# Characteristics of

Production Functions

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 3

## Today's Agenda

Part 1: Elasticity of Substitution

Part 2: Scaling

Production functions and their MRTS

Elasticity of substitution

Scaling in the long run (returns to scale)

Scaling in the short run (holding K fixed)

*What happens as you move
along an isoquant*

*What happens as you move
between isoquants*

# Examples of Production Functions

# Linear

# Leontief

(Fixed Proportions)

# Cobb-Douglas

# Constant Elasticity of Substitution (CES)

# Linear Production Function

# Leontief (Fixed Proportions) Production Function

# Cobb-Douglas Production Function

# CES Production Function

# MRTS for Different Production Functions

# Linear

# Leontief

(Fixed Proportions)

# Cobb-Douglas

# CES

## Interpreting the MRTS

(slope of an isoquant)

"What is the rate at which

one can substitute

one input for another

and keep output the same?"

Another way of thinking about it:

"If we fired one worker and wanted to keep output the same,

how many additional machines would we need to buy?"

## 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 \(f(L\ |\ \overline K)\)

pollev.com/chrismakler

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?

pollev.com/chrismakler

When does the production function

exhibit **constant returns to scale**?

## Next Steps

- Homework for the last two classes is due on
**Saturday night** - Readings (and old videos) for next week have been posted
- Quizzes 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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