The Marginal Rate of Substitution and the Implicit Function Theorem

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 3

Today's Agenda

Thinking about tradeoffs: the marginal rate of substitution (MRS)
Derivatives of multivariate functions and the Implicit Function Theorem
Applying the IFT to the MRS

Math

Econ

previously in Econ 50...

Choices in general

Choices of commodity bundles

Choosing bundles of two goods

Good 1 \((x_1)\)

Good 2 \((x_2)\)

Given any bundle \(A\),
the choice space may be divided
into three regions:

preferred to A

dispreferred to A

indifferent to A

The indifference curve through A connects all the bundles indifferent to A.

Indifference curve
through A

Good 1 - Good 2 Space

x_1

x_2

Two "Goods" (e.g. apples and bananas)

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)

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 =

\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

TW: HORRIBLE STROBE EFFECT!

8 \text{ units of good 2}

4 \text{ units of good 1}

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

\Delta x_2

\Delta x_1

Marginal Rate of Substitution

X = (10,24)

🍏🍏

🍌🍌🍌🍌

Y=(12,20)

🍏🍏

🍌🍌🍌🍌

Suppose you were indifferent between the following two bundles:

Starting at bundle X,
you would be willing
to give up 4 bananas
to get 2 apples

Let apples be good 1, and bananas be good 2.

Starting at bundle Y,
you would be willing
to give up 2 apples
to get 4 bananas

MRS = {4 \text{ bananas} \over 2 \text{ apples}}

= 2\ {\text{bananas} \over \text{apple}}

Visually: the MRS is the magnitude of the slope
of an indifference curve

How do we calculate the MRS from a utility function?

Let's review what we learned last time
about multivariable functions...

Multivariable Functions

f()

z = f(x,y)

[INDEPENDENT VARIABLES]

[DEPENDENT VARIABLE]

\text{Level set for }z=\{(x,y)|f(x,y)=z\}

Derivative of a Univariate Function
at a point \(x\)

{dy \over dx} = \lim_{\Delta x \rightarrow 0}\frac{f(x + \Delta x) - f(x)}{\Delta x}

the height of the function changes

per distance traveled to the right

rate at which

\displaystyle{{df \over dx} = \lim_{\Delta x \rightarrow 0} {f(x + \Delta x) - f(x) \over \Delta x}}

Local Linearization

{dy \over dx} = \lim_{\Delta x \rightarrow 0}\frac{f(x + \Delta x) - f(x)}{\Delta x}

\approx \frac{\Delta y}{\Delta x}

c(q) = 10q + q^2

Example: suppose a firm's cost function is given by

Suppose the firm is already producing \(q = 30\) units of output. Approximately how much would it cost to produce three more?

{dy \over dx}

\frac{\Delta y}{\Delta x} \approx

\Delta y \approx \Delta x \times {dy \over dx}

Example:

c(q) = 10q + q^2

MC(q) = {dc \over dq} = 10 + 2q

\begin{aligned} c(30) &= 10 \times 30 + 30^2 = 1,200\\ c(33) &= 10 \times 33 + 33^2 = 1,419 \end{aligned}

\Delta c = 219

MC(30) = {dc \over dq} = 10 + 2 \times 30 = 70

Pretty close to \(3 \times 70\)!

Local Linearization

Partial Derivatives of a Multivariate Function
at a point \((x,y)\)

{\partial f \over \partial x} = \lim_{\Delta x \rightarrow 0}\frac{f(x + \Delta x,y) - f(x,y)}{\Delta x}

{\partial f \over \partial y} = \lim_{\Delta y \rightarrow 0}\frac{f(x,y + \Delta y) - f(x,y)}{\Delta y}

the height of the function changes

per distance traveled East

rate at which

the height of the function changes

per distance traveled North

rate at which

\displaystyle{{\partial f \over \partial x} = \lim_{\Delta x \rightarrow 0} {f(x + \Delta x, y) - f(x, y) \over \Delta x}}

\displaystyle{{\partial f \over \partial y} = \lim_{\Delta y \rightarrow 0} {f(x, y + \Delta y) - f(x, y) \over \Delta y}}

Univariate Chain Rule

\text{Example: }h(x)=(3x + 2)^2

h(x)=f(g(x))

{dh \over dx} = {df \over dg} \times {dg \over dx}

f(g)=g^2

g(x)=3x+2

Multivariable Chain Rule

\text{Example: }h(x,y)=(3x+y)^2

h(x,y)=f(g(x,y))

