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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)

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

x_1

x_2

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

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

\text{Bundle }X = (x_1,x_2)

x_1 = \text{quantity of good 1 in bundle }X

x_2 = \text{quantity of good 2 in bundle }X

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

A = (4,16)

B = (8,8)

A

B

4

8

12

16

20

4

8

12

16

20

x_1

x_2

What **tradeoff **is represented by moving

from bundle A to bundle B?

\text{Give up }\Delta x_2 =

A

B

4

8

12

16

20

4

8

12

16

20

\text{Gain }\Delta x_1 =

\text{Rate of exchange }=

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!

8 \text{ units of good 2}

4 \text{ units of good 1}

2

\displaystyle{\frac{\text{units of good 2}}{\text{units of good 1}}}

\Delta x_2

\Delta x_1

150

100

50

0

0

3 \times 50

150

450

0

2 \times 150

300

200

2 \times 100

300

100

PPF

L_1 = {1 \over 3}x_1

L_2 = {1 \over 2}x_2

L_1

+

L_2

= 150

{1 \over 3}x_1

{1 \over 2}x_2

L_1 = {1 \over 3}x_1

L_2 = {1 \over 2}x_2

+

= 150

{1 \over 3}x_1

{1 \over 2}x_2

0

10\sqrt{36}

60

100

0

6\sqrt{100}

60

48

6 \sqrt{64}

80

36

PPF

L_1 = {1 \over 100}x_1^2

L_2 = {1 \over 36}x_2^2

L_1

+

L_2

= 100

{1 \over 100}x_1^2

{1 \over 36}x_2^2

L_1 = {1 \over 100}x_1^2

L_2 = {1 \over 36}x_2^2

+

= 100

{1 \over 100}x_1^2

{1 \over 36}x_2^2

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

y = f(x)

f'(x) = {dy \over dx}

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**)

x = f^{-1}(y)

f'(x) = {dy \over dx}

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

[f^{-1}]'(y) = {dx \over dy}

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

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?

m(h) = 30h

{dm \over dh} =

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

{dh \over dm} =

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

h(m) = {1 \over 30}m

30 {\text{miles} \over \text{hour}}

{1 \over 30} {\text{hours} \over \text{mile}}

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?

x_1(L_1) = 10\sqrt{L_1}

{dx_1 \over dL_1} =

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

{dL_1 \over dx_1} =

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

L_1(x_1) = {1 \over 100}x_1^2

{5 \over \sqrt{L_1}} {\text{fish} \over \text{hour}}

{x_1 \over 50} {\text{hours} \over \text{fish}}

MP_{L1} =

x_1(L_1) = 10\sqrt{L_1}

{dx_1 \over dL_1} =

{dL_1 \over dx_1} =

L_1(x_1) = {1 \over 100}x_1^2

{5 \over \sqrt{L_1}} {\text{fish} \over \text{hour}}

{x_1 \over 50} {\text{hours} \over \text{fish}}

MP_{L1} =

={10\sqrt{L_1} \over 50} {\text{hours} \over \text{fish}}

={\sqrt{L_1} \over 5} {\text{hours} \over \text{fish}}

{1 \over MP_{L1}}

Marginal Rate of Transformation (MRT)

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:

\displaystyle{\text{slope of the level set}=-{{\partial f \over \partial x_1} \over {\partial f \over \partial x_2}}}

\displaystyle{=-{{1 \over 3} \over {1 \over 2}}}

\displaystyle{=-{2 \over 3}}

\displaystyle{\Rightarrow MRT ={2 \over 3}}

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?

\Delta_2

\Delta_1

100

98

300

303

How many coconuts do we lose?

\Delta_2 = MP_{L_2} = 2\text{ coconuts}

\Delta_1 = MP_{L_1} = 3\text{ fish}

\left|\text{slope}\right| = \frac{MP_{L_2}}{MP_{L_1}}

x_1 = f_1(L_1) = 3L_1

x_2 = f_2(L_2) = 2L_2

Fish production function

Coconut production function

Resource Constraint

L_1 + L_2 = 150

PPF

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

By the implicit function theorem:

\displaystyle{\text{slope of the level set}=-{{dL_1 \over dx_1} \over {dL_2 \over dx_2}}}

\displaystyle{=-{{1 \over MP_{L1}} \over {1 \over {MP_{L2}}}}}

\displaystyle{=-{MP_{L2} \over MP_{L1}}}

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

\displaystyle{\text{slope of the level set}=-{{\partial f \over \partial x_1} \over {\partial f \over \partial x_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:

\displaystyle{\text{slope of the level set}=-{{\partial f \over \partial x_1} \over {\partial f \over \partial x_2}}}

\displaystyle{=-{{2 \over 100}x_1 \over {2 \over 36}x_2}}

\displaystyle{=-{9x_1 \over 25x_2}}

\displaystyle{\Rightarrow MRT ={9x_1 \over 25x_2}}

Could you do this using the MPL method?

\displaystyle{MRT ={MP_{L2} \over MP_{L1}}}

\displaystyle{={{3 \over \sqrt{L_2}} \over {5 \over \sqrt{L_1}}}}

\displaystyle{MRT ={9x_1 \over 25x_2}}

\displaystyle{={3\sqrt{L_1} \over {5\sqrt{L_2}}}}

\sqrt{L_1} = {1 \over 10}x_1

\sqrt{L_2} = {1 \over 6}x_2

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

\displaystyle{=-{3 \times {1 \over 10}x_1 \over {5 \times {1 \over 6}x_2}}}

\displaystyle{={18x_1 \over 50x_2}}

\displaystyle{={9x_1 \over 25x_2}}

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 \((8,2)\)?

\displaystyle{MRT = {MP_{L2} \over MP_{L1}}}

\displaystyle{= {{2 \over \sqrt{L_2}} \over {1 \over \sqrt{L_1}}}}

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:

\displaystyle{MRT = {{\partial f \over \partial x_1} \over {\partial f \over \partial x_2}}}

\displaystyle{= {6x_1^2 \over 12x_2^3}}

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:

\displaystyle{= {x_1^2 \over 2x_2^3}}

- 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