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# Characteristics of

Production Functions

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 3

Goal for Today: Analyze how different functional forms can be used to **model** different production processes

# Examples of Production Functions

# Linear

# Leontief

(Fixed Proportions)

# Cobb-Douglas

# Constant Elasticity of Substitution (CES)

# Story

# If Chuck uses his bare hands (L), he can catch 2 fish per hour.

# If Chuck uses a net (K),

he can collect 4 fish per net.

# Model

# Fish from L hours of labor = 2L

# Fish from K nets = 4K

# How many fish can he produce altogether if he uses

L hours of labor, and K nets?

# If Chuck uses his bare hands (L), he can catch 2 fish per hour.

# If Chuck uses a net (K),

he can collect 4 fish per net.

Intuition: no matter how many hours he works, and how many nets he uses,

he can catch the same number of fish with 1 net as he can using 2 hours of labor.

# If Chuck uses his bare hands (L), he can catch 2 fish per hour.

# If Chuck uses a net (K),

he can collect 4 fish per net.

What does an isoquant look like?

What does the isoquant for \(q = 20\) look like?

L

K

# Linear Production Function

# Linear Production Function

What happens to an isoquant if \(a\) increases?

L

K

# Story

# Two workers and a garbage truck can collect 16 bins per hour.

# Adding a worker or getting an extra truck doesn't help.

# Model

# However, if you get another truck *and* two more workers, you can collect another 16 bins.

# If you have more than twice as many workers as trucks,

your production depends on the number of trucks you have.

# If you have less than twice as many workers as trucks,

your production depends on the number of workers you have.

# If you have more than twice as many workers as trucks,

your production depends on the number of trucks you have.

# If you have less than twice as many workers as trucks,

your production depends on the number of workers you have.

# Leontief Production Function

What do the isoquants look like?

L

K

# Leontief (Fixed Proportions) Production Function

# Cobb-Douglas Production Function

# Cobb-Douglas Production Function

What does the isoquant for \(q = 4\) look like?

# CES Production Function

# MRTS for Different Production Functions

# Linear

# Leontief

(Fixed Proportions)

# Cobb-Douglas

# CES

## Elasticity of Substitution

- Measures the substitutability of one input for another
- Key to answering the question: "will my job be automated?"
- Formal definition: the inverse of the
**percentage change in the MRTS**

**per percentage change in the ratio of capital to labor, K/L** - Intuitively: how "curved" are the isoquants for a production function?

# Scaling Production

# How does a technology respond to **increasing production**?

#
**Short run**: only some resources can be reallocated

#
**Long run**: all resources can be reallocated

# It depends on the time horizon:

# Scaling Production in the Short Run

# Suppose \(K\) is fixed at some \(\overline K\) in the short run.

Then the production function becomes \(f(L\ |\ \overline K)\)

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When does the production function

exhibit **diminishing** marginal product of labor?

# Scaling Production in the Long Run

# What happens when we increase all inputs **proportionally**?

# For example, what happens if we **double both labor and capital**?

# Does doubling inputs -- i.e., getting \(f(2L,2K)\) -- double output?

#
**Decreasing** Returns to Scale

#
**Constant** Returns to Scale

#
**Increasing** Returns to Scale

Does this exhibit **diminishing, constant or increasing **MPL?

Does this exhibit **decreasing, constant or increasing **returns to scale?

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When does the production function

exhibit **constant returns to scale**?

## Next Steps

- Homework for today's class is due on
**Saturday night** - Readings and quizzes for next week have been posted
- Homework for next week will be posted later today

#### Econ 50 | Lecture 03

By Chris Makler

# Econ 50 | Lecture 03

Characteristics of Production Functions

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