review of math essentials prepared by vera tabakova, east carolina university

Appendix A

Review of Math Essentials

Prepared by Vera Tabakova, East Carolina University

Appendix A: Review of Math Essentials

A.1 Summation

A.2 Some Basics

A.3 Linear Relationships

A.4 Nonlinear Relationships

Slide A-2Principles of Econometrics, 3rd Edition

A.1 Summation

The symbol Σ is the capital Greek letter sigma, and means “the sum of.”

The letter i is called the index of summation. This letter is arbitrary and may also appear as t, j, or k.

The expression is read “the sum of the terms xi, from i equal one to n.”

The numbers 1 and n are the lower limit and upper limit of summation.


1 21

n

i ni

x x x x

1

n

ii

x

A.1 Summation

Rules of summation operation:


1 21

n

i ni

x x x x

1 1

n n

i ii i

ax a x

1

n

i

a a a a na

1 1 1

( )n n n

i i i ii i i

x y x y

A.1 Summation


1 1 1

1 1 2

1 1 1 1 1 1

( )

( ) 0

n n n

i i i ii i i

n

ii n

n n n n n n

i i i i ii i i i i i

ax by a x b y

xx x x

xn n

x x x x x nx x x

A.1 Summation


1 21

( ) ( ) ( ) ( )

( ) ("Sum over all values of the index ")

( ) ("Sum over all possible values of ")

n

i ni

ii

x

f x f x f x f x

f x i

f x X

A.1 Summation


1 1 1 1

1 21 1 1

1 1 1 1

( , ) ( )

( , ) [ ( , ) ( , ) ( , )]

( , ) ( , )

m n m n

i j i ji j i j

m n m

i j i i i ni j i

m n n m

i j i ji j j i

f x y x y

f x y f x y f x y f x y

f x y f x y

A.2 Some Basics

A.2.1Numbers

Integers are the whole numbers, 0, ±1, ±2, ±3, …. .

Rational numbers can be written as a/b, where a and b are integers, and b ≠ 0.

The real numbers can be represented by points on a line. There are an uncountable number of real numbers and they are not all rational. Numbers such as are said to be irrational since they cannot be expressed as ratios, and have only decimal representations. Numbers like are not real numbers.


3.1415927 and 2

2

A.2 Some Basics

The absolute value of a number is denoted . It is the positive part

of the number, so that

Basic rules about Inequalities:


a

3 3 and 3 3.

If , then

if 0If , then

if 0

If and , then

a b a c b c

ac bc ca b

ac bc c

a b b c a c

A.2 Some Basics

A.2.2Exponents

(n terms) if n is a positive integer

x0 = 1 if x ≠ 0. 00 is not defined

Rules for working with exponents, assuming x and y are real, m and n are integers, and a and b are rational:


nx xx x

1 if 0n

nx x

x

A.2 Some Basics


1/

/ 1/

/ 1/

n n

mm n n

mm n n

a b a b

a a

a

x x

x x

x x

x x x

x x

y y

A.2.3 Scientific Notation


5 7

5 7

2

510,000 .00000034 (5.1 10 ) (3.4 10 )

(5.1 3.4) (10 10 )

17.34 10

.1734

5 512

7 7

510,000 5.1 10 5.1 101.5 10

.00000034 3.4 10 3.4 10

A.2.4 Logarithms and the number e


ln ln

Rules:

ln( ) ln( ) ln( )

ln( / ) ln( ) ln( )

ln( ) ln( )

b

a

x e b

xy x y

x y x y

x a x


The exponential function is the antilogarithm because we can recover

the value of x using it.


ln exp lnxx e x



(A.1)

(A.2)

1 2y x

2y x

1 2 1 2 10y x


Figure A.1 A linear relationship




(A.3)

(A.4)

2

dy

dx

1 2 2 3 3y x x

(A.5)2 3

2

3 23

given that is held constant

given that is held constant

yx

xy

xx



(A.6)

1 2 3Q L K

2 given that capital is held constant

, the marginal product of labor inputL

QK

L

MP

2 32 3

,y y

x x

A.3.1 Elasticity


(A.7)

∆y/y = the relative change in y, which is a decimal

%∆y = percentage change in y

(A.8a)

(A.8b)

(A.8c)

yx

y y y x xslope

x x x y y

% 100y

yy


Figure A.2 A nonlinear relationship




(A.9)

(A.10)

2dy dx

yx

dy y dy x xslope

dx x dx y y


Figure A.3 Alternative Functional FormsSlide A-24Principles of Econometrics, 3rd Edition

A.4.1 Quadratic Function

If β3 > 0, then the curve is U-shaped, and representative of average or

marginal cost functions, with increasing marginal effects. If β3 < 0,

then the curve is an inverted-U shape, useful for total product curves,

total revenue curves, and curves that exhibit diminishing marginal

effects.


21 2 3y x x

2 3 2 32 0, or / (2 )dy dx x x

A.4.2 Cubic Function

Cubic functions can have two inflection points, where the function crosses its tangent line, and changes from concave to convex, or vice versa.

Cubic functions can be used for total cost and total product curves in economics. The derivative of total cost is marginal cost, and the derivative of total product is marginal product.

If the “total” curves are cubic then the “marginal” curves are quadratic functions, a U-shaped curve for marginal cost, and an inverted-U shape for marginal product.


2 31 2 3 4y x x x

A.4.3 Reciprocal Function

Example: the Phillips Curve


11 2 1 2

1y x

x

1

1

1 2

% 100

1%

t tt

t

tt

w ww

w

wu

A.4.4 Log-Log Function

In order to use this model all values of y and x must be positive. The slopes of these curves change at every point, but the elasticity is constant and equal to β2.


1 2ln( ) ln( )y x

A.4.5 Log-Linear Function

Both its slope and elasticity change at each point and are the same sign as β2.

The slope at any point is β2y, which for β2 > 0 means that the marginal

effect increases for larger values of y.


1 2ln( )y x

1exp ln( ) exp( )y y x

A.4.6 Approximating Logarithms


(A.11) 1 0 1 00

1ln ln( ) ( )y y y y

y

1 0 1 00 0

ln(1 )

1ln ln( ) ln ( ) = relative change in

x x

yy y y y y y

y y



(A.12)

1 0

0

100 ln( ) 100 ln( ) ln( )

100

% percentage change in

y y y

y

y

y y



10

1 1

1

% 100 100( 1)

1 ln% 100 ln( )% approximation error 100 100

100 ln( ) ln( )

yy y

y

y yy y

y y

A.4.7 Approximating Logarithms in the Log-Linear Model


(A.13)

1 0

2 1 0

2

100 ln( ) ln( ) %

100

(100 )

y y y

x x

x

A.4.8 Linear-Log Function


0 1 2 0

1 1 2 1

ln( )

ln( )

y x

y x

1 0 2 1 0

21 0

2

ln( ) ln( )

100 ln( ) ln( )100

%100

y y y x x

x x

x



1 2

2

ln( ) 0 500ln( )

500% 10 50

100 100

y x x

y x

Keywords


absolute value antilogarithm asymptote ceteris paribus cubic function derivative double summation e elasticity exponential function exponents inequalities integers intercept irrational numbers linear relationship logarithm

log-linear function log-log function marginal effect natural logarithm nonlinear relationship partial derivative percentage change Phillips curve quadratic function rational numbers real numbers reciprocal function relative change scientific notation slope summation sign

review of math essentials prepared by vera tabakova, east carolina university

Documents