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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:

2*y
*3 3

*x *

1.4

Introductory Maths: © Huw Dixon.

**Demand curve**: 1*X P* X=output/quantity: P=price
**Inverse Demand curve: **1*P 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: 2*y x* at x=1, y=1. As *x* becomes very small, the ratio is 2.*dy
dx
*

*x* *y* *y
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 2*y 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

( ) 0*d 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 22*y x* ; the green line is the first derivative ' 2*f x* . This is
*decreasing* in x '' 2 0*f * .
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