices Graphically

Isoquants and MRTS

Isoquant Curves

Isoquant Curves

Isoquant Curves

Marginal Rate of Technical Substitution I

Marginal Rate of Technical Substitution I

Marginal Rate of Technical Substitution II

Marginal Rate of Technical Substitution II

MRTS and Marginal Products

Special Case I: Perfect Substitutes

Special Case II: Perfect Complements

Common Case: Cobb-Douglas Production Functions

Practice

Isocost Lines

Isocost Lines

The Isocost Line, Graphically

The Isocost Line, Graphically

The Isocost Line, Graphically

The Isocost Line, Graphically

The Isocost Line: Example

Interpreting the Isocost Line

Interpreting the Isocost Line

Interpreting the Isocost Line

Interpretting the Slope

Changes in Relative Factor Prices I

Changes in Relative Factor Prices II

Outline

Help

2.2 — Production Technology

2.2 — Production Technology

ECON 306 • Microeconomic Analysis • Spring 2023

Ryan Safner Associate Professor of Economics safner@hood.edu ryansafner/microS23 microS23.classes.ryansafner.com

Outline

The “Runs” of Production

The “Runs” of Production

Production in the Short Run

Production in the Short Run: Example

Production in the Short Run: Example

Marginal Products

Marginal Product of Labor

Marginal Product of Capital

Diminishing Returns

Diminishing Returns

Average Product of Labor (and Capital)

The Firm's Problem: Long Run

The Long Run

The Firm's Problem

The Firm's Two Problems

The Firm's Two Problems

Long Run Production

Mapping Input-Combination Choices Graphically

Isoquants and MRTS

Isoquant Curves

Isoquant Curves

Isoquant Curves

Marginal Rate of Technical Substitution I

Marginal Rate of Technical Substitution I

Marginal Rate of Technical Substitution II

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microS23
microS23.classes.ryansafner.com

Ryan Safner
Associate Professor of Economics
safner@hood.edu
ryansafner/microS23
microS23.classes.ryansafner.com