Introduction to Mathematical Economics Lecture 8

Introduction to Mathematical Economics

Lecture 8

Undergraduate Program Faculty of Economics & Business Universitas Padjadjaran

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Previously

• _{Mid Term Exam} • Matrix Algebra

– _{Transposition} – Determinant – _Inverse

Today

• We must find another time for this subject

• _{Some more topics about functions}

– _Continuity – _Smoothness

– _{Convexity / Concavity}

• _{Differential calculus of 1 independent variable}

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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

Continuity of a function

• _{Function y=2x is}

continuous at every point x 

• _{x=3  f(x)=6} • _{Choose a small}

number,e_>0, there is some value d > 0 such that all the function

values defined on the set of x values

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Discontinuous functions

• f(x) = +1, x  0 = -1, x > 0

Continue VS smooth

Continue functions are

f(x)

x

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Continue VS smooth

• _{smooth function} f(x)

Continue VS smooth

smooth cubic function f(x)

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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

_xa

f(x) exists, i.e

•

Lim

_xa-

f(x) = Lim

_xa+

f(x)

Convexity and Concavity of a

(bivariate) function

• The function z=f(x₁,x₂) 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

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Convexity and Concavity of a

function

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

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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

Derivative: graph analysis

• _{The derivative of a function y=f(x) at the}

point P=(x₁,f(x₁)) is the slope of the tangent line at that point

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Derivative: graph analysis

• _{Secant Line}

– A stright line that can be drawn under (above) a curve, connecting 2 points of that curve

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Definition of Differential

• If f’(x0) is the derivative of the function

y=f(x) at the point x0, then the (total)

differential at a point x0 is

dy=df(x0,dx) = f’(x0) dx

• The differential is a function of both x

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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

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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

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

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Derivative of a constant

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.

'

If y = mx + c, then

dy

y

m

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Examples

• _{y=15-2x }

Power Function Rule

y



x

n



n drops down: then index looses 1

mechanics of the power rule:

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Power Function Rule

• _{In words: The derivative of a power}

function, y= kxn, 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

• _{If y= kx}n 1



knx

n

Examples

y=x3 

y=2-½x3

f(x)=2ax4 

x

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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

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

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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)

Examples

• _{y= x}2+x50+100 

• y=16x4 - 5x3 + b 

• f(x₂)=x₁2+5x₁a 

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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

The Product Rule

• _{The rule is usually written as:}

Suppose y=u·v,u=f(x),v=g(x); then

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Examples

• _y=4x4(3x-1) 

The Quotient Rule

If functions f and g are differentiable at x, and g(x) 0,

then is also differentiable at x.

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

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Generalized Power Function

Rule

• _{The derivative of a function raised to a}

power f(x)=[g(x)]n where g(x) is a

differentiable function, and n is a real number is given by

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

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Examples

• _Y=6X2 and X=2Z+1; dY/dZ?

• Y=2X1/2+14X and X=Z; dY/dZ?

EXPONENTIAL AND

LOGARITHMIC FUNCTIONS

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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

Number e and its rules

•

e

0

=1

•

e

a

(e

b

) = e

a+b

•

(e

a

)

b

=e

ab

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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)=…

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

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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

•

ln(u/v) = ln u – ln v (u,v>0)

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Logarithms Rules

•

_ln(uv

a

)= ln u + ln v

a

= ln u + a ln v

• ln(xy2) = ln x + 2 ln y

•

ln(u



v)

_

ln u



ln v

• ln(e5e2) _ ln (e5)  ln (e2) • _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

Inverse Function Rule

The inverse function is given by x=f(y) so

and

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

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Inverse Function Rule

Exponential and Natural

Logarithmic Rules

( ) x BASIC CONCEPTS

The constant =2.71818... is its own derivative The exponential rule for base :

If y = e

The exponenti

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Exponential and Natural

The exponential function rule for base a : If y = a

It can be seen that exponential function rule for b

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