{\partial h \over \partial x} = {df \over dg} \times {\partial g \over \partial x}

f(g)=g^2

g(x)=3x+y

y(x)=4-0.4x

Total Derivative Along a Path

\text{How does }f(x,y)\text{ change along a path?}

\Delta f \approx

\displaystyle{{\partial f \over \partial x} \times \Delta x}

\displaystyle{{\partial f \over \partial y} \times \Delta y}

\text{Suppose the path is defined by some function }y(x):

\Delta f \approx

\displaystyle{{\partial f \over \partial x} \times \Delta x}

\displaystyle{{\partial f \over \partial y}}

\displaystyle{\times {dy \over dx}}

\displaystyle{\times \Delta x}

\displaystyle{\Delta f \over \Delta x} \approx

\displaystyle{{\partial f \over \partial x}}

\displaystyle{{\partial f \over \partial y}}

\displaystyle{\times {dy \over dx}}

Total Derivative Along a Path

\displaystyle{\Delta f \over \Delta x}\ \ \ \ \ \ \approx

\displaystyle{{\partial f \over \partial x}}

\displaystyle{{\partial f \over \partial y}}

\displaystyle{\times\ \ \ \ \ \ {dy \over dx}}

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 \(x\)]

Derivative Along a Level Set

f(x,y)=z

Take total derivative of both sides with respect to x:

Solve for \(dy/dx\):

\displaystyle{{\partial f \over \partial x}}

\displaystyle{{\partial f \over \partial y}}

\displaystyle{\times \ \ {dy \over dx}}

\displaystyle{{dy \over dx}}

= -

\displaystyle{{\partial f \over \partial x}}

\displaystyle{{\partial f \over \partial y}}

IMPLICIT FUNCTION THEOREM

Derivative Along a Level Set

f(x,y)=z

Total derivative with respect to x:

\displaystyle{{\partial f \over \partial x}}

\displaystyle{{\partial f \over \partial y}}

\displaystyle{\times \ \ {dy \over dx}}

2x + 4y = 20

\displaystyle{\times \ \ {dy \over dx}}

-2

\displaystyle{{dy \over dx}}

= -

\displaystyle{{\partial f \over \partial x}}

\displaystyle{{\partial f \over \partial y}}

IMPLICIT FUNCTION THEOREM

\displaystyle{{dy \over dx}}

= -

\displaystyle{\times \ \ {dy \over dx}}

pollev.com/chrismakler

Consider the multivariable function

f(x,y) = 4x^{1 \over 2}y

What is the slope of the level set passing through the point (1, 5)?

ECONOMICS

Application to Utility Functions: Marginal Utility

MU_1(x_1,x_2) = {\partial u(x_1,x_2) \over \partial x_1}

MU_2(x_1,x_2) = {\partial u(x_1,x_2) \over \partial x_2}

Given a utility function \(u(x_1,x_2)\),
we can interpret the partial derivatives
as the "marginal utility" from
another unit of either good:

UTILS

UNITS OF GOOD 1

UTILS

UNITS OF GOOD 2

Indifference Curves and the MRS

Along an indifference curve, all bundles will produce the same amount of utility

In other words, each indifference curve
is a level set of the utility function.

The slope of an indifference curve is the MRS. By the implicit function theorem,

MRS(x_1,x_2) = \frac{\partial u(x_1,x_2) \over \partial x_1}{\partial u(x_1,x_2) \over \partial x_2} = {MU_1 \over MU_2}

UTILS

UNITS OF GOOD 1

UTILS

UNITS OF GOOD 2

Indifference Curves and the MRS

Along an indifference curve, all bundles will produce the same amount of utility

In other words, each indifference curve
is a level set of the utility function.

The slope of an indifference curve is the MRS. By the implicit function theorem,

MRS(x_1,x_2) = \frac{\partial u(x_1,x_2) \over \partial x_1}{\partial u(x_1,x_2) \over \partial x_2} = {MU_1 \over MU_2}

UNITS OF GOOD 1

UNITS OF GOOD 2

If you give up \(\Delta x_2\) units of good 2, how much utility do you lose?

If you get \(\Delta x_1\) units of good 1, how much utility do you gain?

\Delta u \approx \Delta x_2 \times MU_2

\Delta u \approx \Delta x_1 \times MU_1

If you end up with the same utility as you begin with:

\Delta x_2 \times MU_2 \approx \Delta x_1 \times MU_1

{\Delta x_2 \over \Delta x_1} \approx {MU_1 \over MU_2}

pollev.com/chrismakler

What is the MRS of the utility function \(u(x_1,x_2) = x_1x_2\)?

