Transcript Document

Mathematics for Economics and
Business
Jean Soper
chapter one
Functions in Economics
1
Functions in Economics –
Maths Objectives
• Appreciate why economists use mathematics
• Plot points on graphs and handle negative
values
• Express relationships using linear and power
functions, substitute values and sketch the
functions
• Use the basic rules of algebra and carry out
accurate calculations
• Work with fractions
• Handle powers and indices
• Interpret functions of several variables
2
Functions in Economics –
Economic Application Objectives
• Apply mathematics to economic variables
• Understand the relationship between total and
average revenue
• Obtain and plot various cost functions
• Write an expression for profit
• Depict production functions using isoquants and
find the average product of labour
• Use Excel to plot functions and perform
calculations
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Axes of Graphs
• x axis: the horizontal line along which
values of x are measured
• x values increase from left to right
• y axis: the vertical line up which values
of y are measured
• y values increase from bottom to top
• Origin: the point at which the axes
intersect where x and y are both 0
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Terms used in Plotting Points
• Coordinates: a pair of numbers (x,y) that
represent the position of a point
 the first number is the horizontal distance of the
point from the origin
 the second number is the vertical distance
• Positive quadrant: the area above the x axis
and to the right of the y axis where both x
and y take positive values
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Plotting Negative Values
• To the left of the origin x is negative
• As we move further left x becomes more
negative and smaller
• Example:
– 6 is a smaller number than – 2
and occurs to the left of it on the x axis
• On the y axis negative numbers occur below
the origin
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Variables, Constants and Functions
• Variable: a quantity represented by a
symbol that can take different possible
values
• Constant: a quantity whose value is fixed,
even if we do not know its numerical
amount
• Function: a systematic relationship
between pairs of values of the variables,
written
y = f(x)
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Working with Functions
• If y is a function of x, y = f(x)
• A function is a rule telling us how to
obtain y values from x values
• x is known as the independent variable, y
as the dependent variable
• The independent variable is plotted on
the horizontal axis, the dependent
variable on the vertical axis
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Proportional Relationship
• Each y value is the
same amount times
the corresponding
x value
• All points lie on a
straight line
through the origin
• Example:
y = 6x
9
y
60
y = 6x
40
20
0
0
5
10
x
Linear Relationships
• Linear function: a
relationship in which
all the pairs of
values form points on
a straight line
• Shift: a vertical
movement upwards
or downwards of a
line or curve
• Intercept: the value
at which a function
cuts the y axis
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y = 6x + 20
80
y
60
40
y = 6x
20
0
0
5
10
x
General Form of a Linear Function
• A function with just a term in x and
(perhaps) a constant is a linear function
• It has the general form
y = a + bx
• b is the slope of the line
• a is the intercept
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Power Functions
• Power: an index indicating the number of times
that the item to which it is applied is multiplied
by itself
• Quadratic function: a function in which the
highest power of x is 2
 The only other terms may be a term in x and a
constant
• Cubic function: a function in which the highest
power of x is 3
 The only other terms may be in x 2, x and a
constant
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To Sketch a Function
• Decide what to plot on the x axis and on the y
axis
• List some possible and meaningful x values,
choosing easy ones such as 0, 1, 10
• Find the y values corresponding to each, and list
them alongside
• Find points where an axis is crossed
(x = 0 or y = 0)
• Look for maximum and minimum values at which
the graph turns downward or upwards
• If you are not sure of the correct shape, try one or
two more x values
• Connect the points with a smooth curve
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Writing Algebraic Statements
• The multiplication sign is often omitted, or
sometimes replaced by a dot
• An expression in brackets immediately
preceded or followed by a value implies that
the whole expression in the brackets is to be
multiplied by that value
• Example:
y = 3(5 + 7x) = 15 + 21x
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The Order of Algebraic
Operations is
1.
2.
3.
4.
•
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If there are brackets, do what is inside the
brackets first
Exponentiation, or raising to a power
Multiplication and division
Addition and subtraction
You may like to remember the acronym
BEDMAS, meaning brackets, exponentiation,
division, multiplication, addition, subtraction
Positive and Negative Signs
• When two signs come together
– + (or + –) gives –
– – gives +
• Examples:
11 + (– 7) = 11 – 7 = 4
12 – (– 4) = 12 + 4 = 16
(– 9)  (– 5) = +45
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Multiplying or Dividing by 1
• 1x=x
• (– 1)  x = – x
• x1=x
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Multiplying or Dividing by 0
• Any value multiplied by 0 is 0
• 0 divided by any value except 0 is 0
• Division by 0 gives an infinitely large number
which may be positive or negative
• 0  0 may have a finite value
• Example: When quantity produced Q = 0,
variable cost VC = 0
but average variable cost = VC/Q
may have a finite value
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Brackets
• An expression in brackets written
immediately next to another
expression implies that the expressions
are multiplied together
• Example:
5x (7x – 4) = (5x)  (7x – 4)
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Multiplying out Brackets 1
• One pair: multiply each of the terms in
brackets by the term outside
• Example:
5x (7x – 4) = 35x2 – 20x
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Multiplying out Brackets 2
• Two pairs: multiply each term in the second
bracket by each term in the first bracket
• Examples:
(3x – 2)(11 + 5x) = 33x + 15x2 – 22 – 10x
= 15x2 + 23x – 22
(a – b)(– c + d) = – ac + ad + bc – bd
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Results of Multiplying out Brackets
(a + b)2
= (a + b) (a + b)
= a2 + 2 ab + b2
(a – b)2
= (a – b) (a – b)
= a2 – 2 ab + b2
(a + b)(a – b)
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= a2 – ab + ab – b2
= a2 – b2
Factorizing
• Look for a common factor, or for
expressions that multiply together to
give the original expression
• Example:
45x2 – 60x = 15x (3x – 4)
• Factorizing a quadratic expression may
involve some intelligent guesswork
• Example:
45x2 – 53x – 14 = (9x + 2) (5x – 7)
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Fractions
• Fraction: a part of a whole
• Amount of an item
= fractional share of item  total amount
• Ratio: one quantity divided by another
quantity
• Numerator: the value on the top of a
fraction
• Denominator: the value on the bottom of a
fraction
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Cancelling
• Cancelling is dividing both numerator and
denominator by the same amount
• Examples:
70 7  2  5 2


