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• Introductory Maths: Huw Dixon.

INTRODUCTORY MATHEMATICS FOR ECONOMICS MSCS. LECTURE 1: DIFFRENTIATION.

HUW DAVID DIXON

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

• 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 x0 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:

2y3 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 yx

The derivative is the ratio yx

as x becomes very small.

For example: 2y x at x=1, y=1. As x becomes very small, the ratio is 2.dydx

x y yx

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 axdy anxdx

Examples: 4

2

5

8 9

; ' 23 ; ' 152 ; ' 16

y x f xy x f xy x f x

Note: notation 'dy fdx

Rule 2: Natural Logarithm.

ln

1y axdydx x

Example

ln 31

y xdydx x

Rule 3: Exponential Function.

ax

ax

y edy aedx

Example

; '; '

x x

x x

y e f ey 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 xdy f g g fdx

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 ef x x fg x e g

dy f gdx

xe e x xe x

2e

g f

xx 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 xf x x fg x x g xdy f g g f x x x x x xdx

(c)

1

1

ln( ) ; ' 1( ) ln ; '

' ' ln . ln 1

y x xf x x fg x x g xdy f g g f x x x xdx

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 xdy dy dz f gdx dz dx

Examples: (a)

2

lnln

1 22

y xy zz xdy dy dz xdx 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 ey ez xdy dy dz e x xedx dz dx

(b) (c) reciprocals

2

1 2

24

3

1

;2.2

2

yx

y z z xdy dy dz xz xdx dz dx x

x

1.12

• Introductory Maths: Huw Dixon.

2

2

( )

2

( )

( )

2 2 ( )

ax b

z

z x a

y ey ez ww ax b

dy dy dz dw e wa e a ax bdx 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 xyv x

dy v u u vdx v

1.13

• Introductory Maths: Huw Dixon.

For example: 2

2

2 2

2

2 2 2

2 4

23

3; ' 2

3 ; ' 6. ' . ' 6 6

92( 1)

3

x

x x

x x

x

eyx

u e u ev x v x

dy v u u v x e xedx v x

xex

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 xR x x P x

dR xP Pdx

xPPP

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 dxx 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 PP x a xR x x a x

dR xP P x a xdx

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 xP dxx dP

a xx

As output increases, the elasticity increases;

2 2

( ) 0d x a x adx 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

xPaP dx P a Px 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

• 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 MCAC,

2

( )( )

' 1 '

1 ( )

C xAC xx

dAC xC C CCdx x x x

MC ACx

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

2

2

1( ) 2 .2

2 12

2 12

C x x

MC x

AC xx

dACdx x

And MC>AC MC

• 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