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What color shirt am I wearing?

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Suppose Chuck can use labor

to produce fish (good 1)

or coconuts (good 2).

If we plot his PPF in good 1 - good 2 space, what are the **units** of Chuck's MRT?

Suppose Chuck could initially produce 3 fish (good 1) or 2 coconuts (good 2)

in an hour.

He gets better at fishing, which allows him to produce 4 fish per hour.

What effect will this have on his MRT?

CHECK YOUR UNDERSTANDING

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Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 4 (Part II; Part I was on video)

# Resource Constraints and Production Possibilities

## Today's Agenda

Part 1: From production functions to the PPF

Part 2: The slope of the PPF

Getting situated in "Good 1 - Good 2 space"

Resource constraints and the PPF

Deriving the equation of the short-run PPF

The Marginal Rate of Transformation

Relationship between MPL and MRT

Deriving the expression for the MRT

using the implicit function theorem

### Good 1 - Good 2 Space

Two "Goods" (e.g. fish and coconuts)

A **bundle** is some quantity of each good

Can plot this in a graph with \(x_1\) on the horizontal axis and \(x_2\) on the vertical axis

### Good 1 - Good 2 Space

What **tradeoff **is represented by moving

from bundle A to bundle B?

ANY SLOPE IN GOOD 1 - GOOD 2 SPACE

IS MEASURED IN

UNITS OF GOOD 2 PER UNIT OF GOOD 1

ANY SLOPE IN GOOD 1 - GOOD 2 SPACE

IS MEASURED IN

UNITS OF GOOD 2 PER UNIT OF GOOD 1

TW: HORRIBLE STROBE EFFECT!

PPF

PPF

# Part 2: The MRT

# Mathematical Aside

Suppose you have some function

The rate at which \(y\) changes

due to a change in \(x\) is given by the

derivative of this function

The rate at which \(x\) changes

due to a change in \(y\) is given by the derivative of the inverse function

Now consider the **inverse** of that function

(measured in **units of y per units of x**)

(measured in **units of y per units of x**)

(measured in **units of x per units of y**)

# Mathematical Aside

Suppose the distance in miles \((m)\) you travel from home,

as a function of the number of hours driven \((h\)),

is given by the function

How many **additional miles** do you go in each hour?

How long does it take you to drive an additional mile?

The number of hours it takes you to drive \(m\) miles is given by the **inverse of the that function**

# Mathematical Aside

Suppose the amount of fish \((x_1)\) you can produce using \(L_1\) hours of labor is given by

How many **additional fish** do you catch in the \(L_1^\text{th}\) hour?

How many additional hours does it take to catch the \(x_1^\text{th}\) fish?

The number of hours it takes you to get \(x_1\) fish is given by the **inverse of the that function**

# Mathematical Aside

# 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)

This is just a level set of the function \(f(x_1,x_2) = {1 \over 3}x_1 + {1 \over 2}x_2\)

By the implicit function theorem:

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

This is just a level set of the function \(f(x_1,x_2) = \)

By the implicit function theorem:

Total labor required to produce the bundle \((x_1,x_2)\)

\(L_1(x_1) + L_2(x_2)\)

\({1 \over 3}x_1 + {1 \over 2}x_2\)

Recall: \(MP_{L1} = {dx_1 \over dL_1}, MP_{L2} = {dx_2 \over dL_2}\)

This is a level set of the function \(f(x_1,x_2) = {1 \over 100}x_1^2 + {1 \over 36}x_2^2\)

By the implicit function theorem:

Could you do this using the MPL method?

Remember: we need to express this in terms of \(x_1\) and \(x_2\)!

# 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?

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Charlene has the PPF given by

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

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

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?

Charlene has the PPF given by

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

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

If \(L_1 = L_2\) then this is just 2!

Everything is given in terms of L,

so let's use the \(MP_L\) formula:

Since \(x_1^2 = 64\) and \(x_2^3 = 8\), this is 4.

We have the PPF in terms of \(x_1\) and \(x_2\),

so let's use the implicit function theorem:

- Resource constraints + production functions = production possibilities
- The MRT (slope of PPF) is the
**opportunity cost**of producing good 1

(in terms of good 2) - If there is only one input (labor), the MRT is the ratio of the MPL's
- In general, best way to find the MRT is by using the implicit function theorem
- Homework due
**Saturday night**; includes a more complicated PPF than you've seen! - Next topic:
**preferences**over bundles

# Key Takeaways

#### Econ 50 | Lecture 04 (Spring 2024 - live portion)

By Chris Makler

# Econ 50 | Lecture 04 (Spring 2024 - live portion)

Resource Constraints and Production Possibilities

- 300