mathematics for economics and business jean soper chapter one functions in economics l.
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Mathematics for Economics and Business Jean Soper chapter one Functions in Economics. Functions in Economics – Maths Objectives. Appreciate why economists use mathematics Plot points on graphs and handle negative values

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functions in economics maths objectives
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
functions in economics economic application objectives
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
axes of graphs
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
terms used in plotting points
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
plotting negative values
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
variables constants and functions
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)
working with functions
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
proportional relationship
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
linear relationships
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
general form of a linear function
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
power functions
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
to sketch a function
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
writing algebraic statements
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
the order of algebraic operations is
The Order of Algebraic Operations is

1.If there are brackets, do what is inside the brackets first

2.Exponentiation, or raising to a power

3.Multiplication and division

4.Addition and subtraction

  • You may like to remember the acronym BEDMAS, meaning brackets, exponentiation, division, multiplication, addition, subtraction
positive and negative signs
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

multiplying or dividing by 1
Multiplying or Dividing by 1
  • 1 x = x
  • (– 1) x = – x
  • x¸ 1 = x
multiplying or dividing by 0
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

brackets
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)
multiplying out brackets 1
Multiplying out Brackets 1
  • One pair: multiply each of the terms in brackets by the term outside
  • Example: 5x (7x – 4) = 35x2 – 20x
multiplying out brackets 2
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

results of multiplying out brackets
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) = a2 – ab + ab – b2

= a2 – b2

factorizing
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)

fractions
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
cancelling
Cancelling
  • Cancelling is dividing both numerator and denominator by the same amount
  • Examples:
inequality signs
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
comparisons using common denominator
Comparisons using Common Denominator
  • Example:

To find the bigger of 3/7 and 9/20multiply both numerator and denominator of each fraction by the denominator of the other:

3/7 = (20  3)/(20  7) = 60/1409/20 = (7  9)/(7  20) = 63/140

Since 63/140 > 60/140 9/20 > 3/7

adding and subtracting fractions
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
multiplying and dividing fractions
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
powers and indices
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
working with indices
Working with Indices
  • To multiply, add the indices
  • To divide, subtract the indices
  • x– n = 1/ xn
  • x0 = 1
  • x1/2 =  x
  • (ax)n = anxn
  • (xm)n = xmn
functions of more than 1 variable
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
total and average revenue
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
total and average cost
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
profit
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 – 0.4Q3and collecting like terms we find
  •  = – 120 + 195Q – 19Q2 – 0.4Q3
production functions
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
isoquants
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
formulae in excel
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
arithmetic operators are
Arithmetic Operators are:

( ) brackets

+ add

– subtract

* multiply

/ divide

^ raise to the power of

cell references in formulae
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