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Maths for Economics Cheat Sheet, Cheat Sheet of Econometrics and Mathematical Economics

Principles and formulas of mathematics for economics

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Maths for
Economics
PRINCIPLES AND FORMULAE
Exponential functions
e 2.7183 is the exponential constant
Graph of y= exshowing
exponential growth
Graph of y= exshowing
exponential decay
Quadratic functions y = ax2+ bx + c
Total cost functions
TC =a +bq cq2+dq3
Inverse functions
y =a/x =ax–1
q= a/p= ap–1
(1) b2– 4ac < 0; (2) b2– 4 ac = 0; (3) b2– 4ac > 0
Differentiation Graphs of Common Functions
Integration
The sum–difference rule Constant multiples
The product rule The quotient rule
The chain rule
x
v
x
u
xvxu
xd
d
d
d
))()((
d
d±=±
x
f
kxfk
xd
d
))((
d
d=
x
u
v
x
v
uuv
xd
d
d
d
)(
d
d+=
2d
d
d
d
d
d
v
x
v
u
x
u
v
v
u
x
=
x
u
u
y
x
y
xuuuyy d
d
d
d
d
d
then,)(where,)(If .
===
x
v
x
u
xvxu
xd
d
d
d
))()((
d
d±=±
x
f
kxfk
xd
d
))((
d
d=
x
u
v
x
v
uuv
xd
d
d
d
)(
d
d+=
2d
d
d
d
d
d
v
x
v
u
x
u
v
v
u
x
=
x
u
u
y
x
y
xuuuyy d
d
d
d
d
d
then,)(where,)(If .
===
x
v
x
u
xvxu
xd
d
d
d
))()((
d
d±=±
x
f
kxfk
xd
d
))((
d
d=
x
u
v
x
v
uuv
xd
d
d
d
)(
d
d+=
2d
d
d
d
d
d
v
x
v
u
x
u
v
v
u
x
=
x
u
u
y
x
y
xuuuyy d
d
d
d
d
d
then,)(where,)(If .
===
x
v
x
u
xvxu
xd
d
d
d
))()((
d
d±=±
x
f
kxfk
xd
d
))((
d
d=
x
u
v
x
v
uuv
xd
d
d
d
)(
d
d+=
2d
d
d
d
d
d
v
x
v
u
x
u
v
v
u
x
=
x
u
u
y
x
y
xuuuyy d
d
d
d
d
d
then ,)( where,)( If .
===
xxf d)(x
x
f)(
c
x+
c+
2
2
c
x+
3
3
x
2
e
x
e
kx
x, (n = –1)
n
c
x
x
+
n+1
n+1
c
k
kk, (any) constant c
e
kx
+
c+
e
x
ln x + c
x= 1/x
–1
x
v
x
u
xvxu
xd
d
d
d
))()((
d
d±=±
x
f
kxfk
xd
d
))((
d
d=
x
u
v
x
v
uuv
xd
d
d
d
)(
d
d+=
2d
d
d
d
d
d
v
x
v
u
x
u
v
v
u
x
=
x
u
u
y
x
y
xuuuyy d
d
d
d
d
d
then,)(where,)(If .
===
Positive gradient
(x
1
, y
1
)
(x
2
, y
2
)
12
12
xx
yy
m
=
Negative gradient
(x
1
, y
1
)
(x
2
, y
2
)
12
12
xx
yy
m
=
Positive gradient
(x
1
, y
1
)
(x
2
, y
2
)
12
12
xx
yy
m
=
Negative gradient
(x
1
, y
1
)
(x
2
, y
2
)
12
12
xx
yy
m
=
Linear y= mx + c; m= gradient; c= vertical intercept
y = f(x)
k, constant 0
x1
x22x
x33x2
xn, any constant nnx
n–1
exex= y
ekx kekx = ky
e f(x)f’(x)ef(x)
ln x1/x
ln kx = logekx 1/x
ln f(x)f’(x)/f(x)
dy
dx= f’(x)
for k constant
Studying Economics
Studying Economics
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Maths for

Economics

PRINCIPLES AND FORMULAE

Exponential functions

e ≈ 2.7183 is the exponential constant

Graph of y = e x showing

exponential growth

Graph of y = e

- x showing

exponential decay

Quadratic functions y = ax^2 + bx + c

Total cost functions

TC = a + bqcq^2 + dq^3

Inverse functions

y = a/x = ax–

q = a / p = ap –

(1) b^2 – 4 ac < 0; (2) b^2 – 4 ac = 0; (3) b^2 – 4 ac > 0

Differentiation Graphs of Common Functions

Integration

The sum–difference rule Constant multiples

The product rule The quotient rule

The chain rule

x

v

x

u ux vx x d

d

d

d (() ()) d

d ± = ±

x

f k fx k x d

d ( ()) d

d

x

u v x

v uv u x d

d

d

d ( ) d

d = +

2

d

d

d

d

d

d

v

x

v u x

u v

v

u

x

= 

  

x

u

u

y

x

y y yu u ux d

d

d

d

d

d If =(),where =(),then =.

