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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

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

Good 1 - Good 2 Space

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

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

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)

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?

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}}

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?

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} =

Mathematical Aside

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}}

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:

\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}}

Relationship between MPL's and MRT

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}}

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 \((8,2)\)?

\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

Key Takeaways