Introduction to Mathematical Economics Lecture 8

  Introduction to Mathematical Economics _{Lecture 8}

  Undergraduate Program Faculty of Economics & Business

Universitas Padjadjaran

Ekki Syamsulhakim, SE., MApplEc.

  

Previously

• Mid Term Exam •

  Matrix Algebra

• – Transposition – Determinant – Inverse – Cramer’s Rule

  

Today

• We must find another time for this subject
• Some more topics about functions
• – Continuity – Smoothness – Convexity / Concavity
• Differential calculus of 1 independent variable
• – Review of the basic concept
• – Review of rules

Continuity of Functions

  Important: many mathematical techniques

• only applicable if the function is continuous Continuity of a function  explained easily
• with the aid of a graph

  A function is continuous if the graph of

• – the function has no breaks or jumps A function is continuous if it can be
• – sketched without lifting your pen from the paper

  Continuity of a function

  f(x) Function y=2x is y=2x continuous at every point x 

• •

  •

• •

  6

  6+e x=3  f(x)=6

• •

  6-e

  Choose a small

  e there is number, >0, some value d > 0 such that all the function values defined on the set x of x values

  3-d 3+d

  3 Discontinuous functions f(x) = +1, x  0

• f(x)

  = -1, x > 0

• f(x)=+1, x  0

  The function has = -1, x > 0 an obvious break or jump at the

  1 point x = 0 x

• 1

Continue VS smooth f(x) f(x)=|6x| x

  Continue functions are not necessarily smooth functions

  f(x) x

  Continue VS smooth

• smooth function

Continue VS smooth smooth cubic function f(x) x

  

Formal definition

A function f(x) which is defined on an

• open interval including the point x=a

  is continuous at that point if Lim f(x) exists, i.e

• – xa
• xa- xa+

  Lim f(x) = Lim f(x)

• – xa

  Lim f(x) = f(a)

  

Convexity and Concavity of a

(bivariate) function

_{1 2} The function z=f(x ,x ) is concave (convex)

• iff, for any pair of distinct points M and N on its graph – a surface – line segment MN lies either on or below (above) the surface. The function is strictly concave (convex) iff
• line segment MN lies entirely below (above) the surface, except at M and N

  Convexity and Concavity of a function N M N

  

Differential Calculus

The basic calculus measurement, the rate of

• change of a function at a point, is useful in the

  quantitative analysis of business problems

  A convenient way to express how a change in

• the level of one variable (say x) determines a change in the level of another variable (say y)

  How a change in an additional worker affects a

• – change in output (or profit)? How a change in a tariff rate determines a change in
• – domestic (border) price of importables

  

Differential calculus

In mathematics, calculus concentrates on the

• analysis of rates of change in functions  particularly with the instantaneous rate of

  

change of a function, or the rate of change at a

point of a function The derivative of a function measures the

• instantaneous rate of change in the dependent

  variables in response to an infinitesimally

  small change in the independent variable(s)

  

Derivative: graph analysis

• tangent y

  The derivative of a function y=f(x) at the

  point P=(x ,f(x )) is the slope of the _{1 1} line

  tangent line at that point

  y=f(x) P y ₁ x x ₁

  

Derivative: graph analysis

Secant line y

• A stright line that can be drawn under (above)

  Secant Line

• – y+Dy

  y=f(x) P a curve, connecting 2 points of that curve _Dy y

  Used in visualising a change from a point to

• – another point

  Dx x x x+Dx

  

The Derivative and the Differential (Con’t)

  

The Derivative and the Differential (Con’t)

  

Definition of Differential

  If f’(x ) is the derivative of the function

• y=f(x) at the point x , then the (total) differential at a point x is dy=df(x ,dx) = f’(x ) dx

  The differential is a function of both x

• and dx

Differential

  The differential provides us with a method

• of estimating the effect of a change in x of amount dx= Dx on y, where Dy is the exact change in y, while dy is the approximate change in y

  Conditions for differentiability

Rules of Differentiation

  Learning outcomes

By the end of this topic, you should be able to:

• 1. Use the various rules of differentiation confidently, recognising which is the most suitable rule for each case.

  2. Use the techniques of differentiation in Economic situations, such as working out marginal cost, elasticity, etc.

  

Rules for differentiation

The rules we are going to cover

  1. Derivative of a constant

  2. Derivative of a linear function

  3. Power function rule

  4. Sum and difference rules

  5. Product rule

  6. Quotient rule

  7. Generalised power function rule

  8. Chain rule

  9. Inverse function rule

  10. Natural Exponential Rule

  11. Natural Logarithmic Rule

  

Derivative of a constant

• If y=k where k is a constant, then

   dx dy

• Example: If y=8 then

   dx dy

  

Derivative of a linear function

  The derivative of a linear equation

• y=m.x+c is equal to m, the coefficient of x. The derivative of a variable raise to the
• first power is always equal to the coefficient of the variable.

  dy y ' m If y = mx + c, then   dx

Examples

• y=15-2x 
• f(x) = ½x – 100 
• x
₁ =100+ ¼ x ₂ 

  Power Function Rule y x ⁿ

    n drops down: then index looses 1 mechanics of the power rule: giving

  1 , n

  If y=x then n dy nx dx

    d d y x nx ⁿ

   ^1

  

Power Function Rule

_n

  In words: The derivative of a power

• function, y= kx , where k is a constant and n is any real number, is equal to the coefficient k times the exponent n, multiplied by the variable x raise to the (n- 1) power

  dy n

  1  ⁿ

  If y= kx

  knx

• 

  dx

  

Examples

  3

  y=x  ₃  y=2-½x ₄ f(x)=2ax 

  x y 

  

Hint

  The derivative of a power function always

• yields another function that has smaller power, example:

  f(x) f’(x) Cubic Quadratic Quadratic Linear Linear Constant

Constant zero

  f(x)

x

  

Hint

• If dy/dx>0  the function is increasing
• If dy/dx=0  the function is constant
• If dy/dx<0  the function is decreasing

  

Sum and Difference Rules

The derivative of the sum of two functions is the
• sum of the derivatives of the individual functions.