x

v

x

u ux vx x d

d

d

d (() ()) d

d ± = ±

x

f k fx k x d

d ( ()) d

d

x

u v x

v uv u x d

d

d

d ( ) d

d = +

2

d

d

d

d

d

d

v

x

v u x

u v

v

u

x

= 

  

x

u

u

y

x

y y yu u ux d

d

d

d

d

d If = (),where =(),then =.

x

v

x

u ux vx x d

d

d

d (() ()) d

d ± = ±

x

f k fx k x d

d ( ()) d

d

x

u v x

v uv u x d

d

d

d ( ) d

d = +

2

d

d

d

d

d

d

v

x

v u x

u v

v

u

x

= 

  

x

u

u

y

x

y y yu u ux d

d

d

d

d

d If = (),where = (),then =.

x

v

x

u ux vx x d

d

d

d (() ()) d

d ± = ±

x

f k fx k x d

d ( ()) d

d

x

u v x

v uv u x d

d

d

d ( ) d

d = +

2

d

d

d

d

d

d

v

x

v u x

u v

v

u

x

= 

  

x

u

u

y

x

y y yu u ux d

d

d

d

d

d If = (),where = (),then =.

x ∫ f (x)dx

x

f ( )

c

x

+c

2

c

x

3

3

x

2

e

x

e

kx

x , (n = –1)

n c

x

x

n+

n+

c k

k, (any) constant c k

e

kx

e (^) +c

x

x = 1/x ln x + c

x

v

x

u ux vx x d

d

d

d (() ()) d

d ± = ±

x

f k fx k x d

d ( ()) d

d

x

u v x

v uv u x d

d

d

d ( ) d

d = +

2

d

d

d

d

d

d

v

x

v u x

u v

v

u

x

= 

  

x

u

u

y

x

y y yu u ux d

d

d

d

d

d If =(),where =(),then =.

Positive gradient

(x 1 , y 1 )

(x 2 , y 2 )

2 1

2 1

x x

y y m

Negative gradient

(x 1 , y 1 )

(x 2 , y 2 )

2 1

2 1

x x

y y m

Positiv

(x 1 , y 1 ) 2 1

2 1

x x

y y m

Negative gradient

(x 1 , y 1 )

(x 2 , y 2 )

2 1

2 1

x x

y y m

y = f ( x )^ Linear^ y^ =^ mx^ +^ c ;^ m^ = gradient;^ c^ = vertical intercept

k , constant 0

x 1

x^2 2 x

x 3 3 x 2

xn , any constant n nxn

e x^ e x^ = y

e kx^ ke kx^ = ky

e f^ ( x )^ f’ ( x ) e f(x)

ln x 1/ x

ln kx = loge kx 1/ x

ln f ( x ) f’ ( x )/ f(x)

d y

d x

= f’ ( x )

for k constant

Studying Economics Studying Economics

Arithmetic Algebra

Proportion and Percentage

When multiplying or dividing positive and negative

numbers, the sign of the result is given by:

  • and + gives + e.g. 6 x 3 = 18; 21 ÷ 7 = 3
  • and + gives – e.g. (–6) x 3 = –18 (–21) ÷ 7 = –
  • and – gives – e.g. 6 x (–3) = –18 21 ÷ (–7) = –
  • and – gives + (^) e.g. (–6) x (–3) = 18 (–21) ÷ (–7) = 3

Order of calculation

First: brackets

Second: x and ÷

Third: + and –

Fractions

Fraction =

Adding and subtracting fractions

To add or subtract two fractions, first rewrite each fraction

so that they have the same denominator. Then, the

numerators are added or subtracted as appropriate and

the result is divided by the common denominator: e.g.

Multiplying fractions

To multiply two fractions, multiply their numerators and

then multiply their denominators: e.g.

Dividing fractions

To divide two fractions, invert the second and then

multiply: e.g.

Series (e.g. for discounting)

1 + x + x^2 + x^3 + x^4 + … = 1/(1– x )

1 + x + x^2 + x^3 + … + x k^ = (1– x k+1^ )/(1– x )

(where 0 < x < 1)

numerator

denominator

20

31

20

15

20

16

4

3

5

4

77

15

11

5

7

3

10

9

2

3

5

3

3

2 ÷ 5

3

20

31

20

15

20

16

4

3

5

4

77

15

11

5

7

3

10

9

2

3

5

3

3

2 ÷ 5

3

20

31

20

15

20

16

4

3

5

4

77

15

11

5

7

3

10

9

2

3

5

3

3

2 ÷ 5

3

Removing brackets

a ( b + c ) = ab + ac a ( bc ) = abac

( a + b )( c + d ) = ac + ad + bc + bd

( a + b ) 2 = a 2

  • b 2
  • 2 ab ; ( a - b ) 2 = a 2
  • b 2
  • 2 ab

( a + b )( ab ) = a^2 – b^2

Formula for solving a quadratic equation

Laws of indices

Laws of logarithms

y = log b x means b y^ = x and b is called the base

e.g. log 10 2 = 0.3010 means 100.3010^ = 2.000 to 4 sig figures

Logarithms to base e, denoted log e , or alternatively ln,

are called natural logarithms. The letter e stands for the

exponential constant, which is approximately 2.7183.

