INTRODUCTION TO MATHEMATICS ECONOMICS 1, Formulas and forms for Economics. Benue State University
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INTRODUCTION TO MATHEMATICS ECONOMICS 1, Formulas and forms for Economics. Benue State University

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DIFFERENTIATION MADE EASY FOR ALL ECONOMICS STUDENTS IN SECOND SEMESTER COURSE
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Introductory Mathematics for Economics MSc’s

Introductory Maths: © Huw Dixon.

INTRODUCTORY MATHEMATICS FOR ECONOMICS MSC’S. LECTURE 1: DIFFRENTIATION.

HUW DAVID DIXON

CARDIFF BUSINESS SCHOOL.

SEPTEMBER 2009.

1.1

Introductory Maths: © Huw Dixon.

1. Some Definitions.

 An Integer: A positive or negative whole number, as in “counting. -3,-2,-1,0,1,2,3

 A rational number: the ratio of two integers. 11 2,

23 3 

 An irrational number: cannot be represented as the ratio of two integers. E.g. 2, …..  Real Numbers and the Real line : the set of all rational and irrational numbers. This is a “continuum”:

o ={x:- < x<}=(-,) o + ={x:0 < x < }=[0,)

o Open interval: includes end-points. (0,1) means strictly greater than 0 and strictly less than 1 o Closed interval: includes end-points. [0,1].

o Half open interval: includes one of endpoints, but not the other. [0,1) includes 0, but not 1.

 In economics, use real numbers most of the time: x might be output, price, advertising

expenditure etc. This is an approximation (cannot set a price ), but as in physics, seems to work well.

1.2

Introductory Maths: © Huw Dixon.

2. Functions f.

 takes a real number (input) and transforms it into another real number (output).

 y = f(x). You take a real number input x and it is mapped onto output y.

 f(x): y=2+(3.x) x y=f(x)

-10 -28 10 32 1.2 5.6

 f(x): y=x2 x y=f(x)

-10 100 10 100 1.2 1.44

 A function is 1-1 if each input has a unique output, and each output has a unique input. If

we look above, y=x2 is not 1-1. the output 100 can be generated by two values of x: +10 and -10.

1.3

Introductory Maths: © Huw Dixon.

y=2+3x is monotonic: it is always strictly increasing and y gets larger as x gets larger. It is also 1-1. Each value of y is associated with one value of x and vice versa. y=x2 is non-monotonic. For x<0 y is decreasing in x; for x>0 it is increasing in x. it is not 1-1: for each value of y 0 there are two values of x which generate it, the positive and negative square roots. y=2+3x is invertible: we can write y as a function of x and x as a function of y:

2y 3 3

x  

1.4

Introductory Maths: © Huw Dixon.

Demand curve: 1X P  X=output/quantity: P=price Inverse Demand curve: 1P X  Revenue Function: 2. .(1 )R X P X X X X    

In Economics, we usually deal with the inverse demand curve, and treat price as a function of Quantity (Alfred Marshall, British economist in late C19th) That is why price is on the vertical axis!

1.5

Introductory Maths: © Huw Dixon.

3. Differentiation made simple. Differentiation was invented by Sir Isaac Newton, the great British 17th Century Physicist (Principia Mathematica). If you take a function and differentiate it, you get the SLOPE of the function.

the change in y is called y , the change in x is x . The ratio of the changes is y x  

The derivative is the ratio y x  

as x becomes very small.

For example: 2y x at x=1, y=1. As x becomes very small, the ratio is 2.dy dx

xyy x  

1 (x changes from 1 to 2) 3 (1 to 4) 3 0.1 (x changes from 1 to 1.1) 0.21 (1 to 1.21) 2.1 0.01 (1 to 1.01) 0.0201 (1 to 1.0201) 2.01 0.001 0.002001 (1 to 1.002001) 2.001

This is also called the tangent or slope of the function

1.6

Introductory Maths: © Huw Dixon.

At the point y=1, x=1 the green line is 2y x . The red line is the tangent at that point (1,1) which has the slope 2.dy

dx

1.7

Introductory Maths: © Huw Dixon.

