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  • Introductory Maths: Huw Dixon.

    INTRODUCTORY MATHEMATICS FOR ECONOMICS MSCS. 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

  • 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)

    We start with: 2

    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

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