a

b b ac x 2

2

- ± – If _ax = 2

  • bx + c_ = 0, then

m n m n a a a

=

m n n

m

a a

a (^) –

m n mn (a ) =a

1

0 a = m

m

a

a

1

  • (^) n n a = a

1 / n n m a = a

m/

ln AB = lnA+ln B A B

B

A

ln = ln –ln A n A

n

; ; ln = ln

asapercentageis 8

asapercentageis 8

To convert a fraction into a percentage, multiply by 100

and express the result as a percentage. An example is:

Ratios are simply an alternative way of expressing

fractions. Consider dividing £200 between two people in

the ratio of 3:2. This means that for every £3 the first person

gets, the second person gets £2. So the first gets 3 / 5 of the

total (i.e. £120) and the second gets 2 / 5 (i.e. £80).

Generally, to split a quantity in the ratio m : n , the quantity

is divided into m /( m + n ) and n /( m + n ) of the total.

Some common conversions are

Sigma Notation

The Greek Alphabet

The Greek capital letter sigma ∑ is used as an abbreviation

for a sum. Suppose we have n values: x 1 , x 2 , ... xn and we

wish to add them together. The sum

Note that i runs through all integers (whole numbers) from

1 to n. So, for instance

This notation is often used in statistical calculations. The

mean of the n quantities, x 1 , x 2 , ... and xn is

i.e. the mean of the squares minus the square of the mean

The standard deviation (sd) is the square root of the

variance:

Note that the standard deviation is measured in the same

units as x.

The variance is

Example

2 2 2 2 2

5

1

2

∑ means^1 +^2 +^3 +^4 +^5

i =

i

1 2 3

3

1

xmeans x x x i

i

=

x 1 + x 2 ...xn

=

n

i

xi 1

n

x x x

n

x x n

n i i^

2

2 1

2 1

var( ) x n

x

n

x x x

i

n i i

n i = –

sd( x)^ = var(x )

2 2 2 2 2

5

1

2

∑ means^1 +^2 +^3 +^4 +^5

i =

i

1 2 3

3

1

x means x x x i

i

=

x 1 + x 2 ...xn

=

n

i

xi 1

n

x x x

n

x x

n

n i i + + + = =

2

2 1

2 1

var( ) x n

x

n

x x x

i

n i i

n i = –

sd( x^ )= var(x )

2 2 2 2 2

5

1

2

∑ means^1 +^2 +^3 +^4 +^5

i =

i

1 2 3

3

1

xmeans x x x i

i

=

x 1 + x 2 ...xn

=

n

i

xi 1

n

x x x

n

x x n

n i i^

2

2 1

2 1

var( ) x n

x

n

x x x

i

n i i

n i = –

sd( x)^ = var(x )

2 2 2 2 2

5

1

2

∑ means^1 +^2 +^3 +^4 +^5

i =

i

1 2 3

3

1

x means x x x i

∑ i + +

=

1 2 n x x x

=

n

i

xi 1

n

x x x

n

x x

n

n i i^

2

2 1

2 1

var( ) x n

x

n

x x x

i

n i i

n i = –

sd( x )= var(x )

2 2 2 2 2

5

1

2

∑ means^1 +^2 +^3 +^4 +^5

i =

i

1 2 3

3

1

xmeans x x x i

i

=

x 1 + x 2 ...xn

=

n

i

i x 1

n

x x x

n

x x n

n i i^

2

2 1

2 1

var( ) x n

x

n

x x x

i

n i i

n i = –

sd( x)^ = var(x )

2 2 2 2 2

5

1

2

∑ means^1 +^2 +^3 +^4 +^5

i =

i

1 2 3

3

1

xmeans x x x i

∑ i + +

=

x 1 + x 2 ...xn

=

n

i

xi 1

n

x x x

n

x x

n

n i i + + + = =

2

2 1

2 1

var( ) x n

x

n

x x x

i

n i i

n i = –

sd( x)^ = var(x )

2 2 2 2 2

5

1

2

∑ means^1 +^2 +^3 +^4 +^5

i =

i

1 2 3

3

1

xmeans x x x i

∑ i + +

=

x 1 + x 2 ...xn

=

n

i

i x 1

n

x x x

n

x x

n

n i i^

2

2 1

2 1

var( ) x n

x

n

x x x

i

n i i

n i = –

sd( x) = var(x )

is written

Α α alpha Ι ι iota Ρ ρ rho

Β β beta Κ κ kappa Σ σ sigma

Γ γ gamma Λ λ lambda Τ τ tau

∆ δ delta Μ μ mu Υ υ upsilon

Ε ε epsilon Ν ν nu Φ φ phi

Ζ ζ zeta Ξ ξ xi Χ χ chi

Η η eta Ο ο omicron Ψ ψ psi

Θ θ theta Π π pi Ω ω omega