The Rules of Differentiation: Rule 1: Power rule.

1

n

n

y ax dy anx dx

Examples: 4

2

5

8 9

; ' 2 3 ; ' 15 2 ; ' 16

y x f x y x f x y x f x 

 

 

  

Note: notation 'dy f dx

Rule 2: Natural Logarithm.

ln

1 y ax dy dx x

Example

ln 3 1

y x dy dx x

Rule 3: Exponential Function.

ax

ax

y e dy ae dx

Example

; ' ; '

x x

x x

y e f e y e f e   

  

1.8

Introductory Maths: © Huw Dixon.

These are the basic rules used in economics: we do not need to use trigonometric functions (sin, cosine etc) except for very specialised uses. We then combine Rules 1-3 with further rules: these deal with differentiation of combinations of two functions f(x) and g(x). The aim is to take a complex expression (equation) and break it down into simpler parts f and g. Rules 1-3 can be used to differentiate f and g themselves, Rule 4: Product rule.

( ). ( )

'. '.

y f x g x dy f g g f dx

 

Examples: (a)

This is the function.

We break it up into two simple functions

We apply rules 1 and 3 to the simple functions

We then apply the product rule 4 to the two simple functions

2

2

2

( ) ; ' ( ) ; '

' '

2 [2 ]

x

x x x

y x e f x x f g x e g

dy f g dx

xe e x xe x

2 e

g f

x x x 

  

 

 

 

   

1.9

Introductory Maths: © Huw Dixon.

(b)

we know the answer here: but the product rule gives us the same answer as the rule 1: it shows the rules are consistent!

4

3 2

3 2 3 3 3

4 ( ) 4 ; ' 4 ( ) ; ' 3

' ' 4 4 .3 4 12 16

y x f x x f g x x g x dy f g g f x x x x x x dx

  

 

      

(c)

1

1

ln ( ) ; ' 1 ( ) ln ; '

' ' ln . ln 1

y x x f x x f g x x g x dy f g g f x x x x dx

  

 

     

1.10

Introductory Maths: © Huw Dixon.

Rule 5: Function of a function (Chain rule) This is a widely used rule: you break down the expression so that there are two or more stages.

( ); ( )

. '. '

y f z z g x dy dy dz f g dx dz dx

 

 

Examples: (a)

We start with: 2

2

ln ln

1 22

y x y z z x dy dy dz x dx dz dx z x

 

  

Break it down into two expressions we know how to differentiate

Apply the rule

1.11

Introductory Maths: © Huw Dixon.

2

2

2

2 2

x

z

z x

y e y e z x dy dy dz e x xe dx dz dx

  

(b) (c) reciprocals

2

1 2

2 4

3

1

; 2.2

2

y x

y z z x dy dy dz xz x dx dz dx x

x

 

    

 

1.12

Introductory Maths: © Huw Dixon.

2

2

( )

2

( )

( )

2 2 ( )

ax b

z

z x a

y e y e z w w ax b

dy dy dz dw e wa e a ax b dx dz dw dx

  

   

(e) exponential The last rule is actually implied by the previous two rules, but we state it for convenience! Rule 6: quotient rule.

2

( ) ( ) . ' . '

u xy v x

dy v u u v dx v

 

1.13

Introductory Maths: © Huw Dixon.

For example: 2

2

2 2

2

2 2 2

2 4

2 3

3 ; ' 2

3 ; ' 6 . ' . ' 6 6

9 2( 1)

3

x

x x

x x

x

ey x

u e u e v x v x

dy v u u v x e xe dx v x

xe x

 

 

   

 

To summarise: You take function. You use rules 4-6 to break down the expression into simple parts that you can differentiate using rules 1-3.

1.14

Introductory Maths: © Huw Dixon.

4 Economics applications. (a) Revenue and marginal revenue.

( ) ( ) . ( )

'

'(1 )

11

P P x R x x P x

dR xP P dx

xPP P

P

 

 

 

     

The inverse demand curve: (Price depends on output). Revenue: price times output: MR: differentiate using produce rule Rearrange, using elasticity formula

1 '

P P dx x P x dP

    

Let us look at some particular demand curves: these are ones you will see used in economics often!

