Introduction to Mathematical Economics Lecture 5

Introduction to Mathematical Economics

Lecture 5

Undergraduate Program Faculty of Economics & Business Universitas Padjadjaran

Ekki Syamsulhakim, SE., MApplEc.

Previously

•

_{Quadratic function}

– How to graph

– _{Applications in economics}

•

_{Some exercises}

Today

• Cubic Function: Application

• _{Rational Function: Application}

• _{Other polynomial function  bivariate}

• _{Exponential function}

– _Univariate – _Bivariate

4

Cubic function

• y=ax3+bx2+cx+d

• The maximum power of independent variable(s)

is(are) 3

• There might be a maximum and a minimum in

one plot of a cubic function OR a “turning” point

• _{To find the roots (x}

i) can use factoring

– _{No general rule of factoring exists  trial and error}

• _{To graph  use curve tracing method}

• Can be found in Total Product Curve and Total

Finding Roots?

•

_{Example: y = x}

3

– x

2

– 4x + 4

– Find the roots!

•

_{To find the roots, y=0}

x3 – x2 – 4x + 4 = 0

(x – 1) (x + 2) (x – 2) = 0 x₁=1, x₂=-2 x₃=2

•

_{Good news : no need to find the roots in}

6

•

Example: total production function with

one input

•

Usually

, a<0, b>0,c>0,d = 0; has a

maximum point at positive values of x, has

a minimum value y=0 at x=0

TP= – 0.02L

3

+3L

2

+2L

•

We can analyze the total production

function qualitatively as follows:

– Draw the function first

In Brief: Cubic Function

y x( ) 0.02 x 3 _{3 x} 2 _{2 x}

y x( )

0 50 100 150

Production Function

Cost in the Short Run

• The Determinants of Short-Run Cost

– Increasing returns and cost

• With increasing returns, output is increasing

relative to input and variable cost and total cost will fall relative to output.

– Decreasing returns and cost

• With decreasing returns, output is decreasing

10

•

a>0, b<0,c>0,d > 0 and b

2

< 3.a.c

– Does not have either a maximum or minimum

point (only an inflection point)

•

Example

TC = 0.5 q

3

– 10 q

2

+ 125 q + 400

Cubic Total Cost

0 10 20 30

2 10 3

4 10 3

6 10 3

8 10 3

Rational Function

General Form:

Where a and c ≠ 0

•

_{The plot of this function is a “rectangular}

hyperbola”

Application of Rational Function

•

_{A non-linear (“unitary elasticity”) demand}

curve

– _{A special case of rational function:}

Or specifically for our demand function:

And (interestingly) the inverse demand function is

•

To plot a rectangular hyperbola

•

_{We need to find the important points/line}

– Intersection with x axis

– _{Intersection with y axis}

– _{Vertical asymptot (shadow) line}

Rectangular hyperbola

•

_{Suppose the function is}

•

Intersection with the vertical axis, x = 0

•

_{Intersection with the horisontal axis, y = 0}

;

•

Rectangular Hyperbola

•

_{Vertical Asymptot:}

– As y gets closer to infinity, then cx + d = 0

– _{As the result, our vertical asymptot is}

•

_{Horisontal Asymptot:}

– _{As x gets closer to infinity, then:}

– _{As the result, our vertical asymptot is}

Example

•

_{The plot of a rational function}

– Intersection with x axis, y = 0

x = = 2.5

– _{Intersection with y axis, x = 0  y = 1}

– _{Vertical asymptot, x = 5}

– _{Horisontal asymptot, y = 2}

•

Example

•

_{The plot of a rational function}

– Intersection with x axis, y = 0

x = = -2.5

– _{Intersection with y axis, x = 0  y = 1}

– _{Vertical asymptot, x = – 5}

– _{Horisontal asymptot, y = 2}

•

Example of unitary elasticity

demand curve

22

•

_{The plot of an inverse demand function}

p, q

– _{Intersection with q axis, p = 0  q = ~}

– _{Intersection with p axis, q = 0  p = ~}

– _{Vertical asymptot, q = 0}

– _{Horisontal asymptot, p = 0}

24

Other Polynomial Function

•

_{Degree of polynomial < 1}

•

In economics, this type of function is often

in bivariate form:

where a, b < 1

When a+b = 1 we call this function a

Cobb Douglass function (constant returns

to scale)

Other Polynomial Function

•

_{Example: Suppose that a firm faces a}

production function

in the form of

Q=Q(K,L), explicitly Q=K

0.4

L

0.6

•

The function Q=K

0.4

L

0.6

may be drawn in a

3D space

•

Using the concept of

level set

, we can see

The plot of Q=K

0.4

L

0.6

in 3D

26

The plot of Q=K

0.4

L

0.6

in 2D

28

Exponential function

•

General form:

y = b

x

•

_b

_

_{base of the function}

•

In many cases b takes the number

e=2.718

– _{ea number that has a characteristic of ln}

e=1

Derivation of the number e

• _{Consider the function:}

• _{If larger and larger (positive) values are}

assigned to m, then f(m) will also assume larger values

• _{f(1)=2; f(2)=2,25; f(4)=2,44141;}

f(100)=…;f(1000)=…

• _{f(m) will converge to the number}

e = 2,71828…

Number e and its rules

•

e

0

=1

•

e

a

(e

b

) = e

a+b

•

(e

a

)

b

=e

ab

Logarithmic function

•

_{Logarithmic function is}

_{the inverse}

_of

exponential function

•

General form:

y=

b

log x or y=log

b

x

–

_The

_inverse

_{: x = b}

y

–

_Common

_{log: b = 10}

34

Logarithmic function

Natural log: b = e

•

Natural logarithm, general form: y=log

_e

x

•

_{The log of the base = log e = 1}

•

Example:



ln e = 1

Logarithms Rules

•

_ln(u

a

)=a ln u

•

ln e

15

= 15

•

ln(uv) = ln u + ln v (u,v>0)

•

ln(e

3

.e

2

) = ln e

3

+ ln e

2

= 3+2=5

36

Logarithms Rules

•

_ln(uv

a

)= ln u + ln v

a

= ln u + a ln v

•

ln(xy

2

) = ln x + 2 ln y

•

ln(u



v)

_

ln u



ln v

•

ln(e

5



e

2

)

_

ln (e

5

)



ln (e

2

)

• _ln(e5+e2)=ln(155.8)=5.05

• ln(e5)=5, ln(e2)=2  ln (e5) + ln (e2) = 5 + 2 = 7

Logarithms Rules

Proof

Let

Applications

•

_{Growth model}

•

Population model

•

Production function

•

_etc