The total change in the height of the function due to a small increase in $x$

The amount $f$ changes due to the increase in $x$

[indirect effect through $y$]

The amount $f$ changes due to an increase in $y$

The amount $y$ changes due to an increase in $x$

[direct effect from $x$]

Take total derivative of both sides with respect to x:

Solve for $dy/dx$:

IMPLICIT FUNCTION THEOREM

Economic definition: the rate at which a producer can substitute one input for another while keeping output at the same level

Mathematical definition: slope of an isoquant

Recall: by implicit function theorem,
the slope of a level set is given by

Therefore the formula for the MRTS is

(absolute value)

If Chuck uses his bare hands (L), he can catch 2 fish per hour.

If Chuck uses a net (K),
he can collect 4 fish per net.

Fish from L hours of labor = 2L

Fish from K nets = 4K

How many fish can he produce altogether if he uses
L hours of labor, and K nets?

Intuition: no matter how many hours he works, and how many nets he uses,
he can catch the same number of fish with 1 net as he can using 2 hours of labor.

NOTE THAT THESE ARE FUNCTIONS OF L AND K, NOT JUST OF L!!!

DO NOT CONVERT INTO A UNIVARIATE FUNCTION AND TAKE THE DERIVATIVE!!!

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: as with the MRTS, we will generally treat this as a positive number
(the magnitude of the slope)

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

By the implicit function theorem:

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

$L_1(x_1) + L_2(x_2)$

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:

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 $L_2$.

Even though we show the relationship between the MRT and the MPL's, in general you should use the implicit function theorem to find it!

Since $x_1^2 = 64$ and $x_2^3 = 8$, this is 4.

By the implicit function theorem: