Review of Univariate Calculus

Christopher Makler

Stanford University Department of Economics

Econ 50: Lecture 2

Answer by 11:35 at pollev.com/chrismakler

Suppose \(a\), \(b\), and \(c\) are strictly positive constants.

Which of the following is a graph of \(ax+by=c\)?

x
y
x
y
x
y
x
y

(A)

(B)

(C)

(D)

\text{slope} = -{a \over b}
\text{slope} = {a \over b}
\text{slope} = -{b \over a}
\text{slope} = {b \over a}

Again suppose \(a\), \(b\), and \(c\) are strictly positive constants.

Which of the following shows the effect of an increase in the parameter \(a\)?

x
y
x
y
x
y
x
y

(A)

(B)

(C)

(D)

The four dashed lines below show the line \(ax+by=c\)

Parallel shift out

Parallel shift in

Pivot out

Pivot in

Again suppose \(a\), \(b\), and \(c\) are strictly positive constants.

Which of the following shows the graph of the implicit function \(a\ln(x) + b\ln(y) = c\)?

x
y
x
y
x
y
x
y

(A)

(B)

(C)

(D)

Concave, upward sloping curve

Convex, upware sloping curve

Concave, downward sloping curve

Convex, downward sloping curve

This Week

Wednesday: Univariate & Graphing

Fluency in Math

Derivatives of Functions and their Inverses

The Chain Rule

Linear Functions

Sketching a Function

 

Section & Friday: Multivariate

Multivariable Functions

Level Sets

Partial Derivatives

 

Plus: TA intros!

Section starts tonight!

Office Hours Thurs-Fri-Sat

Fluency in Math

Ways of Describing a Model

Verbally

Mathematically

Visually

Diminishing marginal returns:

as you do more of something
(e.g. study)

you get fewer and fewer results
(e.g. more points on an exam)

x = \text{hours studying}
y = \text{grade}
y = f(x)
y = x^a

Diminishing marginal returns:

as you do more of something
(e.g. study)

you get fewer and fewer results
(e.g. more points on an exam)

x = \text{hours studying}
y = \text{grade}
y = f(x)
y = x^a
\text{Marginal return to studying: }{dy \over dx} = f'(x) = ax^{a - 1}

Diminishing marginal returns:

as you do more of something
(e.g. study)

you get fewer and fewer results
(e.g. more points on an exam)

x = \text{hours studying}
y = \text{grade}
y = f(x)
y = x^a
\text{Marginal return to studying: }{dy \over dx} = f'(x) = ax^{a - 1}

What does "diminishing marginal returns" mean?

Diminishing marginal returns:

as you do more of something
(e.g. study)

you get fewer and fewer results
(e.g. more points on an exam)

x = \text{hours studying}
y = \text{grade}
y = f(x) = x^a
{dy \over dx} = f'(x) = ax^{a - 1}

What does "diminishing marginal returns" mean?

{d^2y \over dx^2} = f''(x) =
(a-1)ax^{a - 2}
\begin{cases}a > 1 &\rightarrow \text{positive}\\ a = 1 &\rightarrow \text{zero}\\ a < 1 &\rightarrow \text{negative}\end{cases}

This has the same sign as \( (a -1) \)

Derivatives of Univariate Functions and their Inverses

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

Rules for Calculating Derivatives

f(x) = c
f'(x) = 0

Constant Rule:

f(x) = x^n
f'(x) = nx^{n-1}

Power Rule:

f(x) = \ln(x)
f'(x) = {1 \over x}

Natural Log Rule

f(x) = g(x)h(x)
f'(x) = g(x)h'(x) + g'(x)h(x)

Multiplication Rule

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

Chain Rule

Examples of Derivatives

f(x) = cx
f'(x) = c

Linear Functions:

f(x) = c_0+c_1x+c_2x^2+c_3x^3+\cdots

Polynomials:

f(x) = 0\ +\ c_1\ +\ c_2x+c_3x^2+\cdots

Square Root:

f(x) = \sqrt{x} = x^{1 \over 2}
f'(x) = {1 \over 2}x^{{1 \over 2}-1} = {1 \over 2\sqrt{x}}

Inverse Functions and their Derivatives

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)

Linear Example

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}}

Non-Linear Example

y = x^2
{dy \over dx} = 2x
x = \sqrt{y}
{dx \over dy} = {1 \over 2\sqrt{y}}

The Chain Rule

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

Linear Functions

\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_B - y_A}{x_B - x_A}
\text{inverse slope} = \frac{\text{run}}{\text{rise}} = \frac{\Delta x}{\Delta y} = \frac{x_B - x_A}{y_B - y_A}

Implicit Functions

\text{The equation }ax + by = c\text{ implicitly defines a line:}
\begin{aligned} ax + by &= c & ax + by &= c\\ by &= c - ax & ax &= c - by\\ y(x) &= {c \over b} - {a \over b}x & x(y) &= {c \over a} - {b \over a}y \end{aligned}

Implicit Differentiation

ax + by(x) = c
a + b \times y'(x) = 0

Let's assert that there just is some function \(y(x)\) without solving for it:

If we take the derivative of both sides we get

We can now solve for y'(x):

\begin{aligned} a + b \times y^\prime(x) &= 0\\ b \times y^\prime(x) &= -a\\ y^\prime(x) &= -{a \over b} \end{aligned}

Sketching a Function

Convexity and Concavity

TA Intros!!!

Multivariable Functions

x
f()
z = f(x,y)
y
z

[INDEPENDENT VARIABLES]

[DEPENDENT VARIABLE]

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

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

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 \(x\)]

y(x)=4-0.4x

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}}
=
0
\displaystyle{{dy \over dx}}
= -
\displaystyle{{\partial f \over \partial x}}
\displaystyle{{\partial f \over \partial y}}

IMPLICIT FUNCTION THEOREM

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)?

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)?

\displaystyle{{\partial f(x,y) \over \partial x}=}
\displaystyle{{\partial f(x,y) \over \partial y}=}

Slope of level set =   —

Production Functions

Production Functions

A mathematical form describing how much output is produced as a function of inputs.

Labor \((L)\)

Capital \((K)\)

Production Function  \(f(L,K)\)

Output (\(q\) or \(x\))

Lecture 2: Production Functions

Labor

Fish

🐟

Capital

Coconuts

🥥

[GOODS]

⏳

⛏

[RESOURCES]

Unit I: Scarcity and Choice

Economics is the study of how
we use scarce resources 
to satisfy our unlimited wants

Resources

Goods

Happiness

🌎

⌚️

🤓

Isoquants

Economic definition: if you want to produce some amount \(q\) of output, what combinations of inputs could you use?

Mathematical definition:
level sets of the production function

\text{Isoquant for }q = \{(L,K)\ |\ f(L,K) = q\}

Isoquant: combinations of inputs that produce a given level of output

Isoquant map: a contour map showing the isoquants for various levels of output

pollev.com/chrismakler

What happens to isoquants after an improvement in technology?

Marginal Products of Labor and Capital

Economic definition: how much more output is produced if you increase labor or capital?

Mathematical definition:
partial derivatives of the production function

\displaystyle{MP_L = {\partial f(L,K) \over \partial L}}

These are both rates: they are measured in terms of units of ouptut per unit of input.

\displaystyle{= \lim_{\Delta L \rightarrow 0} {f(L + \Delta L, K) - f(L, K) \over \Delta L}}
\displaystyle{MP_K = {\partial f(L,K) \over \partial K}}
\displaystyle{=\lim_{\Delta K \rightarrow 0} {f(L, K + \Delta K) - f(L, K) \over \Delta K}}

pollev.com/chrismakler

Consider the production function

f(L,K) = 4L^{1 \over 2}K

What is the expression for the marginal product of labor?

Marginal Rate of Technical Substitution (MRTS)

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

\displaystyle{MRTS = {MP_L \over MP_K}}
\displaystyle{\left.{dy \over dx} \right|_{f(x,y) = z} = -{\partial f/\partial x \over \partial f/\partial y}}

Therefore the formula for the MRTS is

(absolute value)

Labor (L)

Capital (K)

Intuition behind the formula for the MRTS

Example: Cobb-Douglas Production Function

q=f(L,K)=AL^aK^b
MP_L = aAL^{a-1}K^b
MP_K = bAL^aK^{b-1}
MRTS =
\displaystyle{= {a \over b} \times {K \over L}}

Summary

So far we've modeled an important economic activity
using multivariable calculus.

We looked at the economic meaning for
various properties of the production function:

Partial Derivatives

Marginal Products

Level Sets

Isoquants

Slope of a Level Set

Marginal Rate of Technical Substitution

In section and on Friday, we'll analyze how
different functional forms can be used to model different kinds of technologies.