1.15

Introductory Maths: © Huw Dixon.

Linear Demand. The Demand curve:

( ) ( ) .( )

' ( )

2 .

x a P P x a x R x x a x

dR xP P x a x dx

a x

     

     

 

a is the “intercept” term. It is the price when output is zero. Note that marginal revenue always has a slope that is twice that of the linear demand curve. Take the example of a=3.

What happens to the elasticity along the linear demand curve?

1.16

Introductory Maths: © Huw Dixon.

;

( )

P a x P dx x dP

a x x

 

 

 

As output increases, the elasticity increases;

2 2

( ) 0d x a x a dx x x        

If x=0 (P=a), =. If x=a (P=0) =0.

1.17

Introductory Maths: © Huw Dixon.

Demand curve 2: constant elasticity of demand.

Direct demand

1

1.

x aP

xP a P dx P a P x dP aP

   

 

     

   

Inverse Demand Elasticity.

If =1, then we have a rectangular hyperbola: P.x=a.

1.18

Introductory Maths: © Huw Dixon.

The Yellow line represents the unit elastic demand curve. Total expenditure P.X is constant. The red line represents an elastic demand curve: revenue is decreasing with price. The green line represents an inelastic demand curve: revenue is increasing with price.

1.19

Introductory Maths: © Huw Dixon.

Revenue: output times price Revenue as a function of output. Marginal Revenue: zero if =1 (yellow line), strictly positive if >1 (red line) and strictly negative if <1 (green line).

1

1 11

1 1

( ) .

1

xR x x a

a x

MR a x

 

  

     

     

Cost functions. C(x) Total cost of producing output x. For example:

2( ) . 2 cC x F x 

F: overhead cost, does not depend on output.

c is marginal cost (MC): dC c dx

1.20

Introductory Maths: © Huw Dixon.

Average Cost:

Average cost is total output divided by output. The Change in AC as output increases (slope of AC). AC is decreasing if MC<AC; increasing if MC>AC,

2

( )( )

' 1 '

1 ( )

C xAC x x

dAC xC C CC dx x x x

MC AC x

       

 

For example, take the previous cost function with F=2, c=1

2

2

1( ) 2 . 2

2 1 2

2 1 2

C x x

MC x

AC x x

dAC dx x

 

 

  

And MC>AC MC<AC MC=AC

2

1 12 4

2 0

2 0

2 0

dAC dx x

dACx dx

dACx dx

dACx dx

    

  

  

  

1.21

Introductory Maths: © Huw Dixon.

Thus “efficient” least cost production occurs when AC=MC.

1.22

Introductory Maths: © Huw Dixon.

Concavity and Convexity. The second derivative of a differentiable function is obtained by differentiating the first derivate:

y=f(x): the first derivative is written as 'f , the second derivative as ''f

2

2

2 2

22 2

d y d dy dx dx dx

dy d yy x x dx dx

     

    

A function is strictly concave if for all x 2

2 '' d y f dx  <0.

This means that the slope of the function is decreasing (becoming less positive, more negative). Example: The quadratic function

2

' 2 '' 2 0

y A x f x f

      

1.23

Introductory Maths: © Huw Dixon.

The red line is the quadratic 22y x  ; the green line is the first derivative ' 2f x  . This is decreasing in x '' 2 0f    . A concave function has the following shape: if it is increasing in x, the slope is becoming less steep as x increases (see x<0 above); if it is decreasing in x, the slope becomes more negative (steeper – see x>0 above). A concave function is like an upside down bow or Hill….

1.24

Introductory Maths: © Huw Dixon.

A function is strictly convex if the slope (derivative) is increasing in x. Take the quadratic

2

' 2 '' 2 0

y A x f x f

    

The green line is the quadratic (A=-2): the red the first derivative. A convex function is like a bowl. Note that the negative of a concave function is convex: if f(x) is strictly convex, then –f(x) is strictly concave and vice versa.

1.25

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