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 – Applications

  

Cubic function

^{3 2} y=ax

• bx +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 ) can use factoring _i
• No general rule of factoring exists  trial and error
• – To graph  use curve tracing method
• Can be found in Total Product Curve and Total •

  Cost Curve

  

Finding Roots?

• Example: y = x
³ – x ² – 4
• – Find the roots!
• To find the roots, y=0

  x ³ – x ² – 4x + 4 = 0 (x – 1) (x + 2) (x – 2) = 0 x ₁ =1, x ₂ =-2 x ₃ =2

• Good news : no need to find the roots in economic application

  Cubic Function for Total Product

  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
_{3 2} TP= – 0.02L +3L +2L We can analyze the total produ
• function qualitatively as follows:

  Draw the function first

• –

  In Brief: Cubic Function _{y x ( ) 0.02 x 3 x 2 x    3 2} 1.2 104 _{1 104 6000} 8000 y x ( ) ₄₀₀₀ 2000 ₂₀₀₀ 50 100 150

  

Production Function

With One Variable Input

  

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 relative to input and variable cost and total cost will rise relative to output. Cubic Function for Total Cost

• a>0, b<0,c>0,d > 0 and b
² <
• – 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

  10 ^{20 30 2 10 3  4 10 3  6 10 3  8 10 TC q ( ) 3 } _q

  

Rational Function

  General Form:

• Where a and c ≠ 0

  The plot of this function is a “rectangular

• hyperbola” We usually restrict either or both x,y

  

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
• – Horisontal 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

Plot of

  _{20 16 18 14}

  20 ₁₀

  12 _{6 8 2 x  5 - - 0.5 2 3.5 5 6.5 8 9.5 11 12.5 14 15.5 17 18.5 20 - - - - - - 10 8.5 7 5.5 4 2.5 1}

  4 ₂

• _{- 2 - x 5}
₁₀ ⁴
• _{- 8 -}
• _{6 - - -}
₂₀
• _{- 16 -}
¹²
• ₁₈
_{14 10 x} ²⁰
• _{20 -}

  

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

  Plot of

  

 

  Example of unitary elasticity

demand curve

• 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

  Example of unitary elasticity demand curve ₅₀

  50 _{40 45 200 30}

  35 _{q 20}

  25 ₁₀

  15

  5 _{5 10 15 20 25 30 35 40 45 50} ^{q 50}

  

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
_{0.4 0.6} Q=Q(K,L), explicitly Q=K L ^{0.4 0.6} The function Q=K L may be drawn
• 3D space Using the concept of level set, we can see
• the countour plot of the function in 2D

  The plot of Q=K

  0.4 L

  0.6 in 3D

  Q The plot of Q=K

  0.4 L

  0.6 in 2D

  Q

  

Exponential function

• x

  General form:

  y = b

  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

  x y = e

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…

  �

  1 _2.8 ( )

  � � = 1+

_2.788

( )

  � [ ]

_2.765

_2.777

_2.742

2.754

_2.719

2.731

_2.696

2.708

_2.673

2.685

₃

  

2.662

_2.65

₃

  

Number e and its rules

• e =1

  a (e b

  ) = e a+b

• e

  a ) b

  =e ab

• (e

  a /e b

  =e a-b

• e

  

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
• – Natural log: b = e

  

Logarithmic function

Natural log: b = e

_e

  Natural logarithm, general form: y=log x

• The log of the base = log e = 1
• Example:
•  ln e = 1
₃  ln e = 3 a )=a ln u

  

Logarithms Rules

• ln(u

  15 = 15

• ln e
• ln(uv) = ln u + ln v (u,v>0)

  3 .e

  2 ) = ln e

  3

  2 = 3+2=5

• ln e
• ln(e
• ln(u/v) = ln u – ln v (u,v>0)

  2 /c) =2 – ln c

• ln(e

  

Logarithms Rules

• a a

  ln(uv )= ln u + ln v = ln u + a ln v ₂

• ₅
_{2 5 2} ln(u v)  ln u  ln v

  ln(xy ) = ln x + 2 ln y

• ₅
₂ ln(
• ln(e +e )=ln(155.8)=5.05

  e )  ln (e  ln (e

• ⁵
^{2 5 2} ln(e )=5, ln(e )=2  ln (e ) + ln (e ) =
• = 7

  5.05  7

  Logarithms Rules

• Proof Let

  Logarithms Rules b _e b b e _{b e} b _e

  1 log e = (log b) Let u = b

  Log b =(log e)(log b) 1 =(log e)(log b) 1 log e =

  (log b) Proof :

Graph

  _9.21

  10 _{ln 4x ( ) ln 2x ( )} ln x ( ) _{2 4 6 8 10} 2 ln x ( )

• _{- 4 ln x ( ) }
_{10 20 0.02 x} ^15.648 _{10 -}

Applications

• Growth model
• Population model
• Production function
• etc