  Similarly the derivative of the difference of two

• functions is the difference of the derivatives. So, if functions f and g are both differentiable at
• c, then if

  h(x)=f(x)+g(x)  h’(x)=f’(x)+g’(x)

Examples

• y= x
² +x ⁵⁰ +
• y=16x
⁴ - 5x ³
• f(x
₂ )=x ₁ ² +5x 1f(x)= 2ax ⁴ +10ax ² -5a ² 

  

The Product Rule

  If functions f(x) and g(x) are differentiable

• at c, so is their product:

  h(x)=f(x)g(x)

  h’(x)=f(x)g’(x)+g(x) f’(x) So the derivative of the product of two

• functions is the first function times the derivative of the second function plus the second function times the derivative of first function.

  

The Product Rule

  The rule is usually written as:

• Suppose y=u·v,u=f(x),v=g(x); then

  

dy dv du

u v  

dx dx dx

  OR: dy/dx=uv’+vu’ where u and v are functions of x

Examples

• y=4x
⁴ (3x
• f(x)=(x
² +1)(x ³ -2)

  The Quotient Rule If functions f and g are differentiable at x, and g(x) 0,  f x ( ) then is also differentiable at x. h x ( )

   g x ( ) f x ( ) f x g x ( ) ( ) f x g x ( ) ( )

     h x ( ) h x ( )

      ₂ g x ( ) g x ( )

    OR if f(x) = u and g(x) = v, then u v uv ' '

   h x ( )

    ₂ v

  Examples x y

   1 x 

  2 6 x

  5

• y for x 

   2 x

  5

  2 

Examples

  Let a unitary elastic demand function takes

• the form If price increases from $10 to $20, how much would quantity demanded decreases? Compute the price elasticity of demand at those points

  Generalized Power Function

Rule

_n

  The derivative of a function raised to a

• power f(x)=[g(x)] where g(x) is a differentiable function, and n is a real number is given by

  n

  1  f ' ( x ) n [ g ( x )] g ' ( x )

    ^{2 3} Example: If f(x)=(x +8) ; find dy/dx

• 2

  If ; find dy/dx

•  

  y 1 x

  The Chain Rule Given a composite function (a function of a function) where y is a function of u and u is a function of x, i.e. y=f(u), and u=g(x) then y=f(g(x)) and the derivative of y with respect to x is the derivative of the first function w.r.t u times the derivative of the second w.r.t x: dy dy du

    dx du dx

Examples

• Y=6X
² and X=2Z+1; d
• 14X and X=Z; dY/dZ?
• Y=2X ^1/2
• S=3X+6 and X=(2T
² +5)(3T-2); dS/dT?

  Review

  

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

  

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

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

  

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

  

Inverse Function Rule

1 2 1 2 _{1 2}

  1

  1

  2

  2 ² If y=f(x)=x then 2x The inverse function is given by x=f(y) so and

  

What is the relationship between dy/dx and dx/dy?

dy dx x y y dx y dy y ^

    

   

  Inverse Function Rule

  1

  1

  2

  1 1 2

  1 2 1 2 Now x y 2x 2y So,

  Similarly, 2y dx dy x dy dx dy dx dx dy

        

  

Exponential and Natural

Logarithmic Rules

BASIC CONCEPTS The constant =2.71818... is its own derivative e

   The exponential rule for base : e  _x If y = e _x dy d e ( ) _x e

    dx dx

  The exponenti al function rule for base : e 

_f(x)

If y = e _{f x ( )} dy d e ( ) _{f x ( )} e f x '( )

    dx dx

  

Exponential and Natural

Logarithmic Rules

  EXAMPLE _{3 x 2   5 x 4}

  y f x ( ) e

    _{f x ( )}

  dy d e ( ) _{f x ( )} e f x '( )

   

  dx dx _{3 x 2   5 x 4} dy e (6 x 5)

   

  dx

  

Exponential and Natural

Logarithmic Rules

The exponential function rule for base a :  _f(x) If y = a _{f x ( )}

dy d a ( )

_{f x ( )} a (ln ) '( ) a f x

    dx dx

  It can be seen that exponential function rule

for b ase e is just a special case of that for base a

  

Exponential and Natural

Logarithmic Rules

  EXAMPLE _{3 x 2   5 x 4}

  y f x ( ) 4

    ₂

  dy _{3 x   5 x 4}

  4 (ln 4)(6 x 5) 

  

  dx

  Exponential and Natural Logarithmic Rules

  _y e

• Relation between and ln if y = ln x, then x = e _y

  e x  _y d e ( ) d x

    dx dx _y d e ( )

  1   dx _y dy e

  1   dx dy

  1

  1    _y

  

Exponential and Natural

Logarithmic Rules

  2

  2 ₂ 1 '( ) '( )

  ( ) ( ) ln (3 2 6)

  6

  2

  3

  2

  6 BASIC CONCEPT If y=ln f(x) EXAMPLE: then dy f x f x dx f x f x y x x dy x dx x x

         

   

  Summary

  Summary

  Summary Higher Order Derivatives