\displaystyle{MRS = {MU_1 \over MU_2} =}

u(x_1,x_2) = x_1x_2

\displaystyle{MU_1 = {\partial u(x_1,x_2) \over \partial x_1} =}

\displaystyle{MU_2 = {\partial u(x_1,x_2) \over \partial x_2} =}

4 \times 16

A = (4,16)

B = (8,8)

8 \times 8

x_2

x_1

\displaystyle{x_2 \over x_1}

= 64

\displaystyle{16 \over 4}

\displaystyle{8 \over 8}

MRS = 4

MRS = 1

Example: draw the indifference curve for \(u(x_1,x_2) = \frac{1}{2}x_1x_2^2\) passing through (4,6).

Step 1: Evaluate \(u(x_1,x_2)\) at the point

Step 2: Set \(u(x_1,x_2)\) equal to that value.

Step 4: Plug in various values of \(x_1\) and plot!

\(u(4,6) = \frac{1}{2}\times 4 \times 6^2 = 72\)

\(\frac{1}{2}x_1x_2^2 = 72\)

\(x_2^2 = \frac{144}{x}\)

\(x_2 = \frac{12}{\sqrt x_1}\)

How to Draw an Indifference Curve through a Point: Method I

Step 3: Solve for \(x_2\).

How would this have been different if the utility function were \(u(x_1,x_2) = \sqrt{x_1} \times x_2\)?

\(u(4,6) =\sqrt{4} \times 6 = 12\)

\(\sqrt{x_1} \times x_2 = 12\)

\(x_2 = \frac{12}{\sqrt x_1}\)

Example: draw the indifference curve for \(u(x_1,x_2) = \frac{1}{2}x_1x_2^2\) passing through (4,6).

Step 1: Derive \(MRS(x_1,x_2)\). Determine its characteristics: is it smoothly decreasing? Constant?

Step 2: Evaluate \(MRS(x_1,x_2)\) at the point.

Step 4: Sketch the right shape of the curve, so that it's tangent to the line at the point.

How to Draw an Indifference Curve through a Point: Method II

Step 3: Draw a line passing through the point with slope \(-MRS(x_1,x_2)\)

How would this have been different if the utility function were \(u(x_1,x_2) = \sqrt{x_1} \times x_2\)?

\text{In both cases }MRS(x_1,x_2) = \frac{MU_1(x_1,x_2)}{MU_2(x_1,x_2)} = \frac{x_2}{2x_1}

MRS(4,6) = \frac{6}{2 \times 4} = \frac{3}{4}

This is continuously strictly decreasing in \(x_1\) and continuously strictly increasing in \(x_2\),
so the function is smooth and strictly convex and has the "normal" shape.

Summary

MRS(x_1,x_2) = \frac{\partial u(x_1,x_2) \over \partial x_1}{\partial u(x_1,x_2) \over \partial x_2} = {MU_1 \over MU_2}

UNITS OF GOOD 1

UNITS OF GOOD 2

\text{Along the level set }f(x,y)=z,

\displaystyle{{dy \over dx}}

= -

\displaystyle{{\partial f \over \partial x}}

\displaystyle{{\partial f \over \partial y}}

IMPLICIT FUNCTION THEOREM

The Marginal Rate of Substitution is the magnitude of the slope of an indifference curve; so, by the implicit function theorem:

The Marginal Rate of Substitution and the Implicit Function Theorem

Today's Agenda

Choices in general

Choices of commodity bundles

Good 1 - Good 2 Space

Good 1 - Good 2 Space

Good 1 - Good 2 Space

Marginal Rate of Substitution

How do we calculate the MRS from a utility function?

Let's review what we learned last time about multivariable functions...

Multivariable Functions

Derivative of a Univariate Function at a point \(x\)

Local Linearization

Local Linearization

Partial Derivatives of a Multivariate Function at a point \((x,y)\)

Univariate Chain Rule

Multivariable Chain Rule

Total Derivative Along a Path

Total Derivative Along a Path

Derivative Along a Level Set

Derivative Along a Level Set

Application to Utility Functions: Marginal Utility

Indifference Curves and the MRS

Indifference Curves and the MRS

Summary

Let's review what we learned last time
about multivariable functions...

Derivative of a Univariate Function
at a point \(x\)

Partial Derivatives of a Multivariate Function
at a point \((x,y)\)