Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 14

# Resource Constraints and Production Possibilities

## Today's Agenda

Part 1: Resource Constraints and the PPF

Part 2: Optimization

The "Desert Island" Model

Resource constraints and the PPF

Deriving the equation of the short-run PPF

Shifts in the PPF

The Marginal Rate of Transformation

Relationship between MPL and MRT

The "Gravitational Pull" argument

Tangency when calculus works

Corners and kinks

# The “Desert Island" Model

Economics is the study of how

we use **scarce resources**

to satisfy our **unlimited wants**

Resources

Goods

Happiness

🌎

⌚️

🤓

Production functions

Utility functions

Demand

Supply

Equilibrium

🤩

🏪

⚖

### Wednesday: Little Green Pieces of Paper

## Multiple Uses of Resources

Labor

Fish

🐟

Coconuts

🥥

[GOOD 1]

⏳

[GOOD 2]

## Resource Constraint

## Production Possibilities

## Resource Constraint

## A PPF with Linear Technologies

Fish production function

Coconut production function

Resource Constraint

## A PPF with Diminishing \(MP_L\)

Fish production function

Coconut production function

Resource Constraint

- Suppose we want to produce a lot more of something -- ventilators, masks, toilet paper, hand sanitizer
- Some resources can be reallocated quickly; others are more specialized and can't be quickly repurposed
- How can we "scale up" in the short run and the long run?
- How do short-run tradeoffs compare with long-run tradeoffs?

# Shifts in the PPF

Consider an economy with \(\overline L = 100\) units of labor and \(\overline K = 100\) units of capital.

In the **short run**, \(K_1 = 64\) and \(K_2 = 36\).

In the **long run**, capital can be reallocated in any combination between goods 1 and 2.

Max in SR

Max in LR

- Up to now: how a short-run PPF can shift due to changing the allocation of capital,
**holding production functions constant**. - What happens when
**the technology itself**(i.e. the production function) changes?

# Improvements in Technology

*The New York Times, Oct. 29, 2013*

*Insider, July 23, 2020*

Consider an economy with \(\overline L = 100\) units of labor and \(\overline K = 100\) units of capital.

In the **short run**, \(K_1 = 64\) and \(K_2 = 36\).

In the **long run**, capital can be reallocated in any combination between goods 1 and 2.

Max in SR

Max in LR

# The Slope of the PPF

# Slope of the PPF:

Marginal Rate of Transformation (MRT)

## Rate at which one good may be “transformed" into another

## ...by reallocating resources from one to the other.

##
**Opportunity cost** of producing an additional unit of good 1,

in terms of good 2

Note: we will generally treat this as a positive number

(the magnitude of the slope)

Suppose we're allocating 100 units of labor to fish (good 1),

and 50 of labor to coconuts (good 2).

Now suppose we shift

one unit of labor

from coconuts to fish.

How many fish do we gain?

100

98

300

303

How many coconuts do we lose?

## Relationship between MPL's and MRT

Fish production function

Coconut production function

Resource Constraint

PPF

## Diminishing \(MP_L\)'s

and Increasing \(MRT\)

# Important Notes

## The MRT is the slope of the PPF at some **output combination** \((x_1,x_2)\)

## You should therefore write it in terms of \(x_1\) and \(x_2\), __not__ \(L_1\) and \(L_2\).

## You can use two methods to find the MRT:

the ratio of the MPL's, or the implicit function theorem.

CHECK YOUR UNDERSTANDING

Chuck has \(\overline L = 8\) total hours of labor,

and the production functions

\(x_1 = 2 \sqrt{L_1}\) and \(x_2 = 4\sqrt{L_2}\).

What is his MRT if he spends

half his time producing each good?

CHECK YOUR UNDERSTANDING

Charlene has the PPF given by

\(2x_1^3 + 3x_2^4 = 1072\)

What is her MRT if she produces the output combination \((8,2)\)?

# Verbal Analysis: MRS, MRT, and the “Gravitational Pull" towards Optimality

## Fish vs. Coconuts

- Can spend your time catching
**fish**(good 1)

or collecting**coconuts**(good 2) - What is your optimal
**division of labor**

between the two? - Intuitively: if you're optimizing, you

**couldn't reallocate your time**in a way

that would make you better off. - The last hour devoted to fish must

bring you the**same amount of utility**

as the last hour devoted to coconuts

**Marginal Rate of Transformation** (MRT)

- The number of coconuts you
**need**to give up in order to get another fish - Opportunity cost of fish in terms of coconuts

**Marginal Rate of Substitution** (MRS)

- The number of coconuts you are
**willing**to give up in order to get another fish - Willingness to "pay" for fish in terms of coconuts

**Both of these are measured in
coconuts per fish**

(units of good 2/units of good 1)

**Marginal Rate of Transformation** (MRT)

- The number of coconuts you
**need**to give up in order to get another fish - Opportunity cost of fish in terms of coconuts

**Marginal Rate of Substitution** (MRS)

- The number of coconuts you are
**willing**to give up in order to get another fish - Willingness to "pay" for fish in terms of coconuts

Opportunity cost of marginal fish produced is **less** than the number of coconuts

you'd be willing to "pay" for a fish.

Opportunity cost of marginal fish produced is **more** than the number of coconuts

you'd be willing to "pay" for a fish.

Better to spend **less** time fishing

and **more** time making coconuts.

Better to spend **more** time fishing

and **less** time collecting coconuts.

Better to produce

**more good 1**

and **less good 2**.

## “Gravitational Pull" Towards Optimality

Better to produce

**more good 2**

and **less good 1**.

These forces are **always true**.

In **certain circumstances**, optimality occurs where MRS = MRT.

# Graphical Analysis:

PPFs and Indifference Curves

The story so far, in two graphs

Production Possibilities Frontier

Resources, Production Functions → Stuff

Indifference Curves

Stuff → Happiness (utility)

Both of these graphs are in the same "Good 1 - Good 2" space

Better to produce

**more good 1**

and **less good 2**.

Better to produce

**less good 1**

and **more good 2**.

# When Calculus Fails:

Kinks and Corners

What would Lagrange find...?

What would Lagrange find...?

What would Lagrange find...?

# Kinks (Discontinuities)

## Kinked Indifference Curve:

## Kinked Constraint:

Discontinuities in the MRS

(e.g. Perfect Complements utility function)

Discontinuities in the MRT

(e.g. homework question with two factories)

What are some bundles that give the same utility as (4,8)?

What is the MRS at (4,8)? What about in the other place where the indifference curve intersects the PPF?

Here are some more indifference curves.

Where is the optimal bundle?

How can we solve for the optimal bundle?

#### Econ 50 | Fall 2022 | 14 | Resource Allocation and the PPF

By Chris Makler

# Econ 50 | Fall 2022 | 14 | Resource Allocation and the PPF

We examine the tradeoff an economy faces in allocating its resources among different goods, and how an individual would choose the allocate those resources if they were the only consumer.

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