105 5  7  3 3




8x z
4  2 x. x z
2x

 3
2 5
2 2 3
28x z
4  7 x z .z
7z
3 2
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2 2
Inequality Signs
• > sign: the greater than sign
indicates that the value on its left is
greater than the value on its right
• < sign: the less than sign indicates
that the value on its left is less than
the value on its right
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Comparisons using Common
Denominator
• Example:
To find the bigger of 3/7 and 9/20
multiply both numerator and denominator of
each fraction by the denominator of the other:
3/7 = (20  3)/(20  7) = 60/140
9/20 = (7  9)/(7  20) = 63/140
Since 63/140 > 60/140
9/20 > 3/7
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Adding and Subtracting Fractions
• To add or subtract fractions first write
them with a common denominator
and then add or subtract the
numerators
• Lowest common denominator: the
lowest value that is exactly divisible
by all the denominators to which it
refers
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Multiplying and Dividing
Fractions
• Fractions are multiplied by multiplying
together the numerators and also the
denominators
• To divide by a fraction turn it upside
down and multiply by it
• Reciprocal of a value: is 1 divided by
that value
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Powers and Indices
• Index or power: a superscript showing
the number of times the value to which
it is applied is to be multiplied by itself
• Exponent: a superscripted number
representing a power
• Exponentiation: raising to a power
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Working with Indices
•
•
•
•
•
•
•
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To multiply, add the indices
To divide, subtract the indices
x – n = 1/ xn
x0 = 1
x1/2 =  x
(ax)n = an xn
(xm)n = xmn
Functions of More Than 1 Variable
• Multivariate function: the dependent variable,
y, is a function of more than one independent
variable
• If
y = f(x,z)
y is a function of the two variables x and z
• We substitute values for x and z to find the
value of the function
• If we hold one variable constant and
investigate the effect on y of changing the
other, this is a form of comparative statics
analysis
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Total and Average Revenue
• TR = P.Q
• AR = TR/Q = P
• A downward sloping linear demand
curve implies a total revenue curve
which has an inverted U shape
• Symmetric: the shape of one half of
the curve is the mirror image of the
other half
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Total and Average Cost
•
•
•
•
Total Cost is denoted TC
Fixed Cost: FC is the constant term in TC
Variable Cost:
VC = TC – FC
Average Cost per unit output:
AC = TC/Q
• Average Variable Cost:
AVC = VC/Q
• Average Fixed Cost:
AFC = FC/Q
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Profit
Profit =  = TR – TC
If
TC = 120 + 45Q – Q2 + 0.4Q3
and
TR = 240Q – 20Q2
 = TR – TC
substitute using brackets
 = 240Q – 20Q2 – (120 + 45Q – Q2 + 0.4Q3)
Taking the minus sign through the brackets and
applying it to each term in turn gives
•  = 240Q – 20Q2 – 120 – 45Q + Q2 – 0.4Q3
and collecting like terms we find
•
•
•
•
•
•  = – 120 + 195Q – 19Q2 – 0.4Q3
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Production functions
• A production function shows the quantity of
output obtained from specific quantities of
inputs, assuming they are used efficiently
• In the short run the quantity of capital is fixed
• In the long run both labour and capital are
variable
• Plot Q on the vertical axis against L on the
horizontal axis for a short-run production
function
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Isoquants
• An isoquant connects points at which the
same quantity of output is produced using
different combinations of inputs
• Plot K against L and connect points that
generate equal output for an isoquant map
Average Product of Labour
• Average Product of Labour (APL) = Q  L
• Plot APL against L
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Formulae in Excel:
• Are entered in the cell where you
want the result to be displayed
• Start with an equals sign
• Must not contain spaces
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Arithmetic Operators are:
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()
brackets
+
add
–
subtract
*
multiply
/
divide
^
raise to the power of
Cell References in Formulae
• A cell reference, such as A6, tells Excel to use
in its calculation the value in that cell
• Values calculated using cell references
automatically recalculate if the values change
• This facilitates ‘what if’ analysis
• Relative cell addresses (e.g. A6)
change as the formula is copied
• Absolute cell addresses
(e.g. $B$3)
remain fixed as the formula is copied
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